# L10a150

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a150 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (w-1) \left(2 v w^2-2 v w+v-w^2+2 w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $q^{10}-4 q^9+8 q^8-11 q^7+13 q^6-13 q^5+13 q^4-8 q^3+6 q^2-2 q+1$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +z^4 a^{-8} +3 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-8} +3 a^{-2} -4 a^{-4} + a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} +4 z^5 a^{-11} -2 z^3 a^{-11} +8 z^6 a^{-10} -9 z^4 a^{-10} +3 z^2 a^{-10} - a^{-10} +9 z^7 a^{-9} -12 z^5 a^{-9} +5 z^3 a^{-9} -z a^{-9} +5 z^8 a^{-8} +2 z^6 a^{-8} -15 z^4 a^{-8} +11 z^2 a^{-8} -3 a^{-8} +z^9 a^{-7} +13 z^7 a^{-7} -29 z^5 a^{-7} +16 z^3 a^{-7} -3 z a^{-7} +7 z^8 a^{-6} -7 z^6 a^{-6} -13 z^4 a^{-6} +18 z^2 a^{-6} + a^{-6} z^{-2} -7 a^{-6} +z^9 a^{-5} +6 z^7 a^{-5} -18 z^5 a^{-5} +11 z^3 a^{-5} +z a^{-5} -2 a^{-5} z^{-1} +2 z^8 a^{-4} -12 z^4 a^{-4} +16 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +2 z^7 a^{-3} -5 z^5 a^{-3} +2 z^3 a^{-3} +3 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -4 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} -4 a^{-2}$ (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         3 -3
17        51 4
15       63  -3
13      75   2
11     88    0
9    55     0
7   38      5
5  35       -2
3 15        4
1 1         -1
-11          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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