# L10a112

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a112 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{9/2}-\frac{4}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+8 q^{5/2}-\frac{14}{q^{5/2}}-13 q^{3/2}+\frac{16}{q^{3/2}}+\frac{1}{q^{11/2}}+16 \sqrt{q}-\frac{18}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -2 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -2 a^3 z+5 a z-4 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4+4 a^5 z^5-a^5 z^3+9 a^4 z^6+z^6 a^{-4} -8 a^4 z^4-2 z^4 a^{-4} +3 a^4 z^2+z^2 a^{-4} +13 a^3 z^7+4 z^7 a^{-3} -20 a^3 z^5-10 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -4 a^3 z-2 z a^{-3} +10 a^2 z^8+6 z^8 a^{-2} -10 a^2 z^6-13 z^6 a^{-2} -3 a^2 z^4+6 z^4 a^{-2} +3 a^2 z^2+z^2 a^{-2} +3 a z^9+3 z^9 a^{-1} +16 a z^7+7 z^7 a^{-1} -47 a z^5-33 z^5 a^{-1} +34 a z^3+28 z^3 a^{-1} -10 a z-8 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +16 z^8-33 z^6+14 z^4-1$ (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 \
-5-4-3-2-1012345χ
10          1-1
8         3 3
6        51 -4
4       83  5
2      85   -3
0     108    2
-2    810     2
-4   68      -2
-6  38       5
-8 16        -5
-10 3         3
-121          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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