\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\citation{Ohtsuki:TwoLoop}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Kricker:Lines}
\citation{GaroufalidisRozansky:LoopExpansion}
\providecommand \oddpage@label [2]{}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{1}{(document)}{Doc-Start}}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines, GaroufalidisRozansky:LoopExpansion}{{1}{(document)}{Doc-Start}}}
\citation{Alexander:TopologicalInvariants}
\citation{Lalani:ThetaSurvey}
\citation{Self}
\citation{Self}
\citation{Self}
\citation{Self}
\citation{DHOEBL:Random}
\citation{DHOEBL:Random}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Fun}}{2}{section.1}\protected@file@percent }
\@writefile{brf}{\backcite{Alexander:TopologicalInvariants}{{2}{1}{section.1}}}
\@writefile{brf}{\backcite{Lalani:ThetaSurvey}{{2}{1}{section.1}}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces $\Theta $ as a bar code and a QR code, for all the knots in the Rolfsen table.\relax }}{3}{figure.caption.2}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:Rolfsen}{{1.1}{3}{$\Theta $ as a bar code and a QR code, for all the knots in the Rolfsen table.\relax }{figure.caption.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces $\Theta $ of some square weave knots, as computed by\nonbreakingspace \cite  [WeaveKnots.nb]{Self}.\relax }}{4}{figure.caption.3}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{4}{1.2}{figure.caption.3}}}
\newlabel{fig:SquareWeaves}{{1.2}{4}{$\Theta $ of some square weave knots, as computed by~\cite [WeaveKnots.nb]{Self}.\relax }{figure.caption.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces $\Theta $ of a randomized weave knot, as computed by\nonbreakingspace \cite  [WeaveKnots.nb]{Self}. Crossings were chosen to be positive or negative with equal probabilities.\relax }}{4}{figure.caption.4}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{4}{1.3}{figure.caption.4}}}
\newlabel{fig:RandomWeave}{{1.3}{4}{$\Theta $ of a randomized weave knot, as computed by~\cite [WeaveKnots.nb]{Self}. Crossings were chosen to be positive or negative with equal probabilities.\relax }{figure.caption.4}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The Main Theorem}}{4}{section.2}\protected@file@percent }
\newlabel{sec:MainTheorem}{{2}{4}{The Main Theorem}{section.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces An example upright knot diagram.\relax }}{4}{figure.2.1}\protected@file@percent }
\newlabel{fig:SampleDiagram}{{2.1}{4}{An example upright knot diagram.\relax }{figure.2.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces $\theta $ (hexagonal QR code only) of the 15 largest knots that we have computed by September 16, 2024. They are all ``generic'' in as much as we know, and they all have $\geq 300$ crossings. The knots come from\nonbreakingspace \cite  {DHOEBL:Random}. Warning: Some screens/printers may introduce spurious Moir\'e interference patterns. \relax }}{5}{figure.caption.5}\protected@file@percent }
\@writefile{brf}{\backcite{DHOEBL:Random}{{5}{1.4}{figure.caption.5}}}
\newlabel{fig:300}{{1.4}{5}{$\theta $ (hexagonal QR code only) of the 15 largest knots that we have computed by September 16, 2024. They are all ``generic'' in as much as we know, and they all have $\geq 300$ crossings. The knots come from~\cite {DHOEBL:Random}. Warning: Some screens/printers may introduce spurious Moir\'e interference patterns. \relax }{figure.caption.5}{}}
\citation{IType}
\citation{APAI}
\citation{Toronto-241030}
\newlabel{eq:A}{{1}{6}{The Main Theorem}{equation.2.1}{}}
\newlabel{eq:Delta}{{2}{6}{The Main Theorem}{equation.2.2}{}}
\@writefile{brf}{\backcite{IType}{{6}{2}{equation.2.2}}}
\@writefile{brf}{\backcite{APAI, Toronto-241030}{{6}{2}{equation.2.2}}}
\newlabel{eq:F1}{{3}{6}{The Main Theorem}{equation.2.3}{}}
\newlabel{eq:F2}{{4}{6}{The Main Theorem}{equation.2.4}{}}
\newlabel{eq:F3}{{5}{6}{The Main Theorem}{equation.2.5}{}}
\citation{APAI}
\citation{APAI}
\citation{IType}
\newlabel{thm:Main}{{1}{7}{The Main Theorem, proof in Section~\ref {sec:Proof}}{theorem.1}{}}
\newlabel{eq:Main}{{6}{7}{The Main Theorem, proof in Section~\ref {sec:Proof}}{equation.2.6}{}}
\newlabel{com:normalization}{{2}{7}{}{theorem.2}{}}
\newlabel{com:traffic}{{3}{7}{}{theorem.3}{}}
\@writefile{brf}{\backcite{APAI}{{7}{3}{theorem.3}}}
\@writefile{brf}{\backcite{APAI}{{7}{3}{theorem.3}}}
\newlabel{com:FeynmanDiagrams}{{4}{7}{}{theorem.4}{}}
\@writefile{brf}{\backcite{IType}{{7}{4}{theorem.4}}}
\citation{Wolfram:Mathematica}
\citation{Self}
\newlabel{com:Computation}{{5}{8}{}{theorem.5}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Implementation and Examples}}{8}{section.3}\protected@file@percent }
\newlabel{sec:Implementation}{{3}{8}{Implementation and Examples}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Implementation}}{8}{subsection.3.1}\protected@file@percent }
\@writefile{brf}{\backcite{Wolfram:Mathematica}{{8}{3.1}{subsection.3.1}}}
\@writefile{brf}{\backcite{Self}{{8}{3.1}{subsection.3.1}}}
\newlabel{ssec:Examples}{{3.2}{10}{Examples}{subsection.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{Examples}}{10}{subsection.3.2}\protected@file@percent }
\citation{Self}
\citation{APAI}
\citation{APAI}
\@writefile{brf}{\backcite{Self}{{11}{3.2}{figure.caption.6}}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Proof of the Main Theorem, Theorem\nonbreakingspace \ref  {thm:Main}}}{11}{section.4}\protected@file@percent }
\newlabel{sec:Proof}{{4}{11}{Proof of the Main Theorem, Theorem~\ref {thm:Main}}{section.4}{}}
\newlabel{ssec:Invariance}{{4.1}{11}{Proof of Invariance}{subsection.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.1}{Proof of Invariance}}{11}{subsection.4.1}\protected@file@percent }
\@writefile{brf}{\backcite{APAI}{{11}{4.1}{subsection.4.1}}}
\@writefile{brf}{\backcite{APAI}{{11}{4.1}{subsection.4.1}}}
\citation{APAI}
\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces The 132-crossing torus knot $T_{22/7}$ and a plot of its $\Theta $ invariant\relax }}{12}{figure.caption.6}\protected@file@percent }
\newlabel{fig:T227}{{3.1}{12}{The 132-crossing torus knot $T_{22/7}$ and a plot of its $\Theta $ invariant\relax }{figure.caption.6}{}}
\newlabel{eq:Ap}{{7}{12}{Proof of Invariance}{equation.4.7}{}}
\@writefile{brf}{\backcite{APAI}{{12}{4.1}{equation.4.7}}}
\newlabel{eq:CarRules}{{8}{13}{Proof of Invariance}{equation.4.8}{}}
\newlabel{eq:CounterRules}{{9}{13}{Proof of Invariance}{equation.4.9}{}}
\newlabel{eq:tilg}{{10}{13}{Proof of Invariance}{equation.4.10}{}}
\newlabel{eq:NullCarRules}{{11}{13}{}{equation.4.11}{}}
\newlabel{eq:NullCounterRules}{{12}{13}{}{equation.4.12}{}}
\citation{APAI}
\citation{APAI}
\citation{APAI}
\citation{Oaxaca-2210}
\citation{Toronto-241030}
\citation{APAI}
\citation{Polyak:RMoves}
\citation{Polyak:RMoves}
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces  The modified Green function ${\tilde  {g}}_{ab}$ is invariant under Reidemeister moves performed away from where it is measured. \relax }}{14}{figure.caption.7}\protected@file@percent }
