# L11a323

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a323 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^2-2 u^2 v+u^2-3 u v^2+5 u v-3 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{9/2}-4 q^{7/2}+9 q^{5/2}-16 q^{3/2}+21 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{24}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z-2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -3 a^3 z+z a^{-3} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-4 z^3 a^{-1} +4 a z-3 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-7 a^3 z^9-13 a z^9-6 z^9 a^{-1} -10 a^4 z^8-19 a^2 z^8-7 z^8 a^{-2} -16 z^8-8 a^5 z^7-3 a^3 z^7+11 a z^7+2 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+12 a^4 z^6+43 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +42 z^6-a^7 z^5+11 a^5 z^5+23 a^3 z^5+20 a z^5+18 z^5 a^{-1} +9 z^5 a^{-3} +5 a^6 z^4-4 a^4 z^4-27 a^2 z^4-8 z^4 a^{-2} +2 z^4 a^{-4} -28 z^4+a^7 z^3-7 a^5 z^3-20 a^3 z^3-23 a z^3-18 z^3 a^{-1} -7 z^3 a^{-3} -2 a^6 z^2-a^4 z^2+5 a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +6 z^2+2 a^5 z+6 a^3 z+8 a z+6 z a^{-1} +2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         61 -5
4        103  7
2       116   -5
0      1410    4
-2     1213     1
-4    1012      -2
-6   612       6
-8  310        -7
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=-1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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