# L10a162

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a162 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(2)-1) (t(3) t(1)-t(1)+1) (t(1) t(3)-t(3)+1) (t(2) t(3)+1)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^7-3 q^6+6 q^5-9 q^4+12 q^3-11 q^2+12 q-8+6 q^{-1} -3 q^{-2} + q^{-3}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} +4 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +2 a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -13 z^4 a^{-2} -12 z^2 a^{-2} -2 a^{-2} z^{-2} -5 a^{-2} +z^6+4 z^4+5 z^2+ z^{-2} +3$ (db) Kauffman polynomial $z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -3 z^3 a^{-7} +5 z^6 a^{-6} -6 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +6 z^7 a^{-5} -9 z^5 a^{-5} +6 z^3 a^{-5} +5 z^8 a^{-4} -8 z^6 a^{-4} +7 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} +2 a^{-4} +2 z^9 a^{-3} +3 z^7 a^{-3} -13 z^5 a^{-3} +12 z^3 a^{-3} -5 z a^{-3} +2 a^{-3} z^{-1} +9 z^8 a^{-2} +a^2 z^6-26 z^6 a^{-2} -3 a^2 z^4+29 z^4 a^{-2} +2 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} +6 a^{-2} +2 z^9 a^{-1} +3 a z^7-9 a z^5-10 z^5 a^{-1} +6 a z^3+9 z^3 a^{-1} -5 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-12 z^6+12 z^4-9 z^2- z^{-2} +4$ (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         2 -2
11        41 3
9       63  -3
7      63   3
5     67    1
3    65     1
1   37      4
-1  35       -2
-3 14        3
-5 2         -2
-71          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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