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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 L11a303's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X18,8,19,7 X20,14,21,13 X14,22,15,21 X22,18,9,17 X8,20,1,19 X6,9,7,10 X4,15,5,16
Gauss code {1, -2, 3, -11, 4, -10, 5, -9}, {10, -1, 2, -3, 6, -7, 11, -4, 8, -5, 9, -6, 7, -8}
A Braid Representative
A Morse Link Presentation L11a303 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v^5-2 u^3 v^4+2 u^3 v^3-u^3 v^2-u^2 v^5+5 u^2 v^4-8 u^2 v^3+7 u^2 v^2-2 u^2 v-2 u v^4+7 u v^3-8 u v^2+5 u v-u-v^3+2 v^2-2 v+1}{u^{3/2} v^{5/2}} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+7 q^{5/2}-12 q^{3/2}+16 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{18}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+5 a^3 z^5+9 a^3 z^3+6 a^3 z+2 a^3 z^{-1} -a z^9-7 a z^7+z^7 a^{-1} -19 a z^5+5 z^5 a^{-1} -24 a z^3+9 z^3 a^{-1} -14 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-10 a z^9-5 z^9 a^{-1} -6 a^4 z^8-5 a^2 z^8-5 z^8 a^{-2} -4 z^8-5 a^5 z^7+6 a^3 z^7+26 a z^7+12 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+8 a^4 z^6+14 a^2 z^6+13 z^6 a^{-2} -z^6 a^{-4} +17 z^6-a^7 z^5+7 a^5 z^5-7 a^3 z^5-38 a z^5-15 z^5 a^{-1} +8 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-15 a^2 z^4-11 z^4 a^{-2} +3 z^4 a^{-4} -20 z^4+2 a^7 z^3-3 a^5 z^3+9 a^3 z^3+35 a z^3+16 z^3 a^{-1} -5 z^3 a^{-3} -2 a^6 z^2+9 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-4} +14 z^2-a^7 z+a^5 z-7 a^3 z-16 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 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 \
10           1-1
8          2 2
6         51 -4
4        72  5
2       95   -4
0      107    3
-2     910     1
-4    89      -1
-6   49       5
-8  38        -5
-10 15         4
-12 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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

Read me first: Modifying Knot Pages

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

See/edit the Link_Splice_Base (expert).

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