# L11n173

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n173 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+5 t(1) t(2)^3-2 t(2)^3+4 t(1)^2 t(2)^2-9 t(1) t(2)^2+4 t(2)^2-2 t(1)^2 t(2)+5 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $-6 q^{9/2}+9 q^{7/2}-\frac{1}{q^{7/2}}-13 q^{5/2}+\frac{4}{q^{5/2}}+14 q^{3/2}-\frac{8}{q^{3/2}}+2 q^{11/2}-14 \sqrt{q}+\frac{11}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-5} +z^5 a^{-3} +z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-5 z^3 a^{-1} +a z+ a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^9 a^{-1} -2 z^9 a^{-3} -9 z^8 a^{-2} -3 z^8 a^{-4} -6 z^8-7 a z^7-7 z^7 a^{-1} -z^7 a^{-3} -z^7 a^{-5} -4 a^2 z^6+15 z^6 a^{-2} +3 z^6 a^{-4} +8 z^6-a^3 z^5+13 a z^5+17 z^5 a^{-1} -2 z^5 a^{-3} -5 z^5 a^{-5} +6 a^2 z^4-9 z^4 a^{-2} -5 z^4 a^{-4} -3 z^4 a^{-6} -z^4+a^3 z^3-6 a z^3-7 z^3 a^{-1} +9 z^3 a^{-3} +9 z^3 a^{-5} -a^2 z^2+2 z^2 a^{-2} +4 z^2 a^{-4} +3 z^2 a^{-6} +a z-2 z a^{-1} -6 z a^{-3} -3 z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-4-3-2-1012345χ
12         2-2
10        4 4
8       52 -3
6      84  4
4     76   -1
2    77    0
0   58     3
-2  36      -3
-4 15       4
-6 3        -3
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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