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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a353 at Knotilus!

Link Presentations

[edit Notes on L11a353's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X16,6,17,5 X22,8,11,7 X20,15,21,16 X14,21,15,22 X6,14,7,13 X4,20,5,19 X18,9,19,10 X2,11,3,12 X10,17,1,18
Gauss code {1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}
A Braid Representative
A Morse Link Presentation L11a353 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(u^2 v^2-3 u^2 v+2 u^2-u v^2+5 u v-u+2 v^2-3 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+21 q^{3/2}-25 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{21}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+7 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-3 a^3 z+6 a z-7 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-6} +a^5 z^7-3 a^5 z^5+4 z^5 a^{-5} +3 a^5 z^3-a^5 z+4 a^4 z^8-13 a^4 z^6+10 z^6 a^{-4} +14 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+3 z^2 a^{-4} +6 a^3 z^9-16 a^3 z^7+17 z^7 a^{-3} +9 a^3 z^5-25 z^5 a^{-3} +4 a^3 z^3+16 z^3 a^{-3} -2 a^3 z-5 z a^{-3} +3 a^2 z^{10}+7 a^2 z^8+18 z^8 a^{-2} -42 a^2 z^6-29 z^6 a^{-2} +43 a^2 z^4+12 z^4 a^{-2} -11 a^2 z^2-z^2 a^{-2} +17 a z^9+11 z^9 a^{-1} -38 a z^7-4 z^7 a^{-1} +9 a z^5-32 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -6 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^{10}+21 z^8-68 z^6+49 z^4-10 z^2-1 (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         71 -6
6        103  7
4       117   -4
2      1410    4
0     1213     1
-2    912      -3
-4   612       6
-6  39        -6
-8 16         5
-10 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
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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