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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 L10a68's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,18,7,17 X18,12,19,11 X12,20,13,19
Gauss code {1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 9, -10, 5, -6, 7, -5, 8, -9, 10, -8}
A Braid Representative
A Morse Link Presentation L10a68 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-5 u v^2+3 u v-u-v^3+3 v^2-3 v+1}{u v^2} (db)
Jones polynomial q^{9/2}-\frac{3}{q^{9/2}}-2 q^{7/2}+\frac{5}{q^{7/2}}+4 q^{5/2}-\frac{8}{q^{5/2}}-7 q^{3/2}+\frac{9}{q^{3/2}}+\frac{1}{q^{11/2}}+8 \sqrt{q}-\frac{10}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -2 a^3 z+3 z a^{-3} + a^{-3} z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +9 a z^3-8 z^3 a^{-1} +7 a z-9 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -3 a^2 z^8-2 z^8 a^{-2} -5 z^8-4 a^3 z^7-4 a z^7-2 z^7 a^{-1} -2 z^7 a^{-3} -4 a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} -z^6 a^{-4} +12 z^6-3 a^5 z^5+4 a^3 z^5+14 a z^5+14 z^5 a^{-1} +7 z^5 a^{-3} -a^6 z^4+4 a^4 z^4-a^2 z^4+3 z^4 a^{-2} +4 z^4 a^{-4} -7 z^4+4 a^5 z^3-a^3 z^3-17 a z^3-19 z^3 a^{-1} -7 z^3 a^{-3} +a^6 z^2-a^2 z^2-7 z^2 a^{-2} -4 z^2 a^{-4} -3 z^2-a^5 z+9 a z+12 z a^{-1} +4 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} 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 \
10          1-1
8         1 1
6        31 -2
4       41  3
2      43   -1
0     64    2
-2    45     1
-4   45      -1
-6  25       3
-8 13        -2
-10 2         2
-121          -1
Integral Khovanov Homology

(db, data source)

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