# L11n157

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n157 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-t(1) t(2)^4-2 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2-7 t(1) t(2)^2+2 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-2 t(2)-t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $-4 q^{9/2}+7 q^{7/2}-\frac{1}{q^{7/2}}-10 q^{5/2}+\frac{3}{q^{5/2}}+10 q^{3/2}-\frac{6}{q^{3/2}}+2 q^{11/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-5 z^5 a^{-1} +z^5 a^{-3} +3 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +3 a z-5 z a^{-1} +z a^{-5} +a z^{-1} -2 a^{-3} z^{-1} + a^{-5} z^{-1}$ (db) Kauffman polynomial $3 z^4 a^{-6} -6 z^2 a^{-6} +2 a^{-6} +z^7 a^{-5} +z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -4 z^6 a^{-4} +10 z^4 a^{-4} -12 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -2 a^3 z^3+6 z^3 a^{-3} +a^3 z-2 a^{-3} z^{-1} +5 z^8 a^{-2} +3 a^2 z^6-9 z^6 a^{-2} -6 a^2 z^4+9 z^4 a^{-2} +2 a^2 z^2-5 z^2 a^{-2} +3 a^{-2} +z^9 a^{-1} +4 a z^7+5 z^7 a^{-1} -7 a z^5-14 z^5 a^{-1} +3 a z^3+14 z^3 a^{-1} -3 a z-5 z a^{-1} +a z^{-1} +3 z^8-2 z^6-4 z^4+3 z^2-1$ (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        2 2
8       52 -3
6      52  3
4     55   0
2    65    1
0   36     3
-2  35      -2
-4 14       3
-6 2        -2
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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