# L11n77

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n77 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^5-4 u v^4+7 u v^3-3 u v^2-3 v^3+7 v^2-4 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-q^{5/2}+3 q^{3/2}-6 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z a^7+z^5 a^5+4 z^3 a^5+5 z a^5+2 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-11 z a^3-4 a^3 z^{-1} +2 z^5 a+7 z^3 a+7 z a+3 a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^8 z^2+5 a^7 z^3-a^7 z+3 a^6 z^6-2 a^6 z^4+a^6+8 a^5 z^7-26 a^5 z^5+31 a^5 z^3-14 a^5 z+2 a^5 z^{-1} +7 a^4 z^8-22 a^4 z^6+22 a^4 z^4-12 a^4 z^2+2 a^4+2 a^3 z^9+6 a^3 z^7-42 a^3 z^5+53 a^3 z^3-26 a^3 z+4 a^3 z^{-1} +10 a^2 z^8-37 a^2 z^6+38 a^2 z^4-16 a^2 z^2+3 a^2+2 a z^9-a z^7+z^7 a^{-1} -20 a z^5-4 z^5 a^{-1} +33 a z^3+6 z^3 a^{-1} -17 a z-4 z a^{-1} +3 a z^{-1} + a^{-1} z^{-1} +3 z^8-12 z^6+14 z^4-5 z^2+1$ (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 \
-5-4-3-2-101234χ
6         11
4        2 -2
2       41 3
0      42  -2
-2     64   2
-4    55    0
-6   45     -1
-8  35      2
-10 24       -2
-12 4        4
-141         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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