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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 L10a55's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,8,15,7 X16,13,17,14 X6,18,1,17 X18,12,19,11 X12,6,13,5 X20,16,7,15 X4,19,5,20
Gauss code {1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -5, 6, -7, 10, -9}
A Braid Representative
A Morse Link Presentation L10a55 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-4 u^2 v^3+4 u^2 v^2-u^2 v-u v^4+5 u v^3-9 u v^2+5 u v-u-v^3+4 v^2-4 v+1}{u v^2} (db)
Jones polynomial q^{9/2}-3 q^{7/2}+6 q^{5/2}-10 q^{3/2}+12 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-2 a^3 z^3+z^3 a^{-3} +2 z a^{-3} +a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +a z-4 z a^{-1} -a z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -6 a^2 z^8-4 z^8 a^{-2} -10 z^8-8 a^3 z^7-8 a z^7-3 z^7 a^{-1} -3 z^7 a^{-3} -7 a^4 z^6+5 a^2 z^6+10 z^6 a^{-2} -z^6 a^{-4} +23 z^6-4 a^5 z^5+9 a^3 z^5+23 a z^5+19 z^5 a^{-1} +9 z^5 a^{-3} -a^6 z^4+7 a^4 z^4+2 a^2 z^4-6 z^4 a^{-2} +3 z^4 a^{-4} -15 z^4+3 a^5 z^3-a^3 z^3-14 a z^3-18 z^3 a^{-1} -8 z^3 a^{-3} -a^4 z^2-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +3 z^2-2 a^3 z+5 z a^{-1} +3 z a^{-3} -a^2+a^3 z^{-1} +a 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        41 -3
4       62  4
2      64   -2
0     86    2
-2    67     1
-4   57      -2
-6  37       4
-8 14        -3
-10 3         3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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