# L11a353

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a353 at Knotilus!

### 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 \
-6-5-4-3-2-1012345χ
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 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.