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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 L11a282's Link Presentations]

Planar diagram presentation X10,1,11,2 X20,11,21,12 X8,9,1,10 X22,17,9,18 X12,4,13,3 X18,8,19,7 X14,6,15,5 X4,14,5,13 X6,16,7,15 X16,21,17,22 X2,20,3,19
Gauss code {1, -11, 5, -8, 7, -9, 6, -3}, {3, -1, 2, -5, 8, -7, 9, -10, 4, -6, 11, -2, 10, -4}
A Braid Representative
A Morse Link Presentation L11a282 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+4 t(1)^2 t(2)^4-t(1) t(2)^4+2 t(1)^3 t(2)^3-6 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+5 t(1)^2 t(2)^2-6 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-6 q^{9/2}+9 q^{7/2}-13 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^9 a^{-1} -a z^7+7 z^7 a^{-1} -z^7 a^{-3} -5 a z^5+18 z^5 a^{-1} -5 z^5 a^{-3} -8 a z^3+21 z^3 a^{-1} -8 z^3 a^{-3} -5 a z+12 z a^{-1} -5 z a^{-3} -a z^{-1} +3 a^{-1} z^{-1} -2 a^{-3} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10}-5 a z^9-9 z^9 a^{-1} -4 z^9 a^{-3} -5 a^2 z^8+z^8 a^{-2} -4 z^8 a^{-4} -3 a^3 z^7+18 a z^7+35 z^7 a^{-1} +10 z^7 a^{-3} -4 z^7 a^{-5} -a^4 z^6+17 a^2 z^6+7 z^6 a^{-2} +5 z^6 a^{-4} -3 z^6 a^{-6} +17 z^6+9 a^3 z^5-24 a z^5-57 z^5 a^{-1} -17 z^5 a^{-3} +6 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-16 a^2 z^4-18 z^4 a^{-2} +6 z^4 a^{-6} -31 z^4-4 a^3 z^3+17 a z^3+39 z^3 a^{-1} +16 z^3 a^{-3} +2 z^3 a^{-7} -a^4 z^2+7 a^2 z^2+11 z^2 a^{-2} -2 z^2 a^{-6} +17 z^2-6 a z-15 z a^{-1} -8 z a^{-3} -z a^{-7} -a^2-3 a^{-2} -3+a z^{-1} +3 a^{-1} z^{-1} +2 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          2 -2
10         41 3
8        63  -3
6       73   4
4      76    -1
2     87     1
0    58      3
-2   47       -3
-4  25        3
-6 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

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