# L11a237

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a237 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^4-5 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u+v^4-5 v^3+8 v^2-5 v+1}{u v^2}$ (db) Jones polynomial $-11 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{3}{q^{7/2}}-23 q^{5/2}+\frac{8}{q^{5/2}}+25 q^{3/2}-\frac{15}{q^{3/2}}-q^{13/2}+5 q^{11/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +2 z^5 a^{-3} -a^3 z^3+5 a z^3-5 z^3 a^{-1} +3 z^3 a^{-3} -z^3 a^{-5} -2 a^3 z+6 a z-4 z a^{-1} +z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^{10} a^{-2} -2 z^{10}-5 a z^9-13 z^9 a^{-1} -8 z^9 a^{-3} -5 a^2 z^8-21 z^8 a^{-2} -13 z^8 a^{-4} -13 z^8-3 a^3 z^7+2 a z^7+12 z^7 a^{-1} -4 z^7 a^{-3} -11 z^7 a^{-5} -a^4 z^6+8 a^2 z^6+46 z^6 a^{-2} +17 z^6 a^{-4} -5 z^6 a^{-6} +33 z^6+7 a^3 z^5+12 a z^5+16 z^5 a^{-1} +27 z^5 a^{-3} +15 z^5 a^{-5} -z^5 a^{-7} +3 a^4 z^4-a^2 z^4-25 z^4 a^{-2} -4 z^4 a^{-4} +4 z^4 a^{-6} -21 z^4-6 a^3 z^3-16 a z^3-21 z^3 a^{-1} -16 z^3 a^{-3} -5 z^3 a^{-5} -3 a^4 z^2-5 a^2 z^2+2 z^2 a^{-2} +3 a^3 z+10 a z+9 z a^{-1} +2 z a^{-3} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} z^{-1}$ (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 \
-5-4-3-2-10123456χ
14           11
12          4 -4
10         71 6
8        104  -6
6       137   6
4      1311    -2
2     1212     0
0    914      5
-2   611       -5
-4  29        7
-6 16         -5
-8 2          2
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=1$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$ ${\mathbb Z}^{13}$ $r=2$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{13}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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