# L11n92

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n92 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1)^3}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-7 q^{9/2}+9 q^{7/2}-11 q^{5/2}+\frac{1}{q^{5/2}}+11 q^{3/2}-\frac{4}{q^{3/2}}-q^{13/2}+4 q^{11/2}-10 \sqrt{q}+\frac{6}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} +z^5 a^{-3} +z^3 a^{-3} +z^5 a^{-1} -a z^3+z^3 a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -4 z^8 a^{-4} -z^8-6 z^7 a^{-3} -6 z^7 a^{-5} +8 z^6 a^{-2} +4 z^6 a^{-4} -4 z^6 a^{-6} -4 a z^5-4 z^5 a^{-1} +13 z^5 a^{-3} +12 z^5 a^{-5} -z^5 a^{-7} -a^2 z^4-6 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4 a^{-6} -3 z^4+5 a z^3+4 z^3 a^{-1} -6 z^3 a^{-3} -4 z^3 a^{-5} +z^3 a^{-7} +a^2 z^2-2 z^2 a^{-4} -z^2 a^{-6} +2 z^2+1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-3-2-10123456χ
14         11
12        3 -3
10       41 3
8      53  -2
6     64   2
4    55    0
2   56     -1
0  37      4
-2 13       -2
-4 3        3
-61         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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