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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 L11a476's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X14,8,15,7 X22,16,17,15 X16,18,5,17 X18,9,19,10 X20,13,21,14 X12,19,13,20 X8,21,9,22 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {5, -6, 8, -7, 9, -4}, {10, -1, 3, -9, 6, -2, 11, -8, 7, -3, 4, -5}
A Braid Representative
A Morse Link Presentation L11a476 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1)^2 (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial -q^5+4 q^4-10 q^3+16 q^2-20 q+25-22 q^{-1} +20 q^{-2} -14 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +11 z^4+3 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +14 z^2+3 a^4-10 a^2-3 a^{-2} +10+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-z^3 a^{-5} -a^5 z+5 a^4 z^8-9 a^4 z^6+4 z^6 a^{-4} +4 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-4 a^3 z^7+9 z^7 a^{-3} -7 a^3 z^5-14 z^5 a^{-3} +9 a^3 z^3+8 z^3 a^{-3} -7 a^3 z-3 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+12 z^8 a^{-2} -38 a^2 z^6-23 z^6 a^{-2} +43 a^2 z^4+21 z^4 a^{-2} -31 a^2 z^2-12 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+5 a^{-2} +13 a z^9+8 z^9 a^{-1} -20 a z^7-4 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +15 a z^3+20 z^3 a^{-1} -13 a z-10 z a^{-1} +2 a z^{-1} +2 z^{10}+19 z^8-55 z^6+61 z^4-38 z^2- z^{-2} +15 (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 \
11           1-1
9          3 3
7         71 -6
5        93  6
3       117   -4
1      149    5
-1     1013     3
-3    1012      -2
-5   612       6
-7  28        -6
-9 16         5
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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