# L11a311

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a311 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 v^3-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+10 u^2 v^2-11 u^2 v+3 u^2+3 u v^3-11 u v^2+10 u v-3 u-v^3+3 v^2-3 v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-\frac{14}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+4 q^{5/2}-\frac{23}{q^{5/2}}-9 q^{3/2}+\frac{22}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)+3 a^5 z^3+4 a^5 z+a^5 z^{-1} -3 a^3 z^5-8 a^3 z^3-8 a^3 z-a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} +5 a z-2 z a^{-1}$ (db) Kauffman polynomial $-a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-9 a^3 z^9-5 a z^9-6 a^6 z^8-17 a^4 z^8-20 a^2 z^8-9 z^8-4 a^7 z^7-5 a^5 z^7-3 a^3 z^7-10 a z^7-8 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+41 a^4 z^6+42 a^2 z^6-4 z^6 a^{-2} +10 z^6+10 a^7 z^5+30 a^5 z^5+42 a^3 z^5+35 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-6 a^6 z^4-27 a^4 z^4-26 a^2 z^4+5 z^4 a^{-2} -2 z^4-8 a^7 z^3-29 a^5 z^3-41 a^3 z^3-29 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+4 a^4 z^2+5 a^2 z^2-2 z^2 a^{-2} +2 a^7 z+10 a^5 z+13 a^3 z+8 a z+3 z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1}$ (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 \
-7-6-5-4-3-2-101234χ
8           11
6          3 -3
4         61 5
2        93  -6
0       116   5
-2      1210    -2
-4     1110     1
-6    812      4
-8   611       -5
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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