# L11n160

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n160 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u v-u-v+2) (2 u v-u-v+1)}{u v}$ (db) Jones polynomial $q^{9/2}-3 q^{7/2}+5 q^{5/2}-7 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^3 z^3+z^3 a^{-3} +a^3 z+z a^{-3} +a^3 z^{-1} -a z^5-z^5 a^{-1} -2 a z^3-2 z^3 a^{-1} -2 a z-z a^{-1} -a z^{-1}$ (db) Kauffman polynomial $-2 a z^9-2 z^9 a^{-1} -3 a^2 z^8-4 z^8 a^{-2} -7 z^8-a^3 z^7+5 a z^7+3 z^7 a^{-1} -3 z^7 a^{-3} +11 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +26 z^6-3 a z^5+7 z^5 a^{-1} +10 z^5 a^{-3} -3 a^4 z^4-18 a^2 z^4-12 z^4 a^{-2} +3 z^4 a^{-4} -30 z^4-a^5 z^3-a^3 z^3-3 a z^3-10 z^3 a^{-1} -7 z^3 a^{-3} +3 a^4 z^2+8 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} +10 z^2+a^5 z+3 z a^{-1} +2 z a^{-3} -a^2+a^3 z^{-1} +a z^{-1}$ (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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       31 -2
4      42  2
2     43   -1
0    54    1
-2   35     2
-4  34      -1
-6 14       3
-8 2        -2
-101         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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