# L11a329

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a329 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{\left(v^2-v+1\right) (u v+1) (u v-u+1) (u v-v+1)}{u^{3/2} v^{5/2}}$ (db) Jones polynomial $q^{9/2}-3 q^{7/2}+7 q^{5/2}-11 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+5 a^3 z^5+9 a^3 z^3+7 a^3 z+2 a^3 z^{-1} -a z^9-7 a z^7+z^7 a^{-1} -19 a z^5+5 z^5 a^{-1} -25 a z^3+9 z^3 a^{-1} -16 a z+7 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^{10}-2 z^{10}-5 a^3 z^9-10 a z^9-5 z^9 a^{-1} -6 a^4 z^8-4 a^2 z^8-5 z^8 a^{-2} -3 z^8-5 a^5 z^7+9 a^3 z^7+31 a z^7+14 z^7 a^{-1} -3 z^7 a^{-3} -3 a^6 z^6+11 a^4 z^6+19 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +20 z^6-a^7 z^5+8 a^5 z^5-10 a^3 z^5-45 a z^5-18 z^5 a^{-1} +8 z^5 a^{-3} +5 a^6 z^4-9 a^4 z^4-28 a^2 z^4-12 z^4 a^{-2} +3 z^4 a^{-4} -29 z^4+2 a^7 z^3-3 a^5 z^3+7 a^3 z^3+34 a z^3+18 z^3 a^{-1} -4 z^3 a^{-3} -a^6 z^2+4 a^4 z^2+14 a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} +18 z^2-a^7 z+2 a^5 z-6 a^3 z-17 a z-8 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          2 2
6         51 -4
4        62  4
2       85   -3
0      106    4
-2     89     1
-4    79      -2
-6   48       4
-8  37        -4
-10 15         4
-12 2          -2
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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