# L11n364

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n364 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (w-1) \left(v^2 w-2 v w+2 v-1\right)}{\sqrt{u} v w}$ (db) Jones polynomial $-q^8+3 q^7-5 q^6+7 q^5-8 q^4+9 q^3-6 q^2+6 q+ q^{-1} -2$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} +z^6 a^{-4} +4 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -2 z^4 a^{-2} -5 z^2 a^{-2} -2 a^{-2} z^{-2} -4 a^{-2} +z^2+ z^{-2} +2$ (db) Kauffman polynomial $z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -7 z^4 a^{-8} +2 z^2 a^{-8} +4 z^7 a^{-7} -10 z^5 a^{-7} +6 z^3 a^{-7} -z a^{-7} +3 z^8 a^{-6} -7 z^6 a^{-6} +7 z^4 a^{-6} -5 z^2 a^{-6} +2 a^{-6} +z^9 a^{-5} +z^7 a^{-5} -6 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +4 z^8 a^{-4} -14 z^6 a^{-4} +27 z^4 a^{-4} -21 z^2 a^{-4} - a^{-4} z^{-2} +7 a^{-4} +z^9 a^{-3} -3 z^7 a^{-3} +7 z^5 a^{-3} -z^3 a^{-3} -4 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +14 z^4 a^{-2} -17 z^2 a^{-2} -2 a^{-2} z^{-2} +7 a^{-2} +2 z^5 a^{-1} -2 z^3 a^{-1} -2 z a^{-1} +2 a^{-1} z^{-1} +z^4-3 z^2- z^{-2} +3$ (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 \
-2-101234567χ
17         1-1
15        2 2
13       31 -2
11      42  2
9     43   -1
7    54    1
5   36     3
3  33      0
1 15       4
-1 1        -1
-31         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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