# L10a155

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a155 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(w-1) \left(u v w^2-3 u v w+2 u v+4 u w-2 u-2 v w^2+4 v w+2 w^2-3 w+1\right)}{\sqrt{u} \sqrt{v} w^{3/2}}$ (db) Jones polynomial $q^4-4 q^3+10 q^2-12 q+16-16 q^{-1} +15 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6-3 z^2 a^4-2 a^4+3 z^4 a^2+5 z^2 a^2+a^2 z^{-2} +4 a^2-z^6-3 z^4-7 z^2-2 z^{-2} -6+z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2}$ (db) Kauffman polynomial $2 a^3 z^9+2 a z^9+4 a^4 z^8+12 a^2 z^8+8 z^8+3 a^5 z^7+7 a^3 z^7+16 a z^7+12 z^7 a^{-1} +a^6 z^6-7 a^4 z^6-21 a^2 z^6+10 z^6 a^{-2} -3 z^6-8 a^5 z^5-27 a^3 z^5-38 a z^5-15 z^5 a^{-1} +4 z^5 a^{-3} -3 a^6 z^4+4 a^2 z^4-12 z^4 a^{-2} +z^4 a^{-4} -12 z^4+7 a^5 z^3+23 a^3 z^3+17 a z^3+z^3 a^{-1} +3 a^6 z^2+4 a^4 z^2+3 a^2 z^2+8 z^2 a^{-2} +10 z^2-2 a^5 z-6 a^3 z+2 a z+6 z a^{-1} -a^6-2 a^2-5 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-6-5-4-3-2-101234χ
9          11
7         41-3
5        6  6
3       64  -2
1      106   4
-1     88    0
-3    78     -1
-5   48      4
-7  37       -4
-9 15        4
-11 2         -2
-131          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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