# L10a76

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a76 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^4-4 u^2 v^3+5 u^2 v^2-2 u^2 v-u v^4+7 u v^3-11 u v^2+7 u v-u-2 v^3+5 v^2-4 v+1}{u v^2}$ (db) Jones polynomial $-q^{7/2}+4 q^{5/2}-9 q^{3/2}+12 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-5 a^3 z^3+7 a z^3-2 z^3 a^{-1} +a^5 z-4 a^3 z+6 a z-2 z a^{-1} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-3 a^3 z^9-3 a z^9-7 a^4 z^8-15 a^2 z^8-8 z^8-7 a^5 z^7-9 a^3 z^7-10 a z^7-8 z^7 a^{-1} -4 a^6 z^6+8 a^4 z^6+29 a^2 z^6-4 z^6 a^{-2} +13 z^6-a^7 z^5+11 a^5 z^5+28 a^3 z^5+32 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+a^4 z^4-17 a^2 z^4+5 z^4 a^{-2} -7 z^4+a^7 z^3-5 a^5 z^3-23 a^3 z^3-28 a z^3-10 z^3 a^{-1} +z^3 a^{-3} -2 a^6 z^2-3 a^4 z^2-a^2 z^2+2 a^5 z+7 a^3 z+10 a z+5 z a^{-1} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-101234χ
8          11
6         3 -3
4        61 5
2       74  -3
0      105   5
-2     88    0
-4    89     -1
-6   58      3
-8  38       -5
-10 15        4
-12 3         -3
-141          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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