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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 L11a59's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X14,8,15,7 X20,16,21,15 X18,9,19,10 X8,19,9,20 X22,18,5,17 X16,22,17,21 X10,14,11,13 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8, 7, -5, 6, -4, 8, -7}
A Braid Representative
A Morse Link Presentation L11a59 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^5-2 t(2)^5-4 t(1) t(2)^4+6 t(2)^4+7 t(1) t(2)^3-7 t(2)^3-7 t(1) t(2)^2+7 t(2)^2+6 t(1) t(2)-4 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+8 q^{7/2}-13 q^{5/2}+16 q^{3/2}-17 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+2 a^5 z^{-1} +z^5 a^{-3} -3 a^3 z^3+2 z^3 a^{-3} -8 a^3 z+z a^{-3} -4 a^3 z^{-1} -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +10 a z^3-7 z^3 a^{-1} +10 a z-6 z a^{-1} +3 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-2 a^3 z^9-7 a z^9-5 z^9 a^{-1} -2 a^4 z^8-5 a^2 z^8-10 z^8 a^{-2} -13 z^8-a^5 z^7+2 a^3 z^7+12 a z^7-2 z^7 a^{-1} -11 z^7 a^{-3} +7 a^4 z^6+22 a^2 z^6+15 z^6 a^{-2} -8 z^6 a^{-4} +38 z^6+5 a^5 z^5+12 a^3 z^5+9 a z^5+21 z^5 a^{-1} +15 z^5 a^{-3} -4 z^5 a^{-5} -6 a^4 z^4-19 a^2 z^4-7 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -28 z^4-9 a^5 z^3-23 a^3 z^3-22 a z^3-16 z^3 a^{-1} -6 z^3 a^{-3} +2 z^3 a^{-5} -a^4 z^2+a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +4 z^2+7 a^5 z+15 a^3 z+12 a z+4 z a^{-1} +2 a^4+3 a^2+ a^{-2} +3-2 a^5 z^{-1} -4 a^3 z^{-1} -3 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          3 3
8         51 -4
6        83  5
4       85   -3
2      98    1
0     99     0
-2    58      -3
-4   49       5
-6  25        -3
-8 15         4
-10 1          -1
-121           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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See/edit the Link Page master template (intermediate).

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