# L11a241

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a241 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u^2 v^2-6 u^2 v+5 u^2-6 u v^2+13 u v-6 u+5 v^2-6 v+2}{u v}$ (db) Jones polynomial $q^{9/2}-\frac{6}{q^{9/2}}-4 q^{7/2}+\frac{9}{q^{7/2}}+7 q^{5/2}-\frac{14}{q^{5/2}}-11 q^{3/2}+\frac{16}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+15 \sqrt{q}-\frac{16}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^{-1} -3 z a^5-a^5 z^{-1} +3 z^3 a^3+z a^3-z^5 a+z^3 a+z a-z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} +z^3 a^{-3}$ (db) Kauffman polynomial $-a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -3 a^4 z^8-7 a^2 z^8-6 z^8 a^{-2} -10 z^8-3 a^5 z^7+3 a^3 z^7+14 a z^7+4 z^7 a^{-1} -4 z^7 a^{-3} -2 a^6 z^6+20 a^2 z^6+17 z^6 a^{-2} -z^6 a^{-4} +36 z^6-a^7 z^5+3 a^5 z^5-7 a^3 z^5-11 a z^5+11 z^5 a^{-1} +11 z^5 a^{-3} +3 a^6 z^4+5 a^4 z^4-24 a^2 z^4-13 z^4 a^{-2} +2 z^4 a^{-4} -41 z^4+3 a^7 z^3+a^5 z^3+9 a^3 z^3+5 a z^3-12 z^3 a^{-1} -6 z^3 a^{-3} -3 a^4 z^2+10 a^2 z^2+4 z^2 a^{-2} +17 z^2-3 a^7 z-2 a^5 z-3 a^3 z-3 a z+z a^{-1} -a^6+a^7 z^{-1} +a^5 z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         41 -3
4        73  4
2       84   -4
0      87    1
-2     99     0
-4    57      -2
-6   49       5
-8  25        -3
-10  4         4
-1212          -1
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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