{\bf\red Implementation} (sources: \url{http://drorbn.net/bu23/ap}). I like it most when the implementation matches the math perfectly. We failed here.

\noindent\nbpdfInput{Snips/Signatures/1.pdf}
\noindent\nbpdfPrint{Snips/Signatures/2.pdf}
\par{\bf\red Utilities.} The step function, algebraic numbers, canonical forms.

\noindent\nbpdfInput{Snips/Signatures/3.pdf}
\noindent\nbpdfInput{Snips/Signatures/4.pdf}
\noindent\nbpdfInput{Snips/Signatures/5.pdf}
\noindent\nbpdfInput{Snips/Signatures/6.pdf}
\noindent\nbpdfInput{Snips/Signatures/7.pdf}
\noindent\nbpdfInput{Snips/Signatures/8.pdf}
\noindent\nbpdfInput{Snips/Signatures/9.pdf}
\noindent\nbpdfInput{Snips/Signatures/10.pdf}
\noindent\nbpdfInput{Snips/Signatures/11.pdf}
\needspace{32mm}
\par{\bf\red Pretty-Printing.}

\noindent\nbpdfInput{Snips/Signatures/12.pdf}
\par{\bf\red The Face-Centric Core.}

\noindent\nbpdfInput{Snips/Signatures/13.pdf}
\par GT for Gap Touch: \hfill \import{../ICERM-2305}{figs/GT.pdf_t}

\noindent\nbpdfInput{Snips/Signatures/14.pdf}
\par\vskip 1mm\par\Cordon

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\par\needspace{20mm}
{\bf\red Strand Operations.} c for contract, mc for magnetic contract:

\noindent\nbpdfInput{Snips/Signatures/16.pdf}
\noindent\nbpdfInput{Snips/Signatures/17.pdf}
\noindent\nbpdfInput{Snips/Signatures/18.pdf}
\par{\bf\red The Crossings} (and empty strands).

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\noindent\nbpdfInput{Snips/Signatures/20.pdf}
\par{\bf\red Evaluation on Tangles and Knots.}

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\par\needspace{20mm}
\parpic[r]{\import{../ICERM-2305}{figs/R3.pdf_t}}
{\bf\red Reidemeister 3.}

\noindent\nbpdfInput{Snips/Signatures/22.pdf}
\noindent\nbpdfOutput{Snips/Signatures/23.pdf}
\needspace{15mm}

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\par
\needspace{20mm}
\parpic[r]{\import{../ICERM-2305}{figs/R2.pdf_t}}
{\bf\red Reidemeister 2.}

\noindent\nbpdfInput{Snips/Signatures/26.pdf}
\noindent\nbpdfOutput{Snips/Signatures/27.pdf}
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\par
\parpic[r]{\import{../ICERM-2305}{figs/R1.pdf_t}}
{\bf\red Reidemeister 1.}

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\noindent\nbpdfOutput{Snips/Signatures/31.pdf}
\par
\parpic[r]{\includegraphics[width=1in]{../ICERM-2305/8_5.png}}
{\bf\red A Knot.}

\noindent\nbpdfInput{Snips/Signatures/32.pdf}
\noindent\nbpdfMessage{Snips/Signatures/33.pdf}
\noindent\nbpdfOutput{Snips/Signatures/34.pdf}
\par\picskip{0}{
  \def\nbpdfInput#1{\vskip 1mm\par\noindent\includegraphics{#1}}
  \def\nbpdfOutput#1{\hfill\includegraphics[width=0.4\linewidth,valign=t]{#1}}

\noindent\nbpdfInput{Snips/Signatures/35.pdf}
\noindent\nbpdfOutput{Snips/Signatures/36.pdf}
}

\needspace{30mm}
\par\parpic[r]{\includegraphics[width=1.88in]{../ICERM-2305/figs/CKT.pdf}}
{\bf\red The  Conway-Kinoshita-Terasaka Tangles.}

\noindent\nbpdfInput{Snips/Signatures/37.pdf}
\par\needspace{10mm}

\noindent\nbpdfInput{Snips/Signatures/38.pdf}
\noindent\nbpdfOutput{Snips/Signatures/39.pdf}
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\noindent\nbpdfOutput{Snips/Signatures/41.pdf}
\parpic[r]{\includegraphics[width=1.88in]{../ICERM-2305/figs/B1B2.pdf}}
{\bf\red Examples with non-trivial codimension.}

\noindent\nbpdfInput{Snips/Signatures/42.pdf}
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\noindent\nbpdfOutput{Snips/Signatures/44.pdf}
\par\needspace{10mm}

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\noindent\nbpdfOutput{Snips/Signatures/46.pdf}