\newlabel{fig:RelativeInvariance}{{4.1}{14}{The modified Green function $\tilg _{ab}$ is invariant under Reidemeister moves performed away from where it is measured. \relax }{figure.caption.7}{}}
\newlabel{lem:NullVertices}{{7}{14}{}{theorem.7}{}}
\newlabel{rem:F4Null}{{8}{14}{}{theorem.8}{}}
\@writefile{brf}{\backcite{APAI}{{14}{4.1}{theorem.8}}}
\@writefile{brf}{\backcite{APAI}{{14}{4.1}{theorem.8}}}
\newlabel{thm:RelativeInvariant}{{9}{14}{}{theorem.9}{}}
\@writefile{brf}{\backcite{APAI, Oaxaca-2210, Toronto-241030}{{14}{4.1}{figure.caption.7}}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces  A generating set of oriented Reidemeister moves as in\nonbreakingspace \cite  [Figure\nonbreakingspace 6]{Polyak:RMoves}. Aside 1: the braid-like R2b is not needed. Aside 2: yet R2b cannot replace R2c$^\pm $ because in the would-be proof, an unpostulated form of R3 is used (which in itself follows from R2c$^\pm $). \relax }}{15}{figure.caption.8}\protected@file@percent }
\@writefile{brf}{\backcite{Polyak:RMoves}{{15}{4.2}{figure.caption.8}}}
\newlabel{fig:RMoves}{{4.2}{15}{A generating set of oriented Reidemeister moves as in~\cite [Figure~6]{Polyak:RMoves}. Aside 1: the braid-like R2b is not needed. Aside 2: yet R2b cannot replace R2c$^\pm $ because in the would-be proof, an unpostulated form of R3 is used (which in itself follows from R2c$^\pm $). \relax }{figure.caption.8}{}}
\@writefile{brf}{\backcite{APAI}{{15}{4.1}{figure.caption.7}}}
\newlabel{eq:R3LeftOuter}{{13}{16}{Proof of Invariance}{equation.4.13}{}}
\newlabel{eq:R3LeftInner}{{14}{16}{Proof of Invariance}{equation.4.14}{}}
\newlabel{eq:R3RightOuter}{{15}{16}{Proof of Invariance}{equation.4.15}{}}
\newlabel{eq:R3RightInner}{{16}{16}{Proof of Invariance}{equation.4.16}{}}
\citation{BecerraVanHelden:RotationalReidemeister}
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces  The upright Reidemeister moves: The R1 and R3 moves are already upright and remain the same as in Figure\nonbreakingspace \ref  {fig:RMoves}. The crossings in the R2 moves of Figure\nonbreakingspace \ref  {fig:RMoves} are rotated to be upright. We also need two further moves: The null vertex move NV for adding and removing null vertices, and the swirl move Sw which then implies that any two ways of turning a crossing upright are the same. We sometimes indicate rotation numbers symbolically rather than using complicated spirals. \relax }}{17}{figure.caption.9}\protected@file@percent }
\newlabel{fig:UprightRMoves}{{4.3}{17}{The upright Reidemeister moves: The R1 and R3 moves are already upright and remain the same as in Figure~\ref {fig:RMoves}. The crossings in the R2 moves of Figure~\ref {fig:RMoves} are rotated to be upright. We also need two further moves: The null vertex move NV for adding and removing null vertices, and the swirl move Sw which then implies that any two ways of turning a crossing upright are the same. We sometimes indicate rotation numbers symbolically rather than using complicated spirals. \relax }{figure.caption.9}{}}
\newlabel{prop:UprightRMoves}{{10}{17}{}{theorem.10}{}}
\@writefile{brf}{\backcite{BecerraVanHelden:RotationalReidemeister}{{17}{4.1}{figure.caption.9}}}
\newlabel{prop:R3}{{11}{17}{}{theorem.11}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces  The two sides $D^l$ and $D^r$ of the R3b move. The left side $D^l$ consists of 3 distinguished crossings $c^l_1=(1,j,k)$, $c^l_2=(1,i,{k^+})$, $c^l_3=(1,{i^+},{j^+})$ and a collection of further crossings $c_y=(s,m,n)\in Y$, where $Y$ is the set of crossings not participating in the R3b move. The right side $D^r$ consists of $c^r_1=(1,i,j)$, $c^r_2=(1,{i^+},k)$, $c^r_3=(1,{j^+},{k^+})$ and the same set $Y$ of further crossings $c_y$. \relax }}{18}{figure.caption.10}\protected@file@percent }
\newlabel{fig:R3}{{4.4}{18}{The two sides $D^l$ and $D^r$ of the R3b move. The left side $D^l$ consists of 3 distinguished crossings $c^l_1=(1,j,k)$, $c^l_2=(1,i,\kp )$, $c^l_3=(1,\ip ,\jp )$ and a collection of further crossings $c_y=(s,m,n)\in Y$, where $Y$ is the set of crossings not participating in the R3b move. The right side $D^r$ consists of $c^r_1=(1,i,j)$, $c^r_2=(1,\ip ,k)$, $c^r_3=(1,\jp ,\kp )$ and the same set $Y$ of further crossings $c_y$. \relax }{figure.caption.10}{}}
\newlabel{eq:ABC}{{17}{18}{Proof of Invariance}{equation.4.17}{}}
\newlabel{eq:R3A}{{18}{18}{Proof of Invariance}{equation.4.18}{}}
\newlabel{rem:E}{{12}{19}{}{theorem.12}{}}
\newlabel{prop:R2c}{{13}{20}{}{theorem.13}{}}
\newlabel{prop:R1s}{{14}{21}{}{theorem.14}{}}
\newlabel{prop:Sw}{{15}{21}{}{theorem.15}{}}
\newlabel{prop:NV}{{16}{21}{}{theorem.16}{}}
\citation{Self}
\citation{Self}
\citation{Self}
\newlabel{ssec:Residue}{{4.2}{22}{Proof of Polynomiality}{subsection.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.2}{Proof of Polynomiality}}{22}{subsection.4.2}\protected@file@percent }
\newlabel{lem:del}{{17}{22}{}{theorem.17}{}}
\newlabel{eq:delf}{{19}{22}{}{equation.4.19}{}}
\@writefile{brf}{\backcite{Self}{{22}{4.2}{equation.4.19}}}
\citation{Levine:KnotCobordism}
\citation{Tristram:CobordismInvariants}
\citation{Conway:SignaturesSurvey}
\citation{Geneva-231201}
\newlabel{tl:n}{{1}{}{}{}{}}
\newlabel{tl:Ks}{{2}{}{}{}{}}
\newlabel{tl:D}{{3}{}{}{}{}}
\newlabel{tl:s}{{4}{}{}{}{}}
\newlabel{tl:J}{{5}{}{}{}{}}
\newlabel{tl:Kh}{{6}{}{}{}{}}
\newlabel{tl:H}{{7}{}{}{}{}}
\newlabel{tl:V}{{8}{}{}{}{}}
\newlabel{tl:KhHV}{{9}{}{}{}{}}
\newlabel{tl:r1}{{10}{}{}{}{}}
\newlabel{tl:r12}{{11}{}{}{}{}}
\newlabel{tl:r12KhHV}{{12}{}{}{}{}}
\newlabel{tl:Th}{{13}{}{}{}{}}
\newlabel{tl:Thr2}{{14}{}{}{}{}}
\newlabel{tl:Ths}{{15}{}{}{}{}}
\newlabel{tl:ThKh}{{16}{}{}{}{}}
\newlabel{tl:ThH}{{17}{}{}{}{}}
\newlabel{tl:ThV}{{18}{}{}{}{}}
\newlabel{tl:Thr2KhHV}{{19}{}{}{}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces  The separation powers of some knot invariants and combinations of knot invariants (in lines \ref  {tl:D}--\ref  {tl:Thr2KhHV}, smaller numbers are better). The data in this table was assembled by \cite  [Stats.nb]{Self}. \relax }}{23}{table.caption.11}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{23}{5.1}{table.caption.11}}}
\newlabel{tab:Strong}{{5.1}{23}{The separation powers of some knot invariants and combinations of knot invariants (in lines \ref {tl:D}--\ref {tl:Thr2KhHV}, smaller numbers are better). The data in this table was assembled by \cite [Stats.nb]{Self}. \relax }{table.caption.11}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Strong and Meaningful}}{23}{section.5}\protected@file@percent }
\newlabel{sec:SandM}{{5}{23}{Strong and Meaningful}{section.5}{}}
\newlabel{ssec:Strong}{{5.1}{23}{Strong}{subsection.5.1}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Strong}}{23}{subsection.5.1}\protected@file@percent }
\@writefile{brf}{\backcite{Levine:KnotCobordism, Tristram:CobordismInvariants, Conway:SignaturesSurvey}{{23}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Geneva-231201}{{23}{5.1}{table.caption.11}}}
