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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 L11n69's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,16,8,17 X22,18,5,17 X18,12,19,11 X20,14,21,13 X12,20,13,19 X14,22,15,21 X15,8,16,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4}
A Braid Representative
A Morse Link Presentation L11n69 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^5-2 u v^4+3 u v^3-3 u v^2+3 u v-u-v^5+3 v^4-3 v^3+3 v^2-2 v+1}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -3 q^{9/2}+5 q^{7/2}-\frac{2}{q^{7/2}}-7 q^{5/2}+\frac{3}{q^{5/2}}+9 q^{3/2}-\frac{6}{q^{3/2}}+q^{11/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} +3 z^3 a^{-3} +a^3 z+2 z a^{-3} +2 a^3 z^{-1} -z^7 a^{-1} +a z^5-5 z^5 a^{-1} +2 a z^3-8 z^3 a^{-1} -2 a z-3 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} -z^2 a^{-6} +3 z^5 a^{-5} -4 z^3 a^{-5} +4 z^6 a^{-4} -5 z^4 a^{-4} +z^2 a^{-4} +4 z^7 a^{-3} +3 a^3 z^5-6 z^5 a^{-3} -10 a^3 z^3+5 z^3 a^{-3} +8 a^3 z-2 z a^{-3} -2 a^3 z^{-1} +a^2 z^8+3 z^8 a^{-2} -2 a^2 z^6-5 z^6 a^{-2} +2 a^2 z^4+5 z^4 a^{-2} -6 a^2 z^2-z^2 a^{-2} +3 a^2+ a^{-2} +a z^9+z^9 a^{-1} -2 a z^7+2 z^7 a^{-1} +4 a z^5-8 z^5 a^{-1} -10 a z^3+9 z^3 a^{-1} +9 a z-z a^{-1} -3 a z^{-1} - a^{-1} z^{-1} +4 z^8-11 z^6+13 z^4-9 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 \
12         1-1
10        2 2
8       31 -2
6      42  2
4     53   -2
2    44    0
0   46     2
-2  23      -1
-4 14       3
-612        -1
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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