# L11n357

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n357 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(-u v^2 w-u v+u+v^2 w^3-v w^3-w^2\right)}{\sqrt{u} v w^2}$ (db) Jones polynomial $q^{-5} -q^4- q^{-4} +2 q^3+3 q^{-3} -2 q^2-3 q^{-2} +4 q+4 q^{-1} -3$ (db) Signature 2 (db) HOMFLY-PT polynomial $a^4 z^2+a^4 z^{-2} +3 a^4-2 a^2 z^4-z^4 a^{-2} -8 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -8 a^2- a^{-2} +z^6+5 z^4+8 z^2+ z^{-2} +6$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-5 a^3 z^7-2 a z^7+3 z^7 a^{-1} -7 a^4 z^6-24 a^2 z^6+2 z^6 a^{-2} -15 z^6+5 a^3 z^5-8 a z^5-12 z^5 a^{-1} +z^5 a^{-3} +17 a^4 z^4+47 a^2 z^4-6 z^4 a^{-2} +24 z^4+5 a^3 z^3+18 a z^3+11 z^3 a^{-1} -2 z^3 a^{-3} -17 a^4 z^2-38 a^2 z^2+3 z^2 a^{-2} +2 z^2 a^{-4} -20 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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 \
-6-5-4-3-2-10123χ
9         1-1
7        1 1
5       22 0
3      211 2
1     23   1
-1    221   1
-3   23     1
-5  11      0
-7 13       2
-9          0
-111         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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