# L11a297

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a297 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^3 v^2-3 u^3 v+u^3+2 u^2 v^3-7 u^2 v^2+7 u^2 v-3 u^2-3 u v^3+7 u v^2-7 u v+2 u+v^3-3 v^2+2 v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $\frac{16}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{10}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{16}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^9-z a^9+z^5 a^7+z^3 a^7+2 z^5 a^5+4 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a$ (db) Kauffman polynomial $a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+3 a^{11} z^7-9 a^{11} z^5+7 a^{11} z^3-2 a^{11} z+4 a^{10} z^8-10 a^{10} z^6+6 a^{10} z^4-2 a^{10} z^2+3 a^9 z^9-3 a^9 z^7-6 a^9 z^5+7 a^9 z^3-3 a^9 z+a^8 z^{10}+6 a^8 z^8-19 a^8 z^6+17 a^8 z^4-5 a^8 z^2+6 a^7 z^9-9 a^7 z^7-a^7 z^5+10 a^7 z^3-3 a^7 z+a^6 z^{10}+7 a^6 z^8-18 a^6 z^6+16 a^6 z^4-2 a^6 z^2+3 a^5 z^9+2 a^5 z^7-14 a^5 z^5+18 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6+3 a^4 z^4-a^4+5 a^3 z^7-9 a^3 z^5+6 a^3 z^3-5 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          2 -2
-2         51 4
-4        63  -3
-6       84   4
-8      86    -2
-10     88     0
-12    69      3
-14   47       -3
-16  26        4
-18 14         -3
-20 2          2
-221           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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