# L11a221

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a221 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u v-2 u-2 v+2) (2 u v-2 u-2 v+1)}{u v}$ (db) Jones polynomial $-\frac{10}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+3 q^{5/2}-\frac{16}{q^{5/2}}-6 q^{3/2}+\frac{15}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+10 \sqrt{q}-\frac{14}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)+2 a^5 z^3+a^5 z-a^3 z^5+a^3 z^{-1} -z a^{-3} -a z^5+2 z^3 a^{-1} -a z-a z^{-1} +z a^{-1}$ (db) Kauffman polynomial $-a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-6 a^3 z^9-3 a z^9-4 a^6 z^8-6 a^4 z^8-6 a^2 z^8-4 z^8-3 a^7 z^7+3 a^5 z^7+11 a^3 z^7+a z^7-4 z^7 a^{-1} -a^8 z^6+10 a^6 z^6+20 a^4 z^6+15 a^2 z^6-3 z^6 a^{-2} +3 z^6+9 a^7 z^5+6 a^5 z^5-8 a^3 z^5+a z^5+5 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4-6 a^6 z^4-20 a^4 z^4-15 a^2 z^4+6 z^4 a^{-2} +2 z^4-7 a^7 z^3-6 a^5 z^3+3 a^3 z^3+2 z^3 a^{-3} -2 a^8 z^2+2 a^6 z^2+7 a^4 z^2+5 a^2 z^2-3 z^2 a^{-2} -z^2+2 a^7 z+a^5 z-3 a^3 z-a z-z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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        62  -4
0       84   4
-2      87    -1
-4     87     1
-6    69      3
-8   47       -3
-10  26        4
-12 14         -3
-14 2          2
-161           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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