# L11n152

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n152 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^4-t(2)^4-3 t(1) t(2)^3+2 t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $2 q^{7/2}-4 q^{5/2}+5 q^{3/2}-7 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a z^5-a^3 z^3+3 a z^3-3 z^3 a^{-1} -a^3 z+4 a z-6 z a^{-1} +2 z a^{-3} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1}$ (db) Kauffman polynomial $a^4 z^6-3 a^4 z^4+a^4 z^2+2 z^2 a^{-4} - a^{-4} +3 a^3 z^7-11 a^3 z^5+z^5 a^{-3} +9 a^3 z^3+2 z^3 a^{-3} -2 a^3 z-3 z a^{-3} + a^{-3} z^{-1} +3 a^2 z^8+z^8 a^{-2} -10 a^2 z^6-3 z^6 a^{-2} +6 a^2 z^4+4 z^4 a^{-2} +3 z^2 a^{-2} -3 a^{-2} +a z^9+z^9 a^{-1} +a z^7-2 z^7 a^{-1} -14 a z^5-2 z^5 a^{-1} +18 a z^3+11 z^3 a^{-1} -9 a z-10 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +4 z^8-14 z^6+13 z^4-3$ (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 \
-5-4-3-2-10123χ
8        2-2
6       2 2
4      32 -1
2     42  2
0    34   1
-2   33    0
-4  24     2
-6 12      -1
-8 2       2
-101        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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