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

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Link Presentations

[edit Notes on L11a326's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,5,21,6 X16,9,17,10 X18,12,19,11 X22,15,9,16 X6,21,7,22 X14,8,15,7 X4,14,5,13 X8,18,1,17 X2,19,3,20
Gauss code {1, -11, 2, -9, 3, -7, 8, -10}, {4, -1, 5, -2, 9, -8, 6, -4, 10, -5, 11, -3, 7, -6}
A Braid Representative
A Morse Link Presentation L11a326 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-v+1\right) \left(u^2 v^2-u v^2+2 u v-u+1\right)}{u^{3/2} v^{5/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+9 q^{5/2}-15 q^{3/2}+19 \sqrt{q}-\frac{24}{\sqrt{q}}+\frac{23}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+4 a^3 z^5+5 a^3 z^3+2 a^3 z-a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -12 a z^3+5 z^3 a^{-1} -4 a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+a^6 z^2+8 a^5 z^7-12 a^5 z^5+5 a^5 z^3-a^5 z+10 a^4 z^8-16 a^4 z^6+z^6 a^{-4} +10 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2+z^2 a^{-4} +8 a^3 z^9-9 a^3 z^7+4 z^7 a^{-3} +2 a^3 z^5-9 z^5 a^{-3} +a^3 z^3+5 z^3 a^{-3} -z a^{-3} +3 a^2 z^{10}+10 a^2 z^8+7 z^8 a^{-2} -31 a^2 z^6-15 z^6 a^{-2} +28 a^2 z^4+7 z^4 a^{-2} -8 a^2 z^2-z^2 a^{-2} +15 a z^9+7 z^9 a^{-1} -33 a z^7-12 z^7 a^{-1} +27 a z^5+3 z^5 a^{-1} -10 a z^3+2 a z+a z^{-1} + a^{-1} z^{-1} +3 z^{10}+7 z^8-27 z^6+22 z^4-6 z^2-1 (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 \
10           1-1
8          3 3
6         61 -5
4        93  6
2       106   -4
0      149    5
-2     1112     1
-4    912      -3
-6   611       5
-8  39        -6
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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