# L11n296

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n296 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^2 w+u v w^4-u v w^3+u v w^2-2 u v w+u v-u w^4+u w^3-v^2 w+v^2-v w^4+2 v w^3-v w^2+v w-v-w^3}{\sqrt{u} v w^2}$ (db) Jones polynomial $q^6-2 q^5+4 q^4-4 q^3+7 q^2-5 q+6-4 q^{-1} +2 q^{-2} - q^{-3}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} +3 z^2 a^{-4} +2 a^{-4} z^{-2} +3 a^{-4} -z^6 a^{-2} -5 z^4 a^{-2} -a^2 z^2-10 z^2 a^{-2} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^2-10 a^{-2} +2 z^4+7 z^2+4 z^{-2} +9$ (db) Kauffman polynomial $z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +3 z^8 a^{-4} +2 z^8+a z^7-z^7 a^{-1} +2 z^7 a^{-5} -25 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -10 z^6-4 a z^5-10 z^5 a^{-1} -13 z^5 a^{-3} -7 z^5 a^{-5} +2 a^2 z^4+48 z^4 a^{-2} +23 z^4 a^{-4} -4 z^4 a^{-6} +23 z^4+a^3 z^3+12 a z^3+30 z^3 a^{-1} +23 z^3 a^{-3} +4 z^3 a^{-5} -3 a^2 z^2-44 z^2 a^{-2} -22 z^2 a^{-4} +3 z^2 a^{-6} -22 z^2-2 a^3 z-13 a z-27 z a^{-1} -16 z a^{-3} +2 a^2+20 a^{-2} +10 a^{-4} +13+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 z^{-2}$ (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 \
-3-2-10123456χ
13         11
11        21-1
9       2  2
7      22  0
5     52   3
3    13    2
1   54     1
-1  13      2
-3 13       -2
-5 1        1
-71         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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