\citation{CullerDunfieldGoernerWeeks:SnapPy}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Overbay:Thesis}
\citation{APAI}
\citation{Oaxaca-2210}
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces  The three pairs responsible for the deficit of 3 in the column $n\leq 11$ of line\nonbreakingspace \ref  {tl:Th} of Table\nonbreakingspace \ref  {tab:Strong}. They are $(11_{a44},11_{a47})$, $(11_{a57},11_{a231})$, and $(11_{n73},11_{n74})$, and each pair is a pair of mutant Montesinos knots (though $\Theta $ sometimes does separate mutant pairs, as was shown in Section\nonbreakingspace \ref  {ssec:Examples}). \relax }}{24}{figure.caption.12}\protected@file@percent }
\newlabel{fig:DrieParen}{{5.1}{24}{The three pairs responsible for the deficit of 3 in the column $n\leq 11$ of line~\ref {tl:Th} of Table~\ref {tab:Strong}. They are $(11_{a44},11_{a47})$, $(11_{a57},11_{a231})$, and $(11_{n73},11_{n74})$, and each pair is a pair of mutant Montesinos knots (though $\Theta $ sometimes does separate mutant pairs, as was shown in Section~\ref {ssec:Examples}). \relax }{figure.caption.12}{}}
\@writefile{brf}{\backcite{CullerDunfieldGoernerWeeks:SnapPy}{{24}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}{{24}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{APAI}{{24}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Oaxaca-2210}{{24}{5.1}{table.caption.11}}}
\citation{GaroufalidisKashaev:Multivariable}
\citation{GaroufalidisLi:Patterns}
\citation{LivingstonMoore:KnotInfo}
\citation{Self}
\citation{GompfScharlemannThompson:Counterexample}
\citation{GompfScharlemannThompson:Counterexample}
\citation{GompfScharlemannThompson:Counterexample}
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces The 48-crossing Gompf-Scharlemann-Thompson $\mathit  {GST}_{48}$ knot \cite  {GompfScharlemannThompson:Counterexample}. \relax }}{25}{figure.caption.13}\protected@file@percent }
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{25}{5.2}{figure.caption.13}}}
\newlabel{fig:GST48}{{5.2}{25}{The 48-crossing Gompf-Scharlemann-Thompson $\mathit {GST}_{48}$ knot \cite {GompfScharlemannThompson:Counterexample}. \relax }{figure.caption.13}{}}
\@writefile{brf}{\backcite{GaroufalidisKashaev:Multivariable}{{25}{5.1}{figure.caption.12}}}
\@writefile{brf}{\backcite{GaroufalidisLi:Patterns}{{25}{5.1}{figure.caption.12}}}
\newlabel{ssec:Meaningful}{{5.2}{25}{Meaningful}{subsection.5.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Meaningful}}{25}{subsection.5.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{5.2.1}{The Knot Genus}}{25}{subsubsection.5.2.1}\protected@file@percent }
\newlabel{conj:Genus}{{18}{25}{}{theorem.18}{}}
\@writefile{brf}{\backcite{LivingstonMoore:KnotInfo}{{25}{5.2.1}{theorem.18}}}
\@writefile{brf}{\backcite{Self}{{25}{5.2.1}{theorem.18}}}
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{25}{5.2.1}{theorem.18}}}
\citation{GompfScharlemannThompson:Counterexample}
\citation{LopezNeumannVanDerVeen:FibredLinks}
\citation{LivingstonMoore:KnotInfo}
\citation{Self}
\citation{Rolfsen:KnotsAndLinks}
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{26}{5.2.1}{figure.caption.13}}}
\@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{5.2.2}{Fibered Knots}}{26}{subsubsection.5.2.2}\protected@file@percent }
\@writefile{brf}{\backcite{LopezNeumannVanDerVeen:FibredLinks}{{26}{5.2.2}{subsubsection.5.2.2}}}
\newlabel{conj:Fibered}{{19}{26}{}{theorem.19}{}}
\@writefile{brf}{\backcite{LivingstonMoore:KnotInfo}{{26}{5.2.2}{theorem.19}}}
\@writefile{brf}{\backcite{Self}{{26}{5.2.2}{theorem.19}}}
\@writefile{brf}{\backcite{Rolfsen:KnotsAndLinks}{{26}{5.2.2}{theorem.19}}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stories, Conjectures, and Dreams}}{26}{section.6}\protected@file@percent }
\newlabel{sec:CandD}{{6}{26}{Stories, Conjectures, and Dreams}{section.6}{}}
\citation{CrowellFox:KnotTheory}
\citation{Lickorish:KnotTheory}
\citation{Kauffman:RotationalVirtualKnots}
\citation{Kauffman:RotationalVirtualKnots}
\citation{Kauffman:VirtualKnotTheory}
\citation{BDV:OU}
\citation{Kauffman:RotationalVirtualKnots}
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces  The invariant $\Theta $ of the fibered knot $12_{n242}$, also known as the $(-2,3,7)$ pretzel knot, and of the fibered knot $7_7$. For the first, $s(K)>0$ and the bar code visibly matches with the top row of the QR code (though our screens and printers and eyes may not be good enough to detect minor shading differences, so a visual inspection may not be enough). For the second, twice the degree of $\Delta $ is visibly greater than the degree of $\theta $, so $s(K)=0$. \relax }}{27}{figure.caption.14}\protected@file@percent }
\newlabel{fig:FiberedExamples}{{5.3}{27}{The invariant $\Theta $ of the fibered knot $12_{n242}$, also known as the $(-2,3,7)$ pretzel knot, and of the fibered knot $7_7$. For the first, $s(K)>0$ and the bar code visibly matches with the top row of the QR code (though our screens and printers and eyes may not be good enough to detect minor shading differences, so a visual inspection may not be enough). For the second, twice the degree of $\Delta $ is visibly greater than the degree of $\theta $, so $s(K)=0$. \relax }{figure.caption.14}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {6.1}{\ignorespaces  A long version of the rotational virtual knot $KS$ from\nonbreakingspace \cite  {Kauffman:RotationalVirtualKnots}. It has $X=\{(-1,1,6),(-1,2,4),(1,9,3),(-1,7,5),(1,10,8)\}$ and $\varphi =(-1,0,0,1,0,-1,0,0,1,0,0)$. \relax }}{27}{figure.caption.15}\protected@file@percent }
\@writefile{brf}{\backcite{Kauffman:RotationalVirtualKnots}{{27}{6.1}{figure.caption.15}}}
\newlabel{fig:KS}{{6.1}{27}{A long version of the rotational virtual knot $KS$ from~\cite {Kauffman:RotationalVirtualKnots}. It has $X=\{(-1,1,6),(-1,2,4),(1,9,3),(-1,7,5),(1,10,8)\}$ and $\varphi =(-1,0,0,1,0,-1,0,0,1,0,0)$. \relax }{figure.caption.15}{}}
\newlabel{conj:D6}{{20}{27}{}{theorem.20}{}}
\@writefile{brf}{\backcite{CrowellFox:KnotTheory}{{27}{6}{theorem.20}}}
\@writefile{brf}{\backcite{Lickorish:KnotTheory}{{27}{6}{theorem.20}}}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Overbay:Thesis}
\citation{APAI}
\citation{Ohtsuki:TwoLoop}
\citation{GaroufalidisRozansky:LoopExpansion}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Kricker:Lines}
\citation{Ohtsuki:TwoLoop}
\citation{Garoufalidis:Whitehead}
\citation{Ohtsuki:Cabling}
\citation{DPG}
\citation{PG}
\@writefile{brf}{\backcite{Kauffman:VirtualKnotTheory}{{28}{6}{figure.caption.15}}}
\newlabel{conj:Mirror}{{21}{28}{}{theorem.21}{}}
\newlabel{conj:Reverse}{{22}{28}{}{theorem.22}{}}
\newlabel{fact:ConnectedSum}{{23}{28}{}{theorem.23}{}}
\newlabel{conj:rho1}{{24}{28}{}{theorem.24}{}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}{{28}{24}{theorem.24}}}
\@writefile{brf}{\backcite{APAI}{{28}{24}{theorem.24}}}
\newlabel{conj:TwoLoop}{{25}{28}{}{theorem.25}{}}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{28}{25}{theorem.25}}}
\@writefile{brf}{\backcite{GaroufalidisRozansky:LoopExpansion, Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines}{{28}{25}{theorem.25}}}
