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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 L11a256's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X14,5,15,6 X18,10,19,9 X22,18,9,17 X8,21,1,22 X20,15,21,16 X16,8,17,7 X6,20,7,19 X4,11,5,12 X2,13,3,14
Gauss code {1, -11, 2, -10, 3, -9, 8, -6}, {4, -1, 10, -2, 11, -3, 7, -8, 5, -4, 9, -7, 6, -5}
A Braid Representative
A Morse Link Presentation L11a256 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 v^4-2 u^3 v^3+2 u^3 v^2-u^3 v+u^2 v^5-4 u^2 v^4+7 u^2 v^3-8 u^2 v^2+4 u^2 v-u^2-u v^5+4 u v^4-8 u v^3+7 u v^2-4 u v+u-v^4+2 v^3-2 v^2+v}{u^{3/2} v^{5/2}} (db)
Jones polynomial -\frac{12}{q^{9/2}}-q^{7/2}+\frac{16}{q^{7/2}}+4 q^{5/2}-\frac{20}{q^{5/2}}-9 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+14 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+4 a^3 z^5+3 a z^5-z^5 a^{-1} -3 a^5 z^3+6 a^3 z^3+2 a z^3-2 z^3 a^{-1} -2 a^5 z+2 a^3 z-a z-z a^{-1} +a^5 z^{-1} -a^3 z^{-1} (db)
Kauffman polynomial -2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-10 a^3 z^9-6 a z^9-4 a^6 z^8-3 a^4 z^8-8 a^2 z^8-9 z^8-3 a^7 z^7+6 a^5 z^7+21 a^3 z^7+4 a z^7-8 z^7 a^{-1} -a^8 z^6+8 a^6 z^6+9 a^4 z^6+18 a^2 z^6-4 z^6 a^{-2} +14 z^6+9 a^7 z^5-a^5 z^5-22 a^3 z^5+2 a z^5+13 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-2 a^6 z^4-4 a^4 z^4-11 a^2 z^4+5 z^4 a^{-2} -7 z^4-8 a^7 z^3-3 a^5 z^3+13 a^3 z^3+a z^3-6 z^3 a^{-1} +z^3 a^{-3} -2 a^8 z^2-2 a^6 z^2-a^4 z^2+2 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z+4 a^5 z-a^3 z-2 a z+z a^{-1} +a^4-a^5 z^{-1} -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 \
8           11
6          3 -3
4         61 5
2        83  -5
0       106   4
-2      119    -2
-4     99     0
-6    711      4
-8   59       -4
-10  28        6
-12 14         -3
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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