# L10a160

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a160 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^2 w^2+u^2 v w^3-2 u^2 v w^2+u^2 v w-u^2 w^3+u^2 w^2-u v^2 w^2+u v^2 w-u v w^3+2 u v w^2-2 u v w+u v-u w^2+u w-v^2 w+v^2-v w^2+2 v w-v-w}{u v w^{3/2}}$ (db) Jones polynomial $q^{10}-2 q^9+4 q^8-6 q^7+8 q^6-7 q^5+8 q^4-5 q^3+4 q^2-2 q+1$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -6 z^2 a^{-6} +3 z^2 a^{-8} + a^{-2} +3 a^{-4} -6 a^{-6} +2 a^{-8} + a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} -2 z^2 a^{-12} +2 z^5 a^{-11} -3 z^3 a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +5 z^2 a^{-10} - a^{-10} +3 z^7 a^{-9} -6 z^5 a^{-9} +6 z^3 a^{-9} +3 z^8 a^{-8} -10 z^6 a^{-8} +19 z^4 a^{-8} -14 z^2 a^{-8} - a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} +z^7 a^{-7} -9 z^5 a^{-7} +16 z^3 a^{-7} -9 z a^{-7} +2 a^{-7} z^{-1} +5 z^8 a^{-6} -20 z^6 a^{-6} +35 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +11 a^{-6} +z^9 a^{-5} -8 z^5 a^{-5} +12 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +2 z^8 a^{-4} -6 z^6 a^{-4} +5 z^4 a^{-4} -6 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +2 z^7 a^{-3} -7 z^5 a^{-3} +5 z^3 a^{-3} +z^6 a^{-2} -4 z^4 a^{-2} +4 z^2 a^{-2} - a^{-2}$ (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 \
-2-1012345678χ
21          11
19         21-1
17        2  2
15       42  -2
13      42   2
11     34    1
9    54     1
7   25      3
5  23       -1
3 13        2
1 1         -1
-11          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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