\newlabel{disc:TwoLoop}{{26}{28}{}{theorem.26}{}}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{28}{26}{theorem.26}}}
\@writefile{brf}{\backcite{Garoufalidis:Whitehead}{{28}{26}{theorem.26}}}
\@writefile{brf}{\backcite{Ohtsuki:Cabling}{{28}{26}{theorem.26}}}
\@writefile{brf}{\backcite{DPG, PG}{{28}{6}{theorem.26}}}
\@writefile{brf}{\backcite{BDV:OU}{{28}{7}{figure.caption.15}}}
\@writefile{brf}{\backcite{Kauffman:RotationalVirtualKnots}{{28}{8}{figure.caption.15}}}
\citation{Drinfeld:QuantumGroups}
\citation{DPG}
\citation{PG}
\citation{Lawrence:UniversalUsingQG}
\citation{Ohtsuki:QuantumInvariants}
\citation{APAI}
\citation{PG}
\citation{Oaxaca-2210}
\citation{Schaveling:Thesis}
\citation{Harper:Non-Abelian}
\citation{AKT}
\citation{PG}
\citation{Ohtsuki:Cabling}
\newlabel{disc:SolvApp}{{27}{29}{}{theorem.27}{}}
\@writefile{brf}{\backcite{DPG, PG}{{29}{27}{theorem.27}}}
\@writefile{brf}{\backcite{Lawrence:UniversalUsingQG, Ohtsuki:QuantumInvariants}{{29}{27}{theorem.27}}}
\@writefile{brf}{\backcite{APAI, PG, Oaxaca-2210}{{29}{27}{theorem.27}}}
\@writefile{brf}{\backcite{Schaveling:Thesis}{{29}{6}{theorem.27}}}
\newlabel{conj:sl3}{{28}{29}{}{theorem.28}{}}
\@writefile{brf}{\backcite{Harper:Non-Abelian}{{29}{29}{theorem.29}}}
\newlabel{com:sl3genus}{{30}{29}{}{theorem.30}{}}
\@writefile{brf}{\backcite{AKT, PG}{{29}{30}{theorem.30}}}
\newlabel{disc:LieVersed}{{31}{29}{}{theorem.31}{}}
\newlabel{disc:Satellites}{{32}{29}{}{theorem.32}{}}
\@writefile{brf}{\backcite{Ohtsuki:Cabling}{{29}{32}{theorem.32}}}
\@writefile{brf}{\backcite{Drinfeld:QuantumGroups}{{29}{10}{theorem.27}}}
\citation{Polyak:FeynmanDiagrams}
\citation{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}
\citation{IType}
\newlabel{disc:FPGI}{{33}{30}{}{theorem.33}{}}
\@writefile{brf}{\backcite{Polyak:FeynmanDiagrams}{{30}{33}{theorem.33}}}
\newlabel{eq:PGI}{{20}{30}{}{equation.6.20}{}}
\newlabel{eq:FPGI}{{21}{30}{}{equation.6.21}{}}
\@writefile{brf}{\backcite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}{{30}{33}{equation.6.21}}}
\newlabel{fact:PerturbedGaussian}{{34}{30}{}{theorem.34}{}}
\@writefile{brf}{\backcite{IType}{{30}{34}{theorem.34}}}
\newlabel{com:UC}{{35}{30}{}{theorem.35}{}}
\citation{Ohtsuki:TwoLoop}
\citation{Kauffman:OnKnots}
\citation{Bai:Alexander}
\citation{FoxMilnor:CobordismOfKnots}
\citation{Self}
\citation{BurdeZieschang:Knots}
\citation{Self}
\newlabel{dream:Seifert}{{36}{31}{}{theorem.36}{}}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{31}{6}{theorem.36}}}
\newlabel{dream:UniversalSeifert}{{37}{31}{}{theorem.37}{}}
\@writefile{brf}{\backcite{Kauffman:OnKnots}{{31}{6}{theorem.37}}}
\@writefile{brf}{\backcite{Bai:Alexander}{{31}{6}{theorem.37}}}
\@writefile{brf}{\backcite{FoxMilnor:CobordismOfKnots}{{31}{38}{theorem.38}}}
\newlabel{dream:Ribbon}{{39}{31}{}{theorem.39}{}}
\newlabel{disc:Links}{{40}{31}{}{theorem.40}{}}
\@writefile{brf}{\backcite{Self}{{31}{40}{theorem.40}}}
\@writefile{brf}{\backcite{BurdeZieschang:Knots}{{31}{11}{figure.caption.16}}}
\citation{Gassner:OnBraidGroups}
\citation{Kawauchi:Srvery}
\citation{Khovanov:Categorification}
\citation{Bar-Natan:Categorification}
\citation{OzsvathSzabo:IntroductionToHF}
\citation{Manolescu:KnotFloerHomology}
\citation{Juhasz:Survey}
\@writefile{lof}{\contentsline {figure}{\numberline {6.2}{\ignorespaces  $\Theta $ for all the prime links with up to 9 crossings, up to reflections and with arbitrary choices of strand orientations. Empty boxes correspond to links for which $\Delta =0$. \relax }}{32}{figure.caption.16}\protected@file@percent }
\newlabel{fig:Theta4Links}{{6.2}{32}{$\Theta $ for all the prime links with up to 9 crossings, up to reflections and with arbitrary choices of strand orientations. Empty boxes correspond to links for which $\Delta =0$. \relax }{figure.caption.16}{}}
\@writefile{brf}{\backcite{Self}{{32}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Gassner:OnBraidGroups}{{32}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Kawauchi:Srvery}{{32}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Khovanov:Categorification, Bar-Natan:Categorification}{{32}{6}{figure.caption.16}}}
\@writefile{brf}{\backcite{OzsvathSzabo:IntroductionToHF, Manolescu:KnotFloerHomology, Juhasz:Survey}{{32}{6}{theorem.41}}}
\bibcite{Alexander:TopologicalInvariants}{Al}
\bibcite{Bai:Alexander}{Ba}
\bibcite{Bar-Natan:Categorification}{BN1}
\bibcite{DPG}{BN2}
\bibcite{AKT}{BN3}
\bibcite{Oaxaca-2210}{BN4}
\bibcite{Geneva-231201}{BN5}
\bibcite{IType}{BN6}
\bibcite{Toronto-241030}{BN7}
\bibcite{BDV:OU}{BDV}
\bibcite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}{BGRT}
\bibcite{APAI}{BV1}
\bibcite{PG}{BV2}
\bibcite{Self}{BV3}
\bibcite{BecerraVanHelden:RotationalReidemeister}{BVH}
\bibcite{BurdeZieschang:Knots}{BZ}
\bibcite{Conway:SignaturesSurvey}{Co}
\bibcite{CrowellFox:KnotTheory}{CF}
\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Acknowledgement}}{33}{section.7}\protected@file@percent }
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{33}{section*.17}\protected@file@percent }
\bibcite{CullerDunfieldGoernerWeeks:SnapPy}{CDGW}
\bibcite{Drinfeld:QuantumGroups}{Dr}
\bibcite{DHOEBL:Random}{DHOEBL}
\bibcite{FoxMilnor:CobordismOfKnots}{FM}
\bibcite{Garoufalidis:Whitehead}{Gar}
\bibcite{GaroufalidisKashaev:Multivariable}{GK}
\bibcite{GaroufalidisLi:Patterns}{GL}
\bibcite{GaroufalidisRozansky:LoopExpansion}{GR}
\bibcite{Gassner:OnBraidGroups}{Gas}
\bibcite{GompfScharlemannThompson:Counterexample}{GST}
\bibcite{Harper:Non-Abelian}{Ha}
\bibcite{Juhasz:Survey}{Ju}
\bibcite{Kauffman:OnKnots}{Kau1}
\bibcite{Kauffman:VirtualKnotTheory}{Kau2}
\bibcite{Kauffman:RotationalVirtualKnots}{Kau3}
\bibcite{Kawauchi:Srvery}{Kaw}
\bibcite{Khovanov:Categorification}{Kh}
\bibcite{Kricker:Lines}{Kr}
\bibcite{Lalani:ThetaSurvey}{Lal}
\bibcite{Lawrence:UniversalUsingQG}{Law}
\bibcite{Levine:KnotCobordism}{Le}
\bibcite{Lickorish:KnotTheory}{Li}
\bibcite{LivingstonMoore:KnotInfo}{LM}
\bibcite{LopezNeumannVanDerVeen:FibredLinks}{LV}
\bibcite{Manolescu:KnotFloerHomology}{Ma}
\bibcite{Ohtsuki:QuantumInvariants}{Oh1}
\bibcite{Ohtsuki:TwoLoop}{Oh2}
\bibcite{Ohtsuki:Cabling}{Oh3}
\bibcite{OzsvathSzabo:IntroductionToHF}{OS}
\bibcite{Overbay:Thesis}{Ov}
\bibcite{Polyak:FeynmanDiagrams}{Po1}
\bibcite{Polyak:RMoves}{Po2}
\bibcite{Rolfsen:KnotsAndLinks}{Rol}
\bibcite{Rozansky:Contribution}{Roz1}
\bibcite{Rozansky:Burau}{Roz2}
\bibcite{Rozansky:U1RCC}{Roz3}
\bibcite{Schaveling:Thesis}{Sch}
\bibcite{Tristram:CobordismInvariants}{Tr}
\bibcite{Wolfram:Mathematica}{Wo}
\citation{Garoufalidis:Whitehead}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{15.01373pt}
\newlabel{tocindent1}{20.88867pt}
\newlabel{tocindent2}{34.5pt}
\newlabel{tocindent3}{40.80037pt}
\@writefile{brf}{\backcite{Garoufalidis:Whitehead}{{35}{7}{section*.17}}}
\gdef \@abspage@last{35}
\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand\HyField@AuxAddToFields[1]{}
\providecommand\HyField@AuxAddToCoFields[2]{}
\citation{Ohtsuki:TwoLoop}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Kricker:Lines}
