L11n63

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L11n62

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L11n64

Contents

L11n63.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n63's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X20,16,21,15 X18,12,19,11 X12,20,13,19 X22,18,5,17 X16,22,17,21 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -6, -9, 3, 4, -8, 7, -5, 6, -4, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n63 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^5-t(2)^5-2 t(1) t(2)^4+4 t(2)^4+4 t(1) t(2)^3-5 t(2)^3-5 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-2 t(2)-t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{11/2}-3 q^{9/2}+7 q^{7/2}-10 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} +3 z^3 a^{-3} +a^3 z+4 z a^{-3} +a^3 z^{-1} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 a z^3-10 z^3 a^{-1} +a z-8 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -2 z^3 a^{-5} +6 z^6 a^{-4} -8 z^4 a^{-4} +6 z^2 a^{-4} -2 a^{-4} +7 z^7 a^{-3} +3 a^3 z^5-11 z^5 a^{-3} -8 a^3 z^3+9 z^3 a^{-3} +6 a^3 z-4 z a^{-3} -a^3 z^{-1} + a^{-3} z^{-1} +a^2 z^8+4 z^8 a^{-2} +z^6 a^{-2} -2 a^2 z^4-14 z^4 a^{-2} -a^2 z^2+16 z^2 a^{-2} +a^2-5 a^{-2} +a z^9+z^9 a^{-1} +a z^7+8 z^7 a^{-1} -2 a z^5-19 z^5 a^{-1} -4 a z^3+15 z^3 a^{-1} +2 a z-8 z a^{-1} +2 a^{-1} z^{-1} +5 z^8-5 z^6-7 z^4+8 z^2-3 (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
12         1-1
10        2 2
8       51 -4
6      52  3
4     65   -1
2    65    1
0   57     2
-2  35      -2
-4 15       4
-613        -2
-82         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}^{2} {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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