# L10a79

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a79 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2+u v^4-5 u v^3+9 u v^2-5 u v+u-v^4+4 v^3-5 v^2+2 v}{u v^2}$ (db) Jones polynomial $q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^7-a^7 z^{-1} +3 z^3 a^5+5 z a^5+3 a^5 z^{-1} -2 z^5 a^3-5 z^3 a^3-6 z a^3-2 a^3 z^{-1} -z^5 a+z a+z^3 a^{-1}$ (db) Kauffman polynomial $a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +4 a^6 z^8-7 a^6 z^6-2 a^6 z^4+7 a^6 z^2-3 a^6+2 a^5 z^9+6 a^5 z^7-26 a^5 z^5+24 a^5 z^3-10 a^5 z+3 a^5 z^{-1} +11 a^4 z^8-20 a^4 z^6+3 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^9+13 a^3 z^7-36 a^3 z^5+27 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +7 a^2 z^8-4 a^2 z^6-7 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+10 a z^7-14 a z^5+4 z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} -2 a z+8 z^6-8 z^4+2 z^2$ (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 \
-7-6-5-4-3-2-10123χ
6          1-1
4         3 3
2        51 -4
0       73  4
-2      86   -2
-4     86    2
-6    69     3
-8   57      -2
-10  26       4
-12 15        -4
-14 2         2
-161          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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