# L10a161

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a161 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^2 t(3)^3-t(1) t(2)^2 t(3)^3-t(1)^2 t(2) t(3)^3-t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-t(2)^2 t(3)^2+t(1)^2 t(2) t(3)^2-t(1) t(2) t(3)^2+t(1)^2 t(3)+t(2)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+t(1)+t(2)-1}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^{11}-2 q^{10}+3 q^9-4 q^8+5 q^7-4 q^6+5 q^5-3 q^4+3 q^3-q^2+q$ (db) Signature 6 (db) HOMFLY-PT polynomial $z^6 a^{-8} +5 z^4 a^{-8} +7 z^2 a^{-8} + a^{-8} z^{-2} +3 a^{-8} -z^8 a^{-6} -7 z^6 a^{-6} -17 z^4 a^{-6} -18 z^2 a^{-6} -2 a^{-6} z^{-2} -9 a^{-6} +z^6 a^{-4} +6 z^4 a^{-4} +11 z^2 a^{-4} + a^{-4} z^{-2} +6 a^{-4}$ (db) Kauffman polynomial $z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -z^7 a^{-7} +4 z^7 a^{-9} -7 z^6 a^{-4} -23 z^6 a^{-6} -11 z^6 a^{-8} +5 z^6 a^{-10} +5 z^5 a^{-5} -11 z^5 a^{-7} -12 z^5 a^{-9} +4 z^5 a^{-11} +17 z^4 a^{-4} +43 z^4 a^{-6} +9 z^4 a^{-8} -14 z^4 a^{-10} +3 z^4 a^{-12} +6 z^3 a^{-5} +20 z^3 a^{-7} +6 z^3 a^{-9} -6 z^3 a^{-11} +2 z^3 a^{-13} -17 z^2 a^{-4} -32 z^2 a^{-6} -3 z^2 a^{-8} +9 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} -9 z a^{-5} -9 z a^{-7} +7 a^{-4} +11 a^{-6} +3 a^{-8} -2 a^{-10} +2 a^{-5} z^{-1} +2 a^{-7} z^{-1} - a^{-4} z^{-2} -2 a^{-6} z^{-2} - a^{-8} z^{-2}$ (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 \
-2-1012345678χ
23          11
21         21-1
19        1  1
17       32  -1
15      21   1
13     23    1
11    32     1
9   24      2
7  11       0
5 13        2
3           0
11          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=5$ $i=7$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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