L11n93

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L11n92.gif

L11n92

L11n94.gif

L11n94

Contents

L11n93.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n93's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X16,8,17,7 X11,20,12,21 X17,22,18,5 X21,18,22,19 X19,10,20,11 X14,10,15,9 X8,16,9,15 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 8, 7, -4, -2, 11, -8, 9, -3, -5, 6, -7, 4, -6, 5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n93 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+5 q^{5/2}-\frac{7}{q^{5/2}}-8 q^{3/2}+\frac{8}{q^{3/2}}+9 \sqrt{q}-\frac{10}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^3+z^3 a^{-3} +z a^{-3} -a z^5-z^5 a^{-1} -a z^3-2 z^3 a^{-1} +a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -2 a^2 z^8-3 z^8 a^{-2} -5 z^8-a^3 z^7-2 a z^7-4 z^7 a^{-1} -3 z^7 a^{-3} +2 a^2 z^6+7 z^6 a^{-2} -z^6 a^{-4} +10 z^6-3 a^3 z^5+4 a z^5+17 z^5 a^{-1} +10 z^5 a^{-3} -4 a^4 z^4-4 a^2 z^4-z^4 a^{-2} +3 z^4 a^{-4} -4 z^4-a^5 z^3+2 a^3 z^3-a z^3-13 z^3 a^{-1} -9 z^3 a^{-3} +a^4 z^2+2 a^2 z^2-2 z^2 a^{-2} -2 z^2 a^{-4} +z^2+2 a z+4 z a^{-1} +2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-1012345χ
10         1-1
8        2 2
6       31 -2
4      52  3
2     43   -1
0    65    1
-2   46     2
-4  34      -1
-6 14       3
-8 3        -3
-101         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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See/edit the Link Page master template (intermediate).

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