# L11a396

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a396 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u v w^3-3 u v w^2+3 u v w-2 u v-2 u w^3+4 u w^2-4 u w+3 u-3 v w^3+4 v w^2-4 v w+2 v+2 w^3-3 w^2+3 w-2}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $-q^8+4 q^7-8 q^6+11 q^5-14 q^4+15 q^3-14 q^2+12 q-6+5 q^{-1} - q^{-2} + q^{-3}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} -z^2 a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +a^2 z^2+6 z^2 a^{-2} +2 a^2 z^{-2} +4 a^{-2} z^{-2} +3 a^2+8 a^{-2} -2 z^4-7 z^2-5 z^{-2} -9$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +z^9 a^{-1} +5 z^9 a^{-3} +4 z^9 a^{-5} -z^8 a^{-2} +5 z^8 a^{-4} +7 z^8 a^{-6} +z^8+a z^7+z^7 a^{-1} -12 z^7 a^{-3} -5 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6+8 z^6 a^{-2} -11 z^6 a^{-4} -14 z^6 a^{-6} +4 z^6 a^{-8} +2 z^6-a z^5+20 z^5 a^{-3} +5 z^5 a^{-5} -13 z^5 a^{-7} +z^5 a^{-9} -5 a^2 z^4-17 z^4 a^{-2} +8 z^4 a^{-4} +10 z^4 a^{-6} -6 z^4 a^{-8} -14 z^4-6 a z^3-15 z^3 a^{-1} -19 z^3 a^{-3} -5 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} +9 a^2 z^2+15 z^2 a^{-2} +z^2 a^{-4} -3 z^2 a^{-6} +20 z^2+11 a z+21 z a^{-1} +13 z a^{-3} +3 z a^{-5} -7 a^2-10 a^{-2} -2 a^{-4} -14-5 a z^{-1} -9 a^{-1} z^{-1} -5 a^{-3} z^{-1} - a^{-5} z^{-1} +2 a^2 z^{-2} +4 a^{-2} z^{-2} + a^{-4} z^{-2} +5 z^{-2}$ (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 \
-4-3-2-101234567χ
17           1-1
15          3 3
13         51 -4
11        63  3
9       85   -3
7      76    1
5     78     1
3    57      -2
1   410       6
-1  12        -1
-3  4         4
-511          0
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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