\citation{GaroufalidisRozansky:LoopExpansion}
\providecommand \oddpage@label [2]{}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{1}{(document)}{Doc-Start}}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines, GaroufalidisRozansky:LoopExpansion}{{1}{(document)}{Doc-Start}}}
\citation{Alexander:TopologicalInvariants}
\citation{Lalani:ThetaSurvey}
\citation{Self}
\citation{Self}
\citation{Self}
\citation{Self}
\citation{DHOEBL:Random}
\citation{DHOEBL:Random}
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Fun}}{3}{section.1}\protected@file@percent }
\@writefile{brf}{\backcite{Alexander:TopologicalInvariants}{{3}{1}{section.1}}}
\@writefile{brf}{\backcite{Lalani:ThetaSurvey}{{3}{1}{section.1}}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces $\Theta $ as a bar code and a QR code, for all the knots in the Rolfsen table.\relax }}{4}{figure.caption.2}\protected@file@percent }
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:Rolfsen}{{1.1}{4}{$\Theta $ as a bar code and a QR code, for all the knots in the Rolfsen table.\relax }{figure.caption.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces $\Theta $ of some square weave knots, as computed by\nonbreakingspace \cite  [WeaveKnots.nb]{Self}.\relax }}{5}{figure.caption.3}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{5}{1.2}{figure.caption.3}}}
\newlabel{fig:SquareWeaves}{{1.2}{5}{$\Theta $ of some square weave knots, as computed by~\cite [WeaveKnots.nb]{Self}.\relax }{figure.caption.3}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces $\Theta $ of a randomized weave knot, as computed by\nonbreakingspace \cite  [WeaveKnots.nb]{Self}. Crossings were chosen to be positive or negative with equal probabilities.\relax }}{5}{figure.caption.4}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{5}{1.3}{figure.caption.4}}}
\newlabel{fig:RandomWeave}{{1.3}{5}{$\Theta $ of a randomized weave knot, as computed by~\cite [WeaveKnots.nb]{Self}. Crossings were chosen to be positive or negative with equal probabilities.\relax }{figure.caption.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces $\theta $ (hexagonal QR code only) of the 15 largest knots that we have computed by September 16, 2024. They are all ``generic'' in as much as we know, and they all have $\geq 300$ crossings. The knots come from\nonbreakingspace \cite  {DHOEBL:Random}. Warning: Some screens/printers may introduce spurious Moir\'e interference patterns. \relax }}{6}{figure.caption.5}\protected@file@percent }
\@writefile{brf}{\backcite{DHOEBL:Random}{{6}{1.4}{figure.caption.5}}}
\newlabel{fig:300}{{1.4}{6}{$\theta $ (hexagonal QR code only) of the 15 largest knots that we have computed by September 16, 2024. They are all ``generic'' in as much as we know, and they all have $\geq 300$ crossings. The knots come from~\cite {DHOEBL:Random}. Warning: Some screens/printers may introduce spurious Moir\'e interference patterns. \relax }{figure.caption.5}{}}
\citation{IType}
\citation{APAI}
\citation{Toronto-241030}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The Main Theorem}}{7}{section.2}\protected@file@percent }
\newlabel{sec:MainTheorem}{{2}{7}{The Main Theorem}{section.2}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {2.1}{\ignorespaces An example upright knot diagram.\relax }}{7}{figure.2.1}\protected@file@percent }
\newlabel{fig:SampleDiagram}{{2.1}{7}{An example upright knot diagram.\relax }{figure.2.1}{}}
\newlabel{eq:A}{{1}{7}{The Main Theorem}{equation.2.1}{}}
\newlabel{eq:Delta}{{2}{7}{The Main Theorem}{equation.2.2}{}}
\@writefile{brf}{\backcite{IType}{{7}{2}{equation.2.2}}}
\@writefile{brf}{\backcite{APAI, Toronto-241030}{{7}{2}{equation.2.2}}}
\citation{APAI}
\newlabel{eq:F1}{{3}{8}{The Main Theorem}{equation.2.3}{}}
\newlabel{eq:F2}{{4}{8}{The Main Theorem}{equation.2.4}{}}
\newlabel{eq:F3}{{5}{8}{The Main Theorem}{equation.2.5}{}}
\newlabel{thm:Main}{{1}{8}{The Main Theorem, proof in Section~\ref {sec:Proof}}{theorem.1}{}}
\newlabel{eq:Main}{{6}{8}{The Main Theorem, proof in Section~\ref {sec:Proof}}{equation.2.6}{}}
\newlabel{com:normalization}{{2}{8}{}{theorem.2}{}}
\newlabel{com:traffic}{{3}{8}{}{theorem.3}{}}
\@writefile{brf}{\backcite{APAI}{{8}{3}{theorem.3}}}
\citation{APAI}
\citation{IType}
\@writefile{brf}{\backcite{APAI}{{9}{3}{theorem.3}}}
\newlabel{com:FeynmanDiagrams}{{4}{9}{}{theorem.4}{}}
\@writefile{brf}{\backcite{IType}{{9}{4}{theorem.4}}}
\newlabel{com:Computation}{{5}{9}{}{theorem.5}{}}
\citation{Wolfram:Mathematica}
\citation{Self}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Implementation and Examples}}{11}{section.3}\protected@file@percent }
\newlabel{sec:Implementation}{{3}{11}{Implementation and Examples}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Implementation}}{11}{subsection.3.1}\protected@file@percent }
\@writefile{brf}{\backcite{Wolfram:Mathematica}{{11}{3.1}{subsection.3.1}}}
\@writefile{brf}{\backcite{Self}{{11}{3.1}{subsection.3.1}}}
\newlabel{ssec:Examples}{{3.2}{12}{Examples}{subsection.3.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{Examples}}{12}{subsection.3.2}\protected@file@percent }
\citation{Self}
\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces The 132-crossing torus knot $T_{22/7}$ and a plot of its $\Theta $ invariant\relax }}{14}{figure.caption.6}\protected@file@percent }
\newlabel{fig:T227}{{3.1}{14}{The 132-crossing torus knot $T_{22/7}$ and a plot of its $\Theta $ invariant\relax }{figure.caption.6}{}}
\@writefile{brf}{\backcite{Self}{{14}{3.2}{figure.caption.6}}}
\citation{APAI}
\citation{APAI}
\citation{APAI}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Proof of the Main Theorem, Theorem\nonbreakingspace \ref  {thm:Main}}}{17}{section.4}\protected@file@percent }
\newlabel{sec:Proof}{{4}{17}{Proof of the Main Theorem, Theorem~\ref {thm:Main}}{section.4}{}}
\newlabel{ssec:Invariance}{{4.1}{17}{Proof of Invariance}{subsection.4.1}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.1}{Proof of Invariance}}{17}{subsection.4.1}\protected@file@percent }
\@writefile{brf}{\backcite{APAI}{{17}{4.1}{subsection.4.1}}}
\@writefile{brf}{\backcite{APAI}{{17}{4.1}{subsection.4.1}}}
\newlabel{eq:Ap}{{7}{17}{Proof of Invariance}{equation.4.7}{}}
\@writefile{brf}{\backcite{APAI}{{17}{4.1}{equation.4.7}}}
\newlabel{eq:CarRules}{{8}{17}{Proof of Invariance}{equation.4.8}{}}
\newlabel{eq:CounterRules}{{9}{17}{Proof of Invariance}{equation.4.9}{}}
\newlabel{eq:tilg}{{10}{17}{Proof of Invariance}{equation.4.10}{}}
\citation{APAI}
\citation{APAI}
\newlabel{eq:NullCarRules}{{11}{18}{}{equation.4.11}{}}
\newlabel{eq:NullCounterRules}{{12}{18}{}{equation.4.12}{}}
\newlabel{lem:NullVertices}{{7}{18}{}{theorem.7}{}}
\newlabel{rem:F4Null}{{8}{18}{}{theorem.8}{}}
\@writefile{brf}{\backcite{APAI}{{18}{4.1}{theorem.8}}}
\@writefile{brf}{\backcite{APAI}{{18}{4.1}{theorem.8}}}
\newlabel{thm:RelativeInvariant}{{9}{18}{}{theorem.9}{}}
\citation{APAI}
\citation{Oaxaca-2210}
\citation{Toronto-241030}
\citation{APAI}
\citation{Polyak:RMoves}
\citation{Polyak:RMoves}
