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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 L10a71's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X14,6,15,5 X16,8,17,7 X10,4,11,3 X20,12,7,11 X12,15,13,16 X6,14,1,13 X4,19,5,20 X2,18,3,17
Gauss code {1, -10, 5, -9, 3, -8}, {4, -1, 2, -5, 6, -7, 8, -3, 7, -4, 10, -2, 9, -6}
A Braid Representative
A Morse Link Presentation L10a71 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-4 u^2 v^3+5 u^2 v^2-3 u^2 v+u^2-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u+v^4-3 v^3+5 v^2-4 v+1}{u v^2} (db)
Jones polynomial -q^{13/2}+4 q^{11/2}-9 q^{9/2}+14 q^{7/2}-18 q^{5/2}+19 q^{3/2}-19 \sqrt{q}+\frac{14}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} +a z^3-2 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -a z+2 z a^{-1} +z a^{-3} -z a^{-5} + a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -3 z^9 a^{-1} -3 z^9 a^{-3} -17 z^8 a^{-2} -8 z^8 a^{-4} -9 z^8-10 a z^7-17 z^7 a^{-1} -15 z^7 a^{-3} -8 z^7 a^{-5} -5 a^2 z^6+25 z^6 a^{-2} +8 z^6 a^{-4} -4 z^6 a^{-6} +8 z^6-a^3 z^5+16 a z^5+42 z^5 a^{-1} +39 z^5 a^{-3} +13 z^5 a^{-5} -z^5 a^{-7} +5 a^2 z^4-4 z^4 a^{-2} +z^4 a^{-4} +5 z^4 a^{-6} +5 z^4-6 a z^3-22 z^3 a^{-1} -26 z^3 a^{-3} -9 z^3 a^{-5} +z^3 a^{-7} -2 z^2 a^{-2} -2 z^2 a^{-4} -2 z^2 a^{-6} -2 z^2-a z+4 z a^{-3} +3 z a^{-5} - a^{-2} + 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 \
14          11
12         3 -3
10        61 5
8       83  -5
6      106   4
4     109    -1
2    99     0
0   611      5
-2  48       -4
-4 16        5
-6 4         -4
-81          1
Integral Khovanov Homology

(db, data source)

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