# L10a104

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a104 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 \left(-v^3\right)+2 u^3 v^2-u^3 v+3 u^2 v^3-7 u^2 v^2+7 u^2 v-2 u^2-2 u v^3+7 u v^2-7 u v+3 u-v^2+2 v-1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-7 q^{9/2}+11 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{4}{q^{5/2}}+15 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+3 q^{11/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +6 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +a z-3 z a^{-1}$ (db) Kauffman polynomial $z^5 a^{-7} -2 z^3 a^{-7} +z a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +5 z^7 a^{-5} -8 z^5 a^{-5} +6 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -5 z^6 a^{-4} +2 z^2 a^{-4} - a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} +a^3 z^5-25 z^5 a^{-3} -a^3 z^3+23 z^3 a^{-3} -9 z a^{-3} + a^{-3} z^{-1} +11 z^8 a^{-2} +4 a^2 z^6-17 z^6 a^{-2} -6 a^2 z^4+6 z^4 a^{-2} +2 a^2 z^2+z^2 a^{-2} +2 z^9 a^{-1} +7 a z^7+11 z^7 a^{-1} -12 a z^5-29 z^5 a^{-1} +6 a z^3+22 z^3 a^{-1} -2 a z-6 z a^{-1} +6 z^8-5 z^6-5 z^4+3 z^2$ (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 \
-4-3-2-10123456χ
14          11
12         2 -2
10        51 4
8       73  -4
6      84   4
4     77    0
2    88     0
0   58      3
-2  37       -4
-4 15        4
-6 3         -3
-81          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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