\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces  The modified Green function ${\tilde  {g}}_{ab}$ is invariant under Reidemeister moves performed away from where it is measured. \relax }}{19}{figure.caption.7}\protected@file@percent }
\newlabel{fig:RelativeInvariance}{{4.1}{19}{The modified Green function $\tilg _{ab}$ is invariant under Reidemeister moves performed away from where it is measured. \relax }{figure.caption.7}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.2}{\ignorespaces  A generating set of oriented Reidemeister moves as in\nonbreakingspace \cite  [Figure\nonbreakingspace 6]{Polyak:RMoves}. Aside 1: the braid-like R2b is not needed. Aside 2: yet R2b cannot replace R2c$^\pm $ because in the would-be proof, an unpostulated form of R3 is used (which in itself follows from R2c$^\pm $). \relax }}{19}{figure.caption.8}\protected@file@percent }
\@writefile{brf}{\backcite{Polyak:RMoves}{{19}{4.2}{figure.caption.8}}}
\newlabel{fig:RMoves}{{4.2}{19}{A generating set of oriented Reidemeister moves as in~\cite [Figure~6]{Polyak:RMoves}. Aside 1: the braid-like R2b is not needed. Aside 2: yet R2b cannot replace R2c$^\pm $ because in the would-be proof, an unpostulated form of R3 is used (which in itself follows from R2c$^\pm $). \relax }{figure.caption.8}{}}
\@writefile{brf}{\backcite{APAI, Oaxaca-2210, Toronto-241030}{{19}{4.1}{figure.caption.7}}}
\@writefile{brf}{\backcite{APAI}{{19}{4.1}{figure.caption.7}}}
\newlabel{eq:R3LeftOuter}{{13}{20}{Proof of Invariance}{equation.4.13}{}}
\newlabel{eq:R3LeftInner}{{14}{20}{Proof of Invariance}{equation.4.14}{}}
\newlabel{eq:R3RightOuter}{{15}{20}{Proof of Invariance}{equation.4.15}{}}
\newlabel{eq:R3RightInner}{{16}{20}{Proof of Invariance}{equation.4.16}{}}
\citation{BecerraVanHelden:RotationalReidemeister}
\newlabel{prop:UprightRMoves}{{10}{21}{}{theorem.10}{}}
\@writefile{brf}{\backcite{BecerraVanHelden:RotationalReidemeister}{{21}{4.1}{figure.caption.9}}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.3}{\ignorespaces  The upright Reidemeister moves: The R1 and R3 moves are already upright and remain the same as in Figure\nonbreakingspace \ref  {fig:RMoves}. The crossings in the R2 moves of Figure\nonbreakingspace \ref  {fig:RMoves} are rotated to be upright. We also need two further moves: The null vertex move NV for adding and removing null vertices, and the swirl move Sw which then implies that any two ways of turning a crossing upright are the same. We sometimes indicate rotation numbers symbolically rather than using complicated spirals. \relax }}{22}{figure.caption.9}\protected@file@percent }
\newlabel{fig:UprightRMoves}{{4.3}{22}{The upright Reidemeister moves: The R1 and R3 moves are already upright and remain the same as in Figure~\ref {fig:RMoves}. The crossings in the R2 moves of Figure~\ref {fig:RMoves} are rotated to be upright. We also need two further moves: The null vertex move NV for adding and removing null vertices, and the swirl move Sw which then implies that any two ways of turning a crossing upright are the same. We sometimes indicate rotation numbers symbolically rather than using complicated spirals. \relax }{figure.caption.9}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {4.4}{\ignorespaces  The two sides $D^l$ and $D^r$ of the R3b move. The left side $D^l$ consists of 3 distinguished crossings $c^l_1=(1,j,k)$, $c^l_2=(1,i,{k^+})$, $c^l_3=(1,{i^+},{j^+})$ and a collection of further crossings $c_y=(s,m,n)\in Y$, where $Y$ is the set of crossings not participating in the R3b move. The right side $D^r$ consists of $c^r_1=(1,i,j)$, $c^r_2=(1,{i^+},k)$, $c^r_3=(1,{j^+},{k^+})$ and the same set $Y$ of further crossings $c_y$. \relax }}{22}{figure.caption.10}\protected@file@percent }
\newlabel{fig:R3}{{4.4}{22}{The two sides $D^l$ and $D^r$ of the R3b move. The left side $D^l$ consists of 3 distinguished crossings $c^l_1=(1,j,k)$, $c^l_2=(1,i,\kp )$, $c^l_3=(1,\ip ,\jp )$ and a collection of further crossings $c_y=(s,m,n)\in Y$, where $Y$ is the set of crossings not participating in the R3b move. The right side $D^r$ consists of $c^r_1=(1,i,j)$, $c^r_2=(1,\ip ,k)$, $c^r_3=(1,\jp ,\kp )$ and the same set $Y$ of further crossings $c_y$. \relax }{figure.caption.10}{}}
\newlabel{prop:R3}{{11}{22}{}{theorem.11}{}}
\newlabel{eq:ABC}{{17}{22}{Proof of Invariance}{equation.4.17}{}}
\newlabel{eq:R3A}{{18}{23}{Proof of Invariance}{equation.4.18}{}}
\newlabel{rem:E}{{12}{24}{}{theorem.12}{}}
\newlabel{prop:R2c}{{13}{24}{}{theorem.13}{}}
\newlabel{prop:R1s}{{14}{25}{}{theorem.14}{}}
\newlabel{prop:Sw}{{15}{26}{}{theorem.15}{}}
\newlabel{prop:NV}{{16}{26}{}{theorem.16}{}}
\citation{Self}
\newlabel{ssec:Residue}{{4.2}{27}{Proof of Polynomiality}{subsection.4.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.2}{Proof of Polynomiality}}{27}{subsection.4.2}\protected@file@percent }
\newlabel{lem:del}{{17}{27}{}{theorem.17}{}}
\newlabel{eq:delf}{{19}{27}{}{equation.4.19}{}}
\@writefile{brf}{\backcite{Self}{{27}{4.2}{equation.4.19}}}
\citation{Self}
\citation{Self}
\citation{Levine:KnotCobordism}
\citation{Tristram:CobordismInvariants}
\citation{Conway:SignaturesSurvey}
\citation{Geneva-231201}
\newlabel{tl:n}{{1}{}{}{}{}}
\newlabel{tl:Ks}{{2}{}{}{}{}}
\newlabel{tl:D}{{3}{}{}{}{}}
\newlabel{tl:s}{{4}{}{}{}{}}
\newlabel{tl:J}{{5}{}{}{}{}}
\newlabel{tl:Kh}{{6}{}{}{}{}}
\newlabel{tl:H}{{7}{}{}{}{}}
\newlabel{tl:V}{{8}{}{}{}{}}
\newlabel{tl:KhHV}{{9}{}{}{}{}}
\newlabel{tl:r1}{{10}{}{}{}{}}
\newlabel{tl:r12}{{11}{}{}{}{}}
\newlabel{tl:r12KhHV}{{12}{}{}{}{}}
\newlabel{tl:Th}{{13}{}{}{}{}}
\newlabel{tl:Thr2}{{14}{}{}{}{}}
\newlabel{tl:Ths}{{15}{}{}{}{}}
\newlabel{tl:ThKh}{{16}{}{}{}{}}
\newlabel{tl:ThH}{{17}{}{}{}{}}
\newlabel{tl:ThV}{{18}{}{}{}{}}
\newlabel{tl:Thr2KhHV}{{19}{}{}{}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5.1}{\ignorespaces  The separation powers of some knot invariants and combinations of knot invariants (in lines \ref  {tl:D}--\ref  {tl:Thr2KhHV}, smaller numbers are better). The data in this table was assembled by \cite  [Stats.nb]{Self}. \relax }}{29}{table.caption.11}\protected@file@percent }
\@writefile{brf}{\backcite{Self}{{29}{5.1}{table.caption.11}}}
\newlabel{tab:Strong}{{5.1}{29}{The separation powers of some knot invariants and combinations of knot invariants (in lines \ref {tl:D}--\ref {tl:Thr2KhHV}, smaller numbers are better). The data in this table was assembled by \cite [Stats.nb]{Self}. \relax }{table.caption.11}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Strong and Meaningful}}{29}{section.5}\protected@file@percent }
\newlabel{sec:SandM}{{5}{29}{Strong and Meaningful}{section.5}{}}
\newlabel{ssec:Strong}{{5.1}{29}{Strong}{subsection.5.1}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Strong}}{29}{subsection.5.1}\protected@file@percent }
\@writefile{brf}{\backcite{Levine:KnotCobordism, Tristram:CobordismInvariants, Conway:SignaturesSurvey}{{29}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Geneva-231201}{{29}{5.1}{table.caption.11}}}
