# L10a152

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a152 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) \left(-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-2 t(2) t(3)+t(3)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $q^7-3 q^6+7 q^5-9 q^4+12 q^3+ q^{-3} -12 q^2-2 q^{-2} +11 q+6 q^{-1} -8$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^2 a^{-6} + a^{-6} -2 z^4 a^{-4} -3 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} +z^6 a^{-2} +2 z^4 a^{-2} +a^2 z^2-z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-4 a^{-2} -2 z^4-4 z^2+ z^{-2}$ (db) Kauffman polynomial $z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -9 z^4 a^{-6} +8 z^2 a^{-6} -3 a^{-6} +6 z^7 a^{-5} -6 z^5 a^{-5} +z^3 a^{-5} +z a^{-5} +4 z^8 a^{-4} -7 z^4 a^{-4} +3 z^2 a^{-4} - a^{-4} z^{-2} +2 a^{-4} +z^9 a^{-3} +10 z^7 a^{-3} -27 z^5 a^{-3} +24 z^3 a^{-3} -10 z a^{-3} +2 a^{-3} z^{-1} +7 z^8 a^{-2} +a^2 z^6-15 z^6 a^{-2} -4 a^2 z^4+15 z^4 a^{-2} +5 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+10 a^{-2} +z^9 a^{-1} +2 a z^7+6 z^7 a^{-1} -5 a z^5-23 z^5 a^{-1} +2 a z^3+23 z^3 a^{-1} +a z-10 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-8 z^6+8 z^4-7 z^2- z^{-2} +4$ (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
15          11
13         31-2
11        4  4
9       53  -2
7      74   3
5     66    0
3    56     -1
1   47      3
-1  24       -2
-3  4        4
-512         -1
-71          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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