# L11a324

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a324 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^3 v^3-2 u^3 v^2-2 u^2 v^3+6 u^2 v^2-3 u^2 v-3 u v^2+6 u v-2 u-2 v+2}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $q^{7/2}-2 q^{5/2}+4 q^{3/2}-7 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^3 z^7-a z^7+a^5 z^5-5 a^3 z^5-5 a z^5+z^5 a^{-1} +4 a^5 z^3-8 a^3 z^3-8 a z^3+4 z^3 a^{-1} +4 a^5 z-5 a^3 z-5 a z+4 z a^{-1} +a^5 z^{-1} -a^3 z^{-1}$ (db) Kauffman polynomial $a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-2 a^7 z^3+a^7 z+4 a^6 z^6-5 a^6 z^4+2 a^6 z^2+5 a^5 z^7-11 a^5 z^5+8 a^5 z^3-5 a^5 z+a^5 z^{-1} +5 a^4 z^8-14 a^4 z^6+10 a^4 z^4-3 a^4 z^2-a^4+3 a^3 z^9-5 a^3 z^7-11 a^3 z^5+18 a^3 z^3-8 a^3 z+a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8+z^8 a^{-2} -24 a^2 z^6-6 z^6 a^{-2} +29 a^2 z^4+12 z^4 a^{-2} -9 a^2 z^2-8 z^2 a^{-2} +5 a z^9+2 z^9 a^{-1} -21 a z^7-11 z^7 a^{-1} +23 a z^5+20 z^5 a^{-1} -8 a z^3-15 z^3 a^{-1} +4 a z+5 z a^{-1} +z^{10}-z^8-12 z^6+24 z^4-11 z^2$ (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 \
-6-5-4-3-2-1012345χ
8           1-1
6          1 1
4         31 -2
2        41  3
0       33   0
-2      64    2
-4     44     0
-6    45      -1
-8   24       2
-10  24        -2
-12 13         2
-14 1          -1
-161           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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