\citation{CullerDunfieldGoernerWeeks:SnapPy}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Overbay:Thesis}
\citation{APAI}
\citation{Oaxaca-2210}
\@writefile{lof}{\contentsline {figure}{\numberline {5.1}{\ignorespaces  The three pairs responsible for the deficit of 3 in the column $n\leq 11$ of line\nonbreakingspace \ref  {tl:Th} of Table\nonbreakingspace \ref  {tab:Strong}. They are $(11_{a44},11_{a47})$, $(11_{a57},11_{a231})$, and $(11_{n73},11_{n74})$, and each pair is a pair of mutant Montesinos knots (though $\Theta $ sometimes does separate mutant pairs, as was shown in Section\nonbreakingspace \ref  {ssec:Examples}). \relax }}{30}{figure.caption.12}\protected@file@percent }
\newlabel{fig:DrieParen}{{5.1}{30}{The three pairs responsible for the deficit of 3 in the column $n\leq 11$ of line~\ref {tl:Th} of Table~\ref {tab:Strong}. They are $(11_{a44},11_{a47})$, $(11_{a57},11_{a231})$, and $(11_{n73},11_{n74})$, and each pair is a pair of mutant Montesinos knots (though $\Theta $ sometimes does separate mutant pairs, as was shown in Section~\ref {ssec:Examples}). \relax }{figure.caption.12}{}}
\@writefile{brf}{\backcite{CullerDunfieldGoernerWeeks:SnapPy}{{30}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}{{30}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{APAI}{{30}{5.1}{table.caption.11}}}
\@writefile{brf}{\backcite{Oaxaca-2210}{{30}{5.1}{table.caption.11}}}
\citation{GaroufalidisKashaev:Multivariable}
\citation{GaroufalidisLi:Patterns}
\citation{LivingstonMoore:KnotInfo}
\citation{Self}
\citation{GompfScharlemannThompson:Counterexample}
\citation{GompfScharlemannThompson:Counterexample}
\citation{GompfScharlemannThompson:Counterexample}
\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces The 48-crossing Gompf-Scharlemann-Thompson $\mathit  {GST}_{48}$ knot \cite  {GompfScharlemannThompson:Counterexample}. \relax }}{31}{figure.caption.13}\protected@file@percent }
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{31}{5.2}{figure.caption.13}}}
\newlabel{fig:GST48}{{5.2}{31}{The 48-crossing Gompf-Scharlemann-Thompson $\mathit {GST}_{48}$ knot \cite {GompfScharlemannThompson:Counterexample}. \relax }{figure.caption.13}{}}
\@writefile{brf}{\backcite{GaroufalidisKashaev:Multivariable}{{31}{5.1}{figure.caption.12}}}
\@writefile{brf}{\backcite{GaroufalidisLi:Patterns}{{31}{5.1}{figure.caption.12}}}
\newlabel{ssec:Meaningful}{{5.2}{31}{Meaningful}{subsection.5.2}{}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Meaningful}}{31}{subsection.5.2}\protected@file@percent }
\@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{5.2.1}{The Knot Genus}}{31}{subsubsection.5.2.1}\protected@file@percent }
\newlabel{conj:Genus}{{18}{31}{}{theorem.18}{}}
\@writefile{brf}{\backcite{LivingstonMoore:KnotInfo}{{31}{5.2.1}{theorem.18}}}
\@writefile{brf}{\backcite{Self}{{31}{5.2.1}{theorem.18}}}
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{31}{5.2.1}{theorem.18}}}
\citation{GompfScharlemannThompson:Counterexample}
\citation{LopezNeumannVanDerVeen:FibredLinks}
\citation{LivingstonMoore:KnotInfo}
\citation{Self}
\citation{Rolfsen:KnotsAndLinks}
\@writefile{brf}{\backcite{GompfScharlemannThompson:Counterexample}{{32}{5.2.1}{figure.caption.13}}}
\@writefile{toc}{\contentsline {subsubsection}{\tocsubsubsection {}{5.2.2}{Fibered Knots}}{32}{subsubsection.5.2.2}\protected@file@percent }
\@writefile{brf}{\backcite{LopezNeumannVanDerVeen:FibredLinks}{{32}{5.2.2}{subsubsection.5.2.2}}}
\newlabel{conj:Fibered}{{19}{32}{}{theorem.19}{}}
\@writefile{brf}{\backcite{LivingstonMoore:KnotInfo}{{32}{5.2.2}{theorem.19}}}
\@writefile{brf}{\backcite{Self}{{32}{5.2.2}{theorem.19}}}
\@writefile{brf}{\backcite{Rolfsen:KnotsAndLinks}{{32}{5.2.2}{theorem.19}}}
\@writefile{lof}{\contentsline {figure}{\numberline {5.3}{\ignorespaces  The invariant $\Theta $ of the fibered knot $12_{n242}$, also known as the $(-2,3,7)$ pretzel knot, and of the fibered knot $7_7$. For the first, $s(K)>0$ and the bar code visibly matches with the top row of the QR code (though our screens and printers and eyes may not be good enough to detect minor shading differences, so a visual inspection may not be enough). For the second, twice the degree of $\Delta $ is visibly greater than the degree of $\theta $, so $s(K)=0$. \relax }}{33}{figure.caption.14}\protected@file@percent }
\newlabel{fig:FiberedExamples}{{5.3}{33}{The invariant $\Theta $ of the fibered knot $12_{n242}$, also known as the $(-2,3,7)$ pretzel knot, and of the fibered knot $7_7$. For the first, $s(K)>0$ and the bar code visibly matches with the top row of the QR code (though our screens and printers and eyes may not be good enough to detect minor shading differences, so a visual inspection may not be enough). For the second, twice the degree of $\Delta $ is visibly greater than the degree of $\theta $, so $s(K)=0$. \relax }{figure.caption.14}{}}
\citation{CrowellFox:KnotTheory}
\citation{Lickorish:KnotTheory}
\citation{Kauffman:RotationalVirtualKnots}
\citation{Kauffman:RotationalVirtualKnots}
\citation{Kauffman:VirtualKnotTheory}
\citation{BDV:OU}
\citation{Kauffman:RotationalVirtualKnots}
\@writefile{lof}{\contentsline {figure}{\numberline {6.1}{\ignorespaces  A long version of the rotational virtual knot $KS$ from\nonbreakingspace \cite  {Kauffman:RotationalVirtualKnots}. It has $X=\{(-1,1,6),(-1,2,4),(1,9,3),(-1,7,5),(1,10,8)\}$ and $\varphi =(-1,0,0,1,0,-1,0,0,1,0,0)$. \relax }}{35}{figure.caption.15}\protected@file@percent }
\@writefile{brf}{\backcite{Kauffman:RotationalVirtualKnots}{{35}{6.1}{figure.caption.15}}}
\newlabel{fig:KS}{{6.1}{35}{A long version of the rotational virtual knot $KS$ from~\cite {Kauffman:RotationalVirtualKnots}. It has $X=\{(-1,1,6),(-1,2,4),(1,9,3),(-1,7,5),(1,10,8)\}$ and $\varphi =(-1,0,0,1,0,-1,0,0,1,0,0)$. \relax }{figure.caption.15}{}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stories, Conjectures, and Dreams}}{35}{section.6}\protected@file@percent }
\newlabel{sec:CandD}{{6}{35}{Stories, Conjectures, and Dreams}{section.6}{}}
\newlabel{conj:D6}{{20}{35}{}{theorem.20}{}}
\@writefile{brf}{\backcite{CrowellFox:KnotTheory}{{35}{6}{theorem.20}}}
\@writefile{brf}{\backcite{Lickorish:KnotTheory}{{35}{6}{theorem.20}}}
\@writefile{brf}{\backcite{Kauffman:VirtualKnotTheory}{{35}{6}{figure.caption.15}}}
\@writefile{brf}{\backcite{BDV:OU}{{35}{7}{figure.caption.15}}}
\@writefile{brf}{\backcite{Kauffman:RotationalVirtualKnots}{{35}{8}{figure.caption.15}}}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Overbay:Thesis}
\citation{APAI}
\citation{Ohtsuki:TwoLoop}
\citation{GaroufalidisRozansky:LoopExpansion}
\citation{Rozansky:Contribution}
\citation{Rozansky:Burau}
\citation{Rozansky:U1RCC}
\citation{Kricker:Lines}
\citation{Ohtsuki:TwoLoop}
\citation{Garoufalidis:Whitehead}
\citation{Ohtsuki:Cabling}
\citation{DPG}
\citation{PG}
\citation{Drinfeld:QuantumGroups}
\citation{DPG}
\citation{PG}
\citation{Lawrence:UniversalUsingQG}
\citation{Ohtsuki:QuantumInvariants}
\newlabel{conj:Mirror}{{21}{36}{}{theorem.21}{}}
\newlabel{conj:Reverse}{{22}{36}{}{theorem.22}{}}
