# L10a135

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a135 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^2 w^2-3 u v^2 w+u v^2-2 u v w^2+6 u v w-3 u v-2 u w+2 u-2 v^2 w^2+2 v^2 w+3 v w^2-6 v w+2 v-w^2+3 w-2}{\sqrt{u} v w}$ (db) Jones polynomial $q^6-4 q^5+7 q^4+ q^{-4} -10 q^3-3 q^{-3} +14 q^2+7 q^{-2} -13 q-10 q^{-1} +14$ (db) Signature 0 (db) HOMFLY-PT polynomial $-z^6 a^{-2} -z^6+a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -3 z^4+2 a^2 z^2+z^2 a^{-2} +z^2 a^{-4} -5 z^2+2 a^2+4 a^{-2} - a^{-4} -5+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $z^6 a^{-6} -2 z^4 a^{-6} +z^2 a^{-6} +4 z^7 a^{-5} -11 z^5 a^{-5} +7 z^3 a^{-5} +z a^{-5} +5 z^8 a^{-4} -12 z^6 a^{-4} +a^4 z^4+6 z^4 a^{-4} -a^4 z^2+z^2 a^{-4} -2 a^{-4} +2 z^9 a^{-3} +5 z^7 a^{-3} +3 a^3 z^5-20 z^5 a^{-3} -2 a^3 z^3+8 z^3 a^{-3} +3 z a^{-3} +10 z^8 a^{-2} +6 a^2 z^6-18 z^6 a^{-2} -8 a^2 z^4+8 a^2 z^2+11 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -4 a^2-8 a^{-2} +2 z^9 a^{-1} +7 a z^7+8 z^7 a^{-1} -8 a z^5-20 z^5 a^{-1} +3 a z^3+6 z^3 a^{-1} +3 a z+5 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +5 z^8+z^6-17 z^4+20 z^2+2 z^{-2} -9$ (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 \
-4-3-2-10123456χ
13          11
11         3 -3
9        41 3
7       63  -3
5      84   4
3     67    1
1    87     1
-1   48      4
-3  36       -3
-5 15        4
-7 2         -2
-91          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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