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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 L10a90's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X6,20,7,19 X16,7,17,8 X18,15,19,16 X8,17,1,18
Gauss code {1, -5, 2, -6, 4, -7, 8, -10}, {5, -1, 3, -2, 6, -4, 9, -8, 10, -9, 7, -3}
A Braid Representative
A Morse Link Presentation L10a90 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1)^2 (t(2)-1)^2 (t(1) t(2)+1)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -3 q^{9/2}+\frac{1}{q^{9/2}}+5 q^{7/2}-\frac{3}{q^{7/2}}-8 q^{5/2}+\frac{5}{q^{5/2}}+10 q^{3/2}-\frac{8}{q^{3/2}}+q^{11/2}-11 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -2 a^3 z+2 z a^{-3} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +7 a z^3-9 z^3 a^{-1} +6 a z-6 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-5 a z^7-6 z^7 a^{-1} -4 z^7 a^{-3} -a^4 z^6+7 a^2 z^6+z^6 a^{-2} -4 z^6 a^{-4} +13 z^6+10 a^3 z^5+22 a z^5+18 z^5 a^{-1} +3 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4-a^2 z^4+3 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} -6 z^4-9 a^3 z^3-22 a z^3-17 z^3 a^{-1} +4 z^3 a^{-5} -2 a^4 z^2-2 a^2 z^2-3 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -z^2+3 a^3 z+8 a z+6 z a^{-1} -z a^{-5} +1-a z^{-1} - a^{-1} 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 \
12          1-1
10         2 2
8        31 -2
6       52  3
4      53   -2
2     65    1
0    57     2
-2   34      -1
-4  25       3
-6 13        -2
-8 2         2
-101          -1
Integral Khovanov Homology

(db, data source)

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