\newlabel{fact:ConnectedSum}{{23}{36}{}{theorem.23}{}}
\newlabel{conj:rho1}{{24}{36}{}{theorem.24}{}}
\@writefile{brf}{\backcite{Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Overbay:Thesis}{{36}{24}{theorem.24}}}
\@writefile{brf}{\backcite{APAI}{{36}{24}{theorem.24}}}
\newlabel{conj:TwoLoop}{{25}{36}{}{theorem.25}{}}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{36}{25}{theorem.25}}}
\@writefile{brf}{\backcite{GaroufalidisRozansky:LoopExpansion, Rozansky:Contribution, Rozansky:Burau, Rozansky:U1RCC, Kricker:Lines}{{36}{25}{theorem.25}}}
\newlabel{disc:TwoLoop}{{26}{36}{}{theorem.26}{}}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{36}{26}{theorem.26}}}
\@writefile{brf}{\backcite{Garoufalidis:Whitehead}{{36}{26}{theorem.26}}}
\@writefile{brf}{\backcite{Ohtsuki:Cabling}{{36}{26}{theorem.26}}}
\@writefile{brf}{\backcite{DPG, PG}{{36}{6}{theorem.26}}}
\newlabel{disc:SolvApp}{{27}{36}{}{theorem.27}{}}
\@writefile{brf}{\backcite{Drinfeld:QuantumGroups}{{36}{10}{theorem.27}}}
\citation{APAI}
\citation{PG}
\citation{Oaxaca-2210}
\citation{Schaveling:Thesis}
\citation{Harper:Non-Abelian}
\citation{AKT}
\citation{PG}
\citation{Ohtsuki:Cabling}
\citation{Polyak:FeynmanDiagrams}
\@writefile{brf}{\backcite{DPG, PG}{{37}{27}{theorem.27}}}
\@writefile{brf}{\backcite{Lawrence:UniversalUsingQG, Ohtsuki:QuantumInvariants}{{37}{27}{theorem.27}}}
\@writefile{brf}{\backcite{APAI, PG, Oaxaca-2210}{{37}{27}{theorem.27}}}
\@writefile{brf}{\backcite{Schaveling:Thesis}{{37}{6}{theorem.27}}}
\newlabel{conj:sl3}{{28}{37}{}{theorem.28}{}}
\@writefile{brf}{\backcite{Harper:Non-Abelian}{{37}{29}{theorem.29}}}
\newlabel{com:sl3genus}{{30}{37}{}{theorem.30}{}}
\@writefile{brf}{\backcite{AKT, PG}{{37}{30}{theorem.30}}}
\newlabel{disc:LieVersed}{{31}{37}{}{theorem.31}{}}
\newlabel{disc:Satellites}{{32}{37}{}{theorem.32}{}}
\@writefile{brf}{\backcite{Ohtsuki:Cabling}{{37}{32}{theorem.32}}}
\newlabel{disc:FPGI}{{33}{37}{}{theorem.33}{}}
\@writefile{brf}{\backcite{Polyak:FeynmanDiagrams}{{37}{33}{theorem.33}}}
\newlabel{eq:PGI}{{20}{37}{}{equation.6.20}{}}
\citation{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}
\citation{IType}
\citation{Ohtsuki:TwoLoop}
\newlabel{eq:FPGI}{{21}{38}{}{equation.6.21}{}}
\@writefile{brf}{\backcite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}{{38}{33}{equation.6.21}}}
\newlabel{fact:PerturbedGaussian}{{34}{38}{}{theorem.34}{}}
\@writefile{brf}{\backcite{IType}{{38}{34}{theorem.34}}}
\newlabel{com:UC}{{35}{38}{}{theorem.35}{}}
\newlabel{dream:Seifert}{{36}{38}{}{theorem.36}{}}
\citation{Kauffman:OnKnots}
\citation{Bai:Alexander}
\citation{FoxMilnor:CobordismOfKnots}
\citation{Self}
\citation{BurdeZieschang:Knots}
\citation{Self}
\citation{Gassner:OnBraidGroups}
\citation{Kawauchi:Srvery}
\citation{Khovanov:Categorification}
\citation{Bar-Natan:Categorification}
\@writefile{brf}{\backcite{Ohtsuki:TwoLoop}{{39}{6}{theorem.36}}}
\newlabel{dream:UniversalSeifert}{{37}{39}{}{theorem.37}{}}
\@writefile{brf}{\backcite{Kauffman:OnKnots}{{39}{6}{theorem.37}}}
\@writefile{brf}{\backcite{Bai:Alexander}{{39}{6}{theorem.37}}}
\@writefile{brf}{\backcite{FoxMilnor:CobordismOfKnots}{{39}{38}{theorem.38}}}
\newlabel{dream:Ribbon}{{39}{39}{}{theorem.39}{}}
\newlabel{disc:Links}{{40}{39}{}{theorem.40}{}}
\@writefile{brf}{\backcite{Self}{{39}{40}{theorem.40}}}
\@writefile{brf}{\backcite{Self}{{39}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Gassner:OnBraidGroups}{{39}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Kawauchi:Srvery}{{39}{40}{figure.caption.16}}}
\@writefile{brf}{\backcite{Khovanov:Categorification, Bar-Natan:Categorification}{{39}{6}{figure.caption.16}}}
\@writefile{brf}{\backcite{BurdeZieschang:Knots}{{39}{11}{figure.caption.16}}}
\citation{OzsvathSzabo:IntroductionToHF}
\citation{Manolescu:KnotFloerHomology}
\citation{Juhasz:Survey}
\@writefile{lof}{\contentsline {figure}{\numberline {6.2}{\ignorespaces  $\Theta $ for all the prime links with up to 9 crossings, up to reflections and with arbitrary choices of strand orientations. Empty boxes correspond to links for which $\Delta =0$. \relax }}{40}{figure.caption.16}\protected@file@percent }
\newlabel{fig:Theta4Links}{{6.2}{40}{$\Theta $ for all the prime links with up to 9 crossings, up to reflections and with arbitrary choices of strand orientations. Empty boxes correspond to links for which $\Delta =0$. \relax }{figure.caption.16}{}}
\@writefile{brf}{\backcite{OzsvathSzabo:IntroductionToHF, Manolescu:KnotFloerHomology, Juhasz:Survey}{{40}{6}{theorem.41}}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Acknowledgement}}{40}{section.7}\protected@file@percent }
\bibcite{Alexander:TopologicalInvariants}{Al}
\bibcite{Bai:Alexander}{Ba}
\bibcite{Bar-Natan:Categorification}{BN1}
\bibcite{DPG}{BN2}
\bibcite{AKT}{BN3}
\bibcite{Oaxaca-2210}{BN4}
\bibcite{Geneva-231201}{BN5}
\bibcite{IType}{BN6}
\bibcite{Toronto-241030}{BN7}
\bibcite{BDV:OU}{BDV}
\bibcite{Bar-NatanGaroufalidisRozanskyThurston:Aarhus}{BGRT}
\bibcite{APAI}{BV1}
\bibcite{PG}{BV2}
\bibcite{Self}{BV3}
\bibcite{BecerraVanHelden:RotationalReidemeister}{BVH}
\bibcite{BurdeZieschang:Knots}{BZ}
\bibcite{Conway:SignaturesSurvey}{Co}
\bibcite{CrowellFox:KnotTheory}{CF}
\bibcite{CullerDunfieldGoernerWeeks:SnapPy}{CDGW}
\bibcite{Drinfeld:QuantumGroups}{Dr}
\bibcite{DHOEBL:Random}{DHOEBL}
\bibcite{FoxMilnor:CobordismOfKnots}{FM}
\bibcite{Garoufalidis:Whitehead}{Gar}
\bibcite{GaroufalidisKashaev:Multivariable}{GK}
\bibcite{GaroufalidisLi:Patterns}{GL}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{41}{section*.17}\protected@file@percent }
\bibcite{GaroufalidisRozansky:LoopExpansion}{GR}
\bibcite{Gassner:OnBraidGroups}{Gas}
\bibcite{GompfScharlemannThompson:Counterexample}{GST}
\bibcite{Harper:Non-Abelian}{Ha}
\bibcite{Juhasz:Survey}{Ju}
\bibcite{Kauffman:OnKnots}{Kau1}
\bibcite{Kauffman:VirtualKnotTheory}{Kau2}
\bibcite{Kauffman:RotationalVirtualKnots}{Kau3}
\bibcite{Kawauchi:Srvery}{Kaw}
\bibcite{Khovanov:Categorification}{Kh}
\bibcite{Kricker:Lines}{Kr}
\bibcite{Lalani:ThetaSurvey}{Lal}
\bibcite{Lawrence:UniversalUsingQG}{Law}
\bibcite{Levine:KnotCobordism}{Le}
\bibcite{Lickorish:KnotTheory}{Li}
\bibcite{LivingstonMoore:KnotInfo}{LM}
\bibcite{LopezNeumannVanDerVeen:FibredLinks}{LV}
\bibcite{Manolescu:KnotFloerHomology}{Ma}
\bibcite{Ohtsuki:QuantumInvariants}{Oh1}
\bibcite{Ohtsuki:TwoLoop}{Oh2}
\bibcite{Ohtsuki:Cabling}{Oh3}
\bibcite{OzsvathSzabo:IntroductionToHF}{OS}
\bibcite{Overbay:Thesis}{Ov}
\bibcite{Polyak:FeynmanDiagrams}{Po1}
\bibcite{Polyak:RMoves}{Po2}
\bibcite{Rolfsen:KnotsAndLinks}{Rol}
\bibcite{Rozansky:Contribution}{Roz1}
\bibcite{Rozansky:Burau}{Roz2}
\bibcite{Rozansky:U1RCC}{Roz3}
\bibcite{Schaveling:Thesis}{Sch}
\bibcite{Tristram:CobordismInvariants}{Tr}
\bibcite{Wolfram:Mathematica}{Wo}
\citation{Garoufalidis:Whitehead}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{15.01373pt}
\newlabel{tocindent1}{20.88867pt}
\newlabel{tocindent2}{34.5pt}
\newlabel{tocindent3}{40.80037pt}
\@writefile{brf}{\backcite{Garoufalidis:Whitehead}{{43}{7}{section*.17}}}
\gdef \@abspage@last{43}
