L10n36

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L10n35

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L10n37

Contents

L10n36.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n36's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X16,13,17,14 X14,9,15,10 X10,15,11,16 X17,5,18,20 X7,19,8,18 X19,9,20,8 X2536 X11,1,12,4
Gauss code {1, -9, -2, 10}, {9, -1, -7, 8, 4, -5, -10, 2, 3, -4, 5, -3, -6, 7, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n36 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial q^{9/2}+\frac{1}{q^{9/2}}-q^{7/2}-\frac{1}{q^{7/2}}+q^{5/2}+\frac{1}{q^{5/2}}-q^{3/2}-\frac{1}{q^{3/2}}-\sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^3 z^3+z^3 a^{-3} -3 a^3 z+3 z a^{-3} -2 a^3 z^{-1} +2 a^{-3} z^{-1} +a z^5-z^5 a^{-1} +6 a z^3-6 z^3 a^{-1} +11 a z-11 z a^{-1} +7 a z^{-1} -7 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^7-a z^7-z^7 a^{-1} -z^7 a^{-3} -a^4 z^6-2 a^2 z^6-2 z^6 a^{-2} -z^6 a^{-4} -2 z^6+5 a^3 z^5+7 a z^5+7 z^5 a^{-1} +5 z^5 a^{-3} +5 a^4 z^4+12 a^2 z^4+12 z^4 a^{-2} +5 z^4 a^{-4} +14 z^4-6 a^3 z^3-14 a z^3-14 z^3 a^{-1} -6 z^3 a^{-3} -6 a^4 z^2-18 a^2 z^2-18 z^2 a^{-2} -6 z^2 a^{-4} -24 z^2+4 a^3 z+14 a z+14 z a^{-1} +4 z a^{-3} +2 a^4+8 a^2+8 a^{-2} +2 a^{-4} +13-2 a^3 z^{-1} -7 a z^{-1} -7 a^{-1} z^{-1} -2 a^{-3} 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 \
-5-4-3-2-1012345χ
10          1-1
8           0
6        11 0
4      11   0
2     1 1   2
0    141    2
-2   1 1     2
-4   11      0
-6 11        0
-8           0
-101          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}
r=1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}
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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L10n35.gif

L10n35

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L10n37