# L11n317

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n317 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^2 w^2-2 u v^2 w+u v^2-3 u v w^2+5 u v w-2 u v+u w^2-3 u w+u-v^2 w^2+3 v^2 w-v^2+2 v w^2-5 v w+3 v-w^2+2 w-1}{\sqrt{u} v w}$ (db) Jones polynomial $-q^2+4 q-7+11 q^{-1} -12 q^{-2} +14 q^{-3} -11 q^{-4} +9 q^{-5} -5 q^{-6} +2 q^{-7}$ (db) Signature -2 (db) HOMFLY-PT polynomial $2 a^6 z^2+a^6 z^{-2} +2 a^6-3 a^4 z^4-7 a^4 z^2-2 a^4 z^{-2} -6 a^4+a^2 z^6+3 a^2 z^4+5 a^2 z^2+a^2 z^{-2} +4 a^2-z^4-z^2$ (db) Kauffman polynomial $3 a^8 z^4-4 a^8 z^2+a^8+a^7 z^7+3 a^7 z^5-4 a^7 z^3+2 a^6 z^8+a^6 z^6-3 a^6 z^4+4 a^6 z^2+a^6 z^{-2} -3 a^6+a^5 z^9+5 a^5 z^7-5 a^5 z^5-4 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +6 a^4 z^8-2 a^4 z^6-13 a^4 z^4+16 a^4 z^2+2 a^4 z^{-2} -8 a^4+a^3 z^9+10 a^3 z^7-19 a^3 z^5+4 a^3 z^3+6 a^3 z-2 a^3 z^{-1} +4 a^2 z^8+a^2 z^6-14 a^2 z^4+11 a^2 z^2+a^2 z^{-2} -5 a^2+6 a z^7-10 a z^5+z^5 a^{-1} +3 a z^3-z^3 a^{-1} +4 z^6-7 z^4+3 z^2$ (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
5         1-1
3        3 3
1       41 -3
-1      73  4
-3     76   -1
-5    75    2
-7   58     3
-9  46      -2
-11 15       4
-1314        -3
-152         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-6$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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