# L11a296

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a296 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3-t(1) t(2)^3+t(2)^3+3 t(1) t(2)^2+t(1)^2 t(2)-t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{9}{q^{9/2}}-q^{7/2}+\frac{11}{q^{7/2}}+3 q^{5/2}-\frac{14}{q^{5/2}}-6 q^{3/2}+\frac{14}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{5}{q^{11/2}}+9 \sqrt{q}-\frac{13}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^7+a z^7-a^5 z^5+5 a^3 z^5+4 a z^5-z^5 a^{-1} -4 a^5 z^3+10 a^3 z^3+4 a z^3-3 z^3 a^{-1} -5 a^5 z+9 a^3 z-2 a z-2 z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $-a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-3 a^6 z^8-3 a^4 z^8-5 a^2 z^8-5 z^8-2 a^7 z^7+10 a^5 z^7+19 a^3 z^7+2 a z^7-5 z^7 a^{-1} -a^8 z^6+9 a^6 z^6+14 a^4 z^6+18 a^2 z^6-3 z^6 a^{-2} +11 z^6+6 a^7 z^5-21 a^5 z^5-35 a^3 z^5+5 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +4 a^8 z^4-10 a^6 z^4-23 a^4 z^4-25 a^2 z^4+6 z^4 a^{-2} -10 z^4-4 a^7 z^3+27 a^5 z^3+35 a^3 z^3-8 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} -4 a^8 z^2+7 a^6 z^2+18 a^4 z^2+11 a^2 z^2-z^2 a^{-2} +3 z^2-10 a^5 z-17 a^3 z-3 a z+4 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (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          2 -2
4         41 3
2        52  -3
0       84   4
-2      87    -1
-4     66     0
-6    58      3
-8   46       -2
-10  15        4
-12 14         -3
-14 1          1
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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