# L10a88

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a88 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^2-2 u^3 v+u^3+u^2 v^3-5 u^2 v^2+7 u^2 v-2 u^2-2 u v^3+7 u v^2-5 u v+u+v^3-2 v^2+v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{5/2}-4 q^{3/2}+7 \sqrt{q}-\frac{10}{\sqrt{q}}+\frac{12}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{11}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^7+2 z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-z^3 a^3-2 z a^3-a^3 z^{-1} -z^5 a-z^3 a-z a+z^3 a^{-1}$ (db) Kauffman polynomial $-a^5 z^9-a^3 z^9-3 a^6 z^8-7 a^4 z^8-4 a^2 z^8-3 a^7 z^7-7 a^5 z^7-11 a^3 z^7-7 a z^7-a^8 z^6+6 a^6 z^6+11 a^4 z^6-3 a^2 z^6-7 z^6+10 a^7 z^5+26 a^5 z^5+27 a^3 z^5+7 a z^5-4 z^5 a^{-1} +3 a^8 z^4+a^4 z^4+13 a^2 z^4-z^4 a^{-2} +8 z^4-10 a^7 z^3-26 a^5 z^3-20 a^3 z^3-a z^3+3 z^3 a^{-1} -2 a^8 z^2-3 a^6 z^2-5 a^4 z^2-6 a^2 z^2-2 z^2+3 a^7 z+10 a^5 z+7 a^3 z+a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-7-6-5-4-3-2-10123χ
6          1-1
4         3 3
2        41 -3
0       63  3
-2      75   -2
-4     65    1
-6    57     2
-8   46      -2
-10  26       4
-12 13        -2
-14 2         2
-161          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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