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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 L11a243's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X22,10,7,9 X16,11,17,12 X10,15,11,16 X2738 X20,17,21,18 X6,14,1,13 X4,20,5,19 X18,6,19,5 X14,21,15,22
Gauss code {1, -6, 2, -9, 10, -8}, {6, -1, 3, -5, 4, -2, 8, -11, 5, -4, 7, -10, 9, -7, 11, -3}
A Braid Representative
A Morse Link Presentation L11a243 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(v^2-v+1\right) (u v-u-2 v+1) (u v-2 u-v+1)}{u v^2} (db)
Jones polynomial q^{9/2}-\frac{9}{q^{9/2}}-4 q^{7/2}+\frac{16}{q^{7/2}}+9 q^{5/2}-\frac{21}{q^{5/2}}-16 q^{3/2}+\frac{24}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+20 \sqrt{q}-\frac{25}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z-2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -3 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-4 z^3 a^{-1} +5 a z-3 z a^{-1} +3 a z^{-1} -2 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+2 a^6 z^2+8 a^5 z^7-11 a^5 z^5+6 a^5 z^3-a^5 z+10 a^4 z^8-13 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} +z^2 a^{-4} -a^4+7 a^3 z^9+a^3 z^7+4 z^7 a^{-3} -16 a^3 z^5-9 z^5 a^{-3} +11 a^3 z^3+7 z^3 a^{-3} -2 a^3 z-2 z a^{-3} +a^3 z^{-1} +2 a^2 z^{10}+18 a^2 z^8+7 z^8 a^{-2} -41 a^2 z^6-14 z^6 a^{-2} +26 a^2 z^4+7 z^4 a^{-2} -3 a^2 z^2-3 a^2+13 a z^9+6 z^9 a^{-1} -14 a z^7-3 z^7 a^{-1} -10 a z^5-15 z^5 a^{-1} +12 a z^3+15 z^3 a^{-1} -5 a z-6 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +2 z^{10}+15 z^8-39 z^6+24 z^4-2 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 \
10           1-1
8          3 3
6         61 -5
4        103  7
2       117   -4
0      149    5
-2     1112     1
-4    1013      -3
-6   611       5
-8  310        -7
-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}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
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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