# L11a39

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a39 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(3 v^2-8 v+3\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-13 q^{9/2}+16 q^{7/2}-\frac{1}{q^{7/2}}-18 q^{5/2}+\frac{2}{q^{5/2}}+18 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-4 q^{13/2}+8 q^{11/2}-15 \sqrt{q}+\frac{10}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-7} -3 z^3 a^{-5} -3 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +4 z^3 a^{-3} +a^3 z+5 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-z^3 a^{-1} -2 a z-2 z a^{-1} -a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -10 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +6 z^8 a^{-6} -13 z^6 a^{-6} +5 z^4 a^{-6} -z^2 a^{-6} +4 z^9 a^{-5} +z^7 a^{-5} -24 z^5 a^{-5} +24 z^3 a^{-5} -9 z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +12 z^8 a^{-4} -34 z^6 a^{-4} +26 z^4 a^{-4} -7 z^2 a^{-4} + a^{-4} +7 z^9 a^{-3} -4 z^7 a^{-3} +a^3 z^5-23 z^5 a^{-3} -3 a^3 z^3+36 z^3 a^{-3} +3 a^3 z-17 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +z^{10} a^{-2} +10 z^8 a^{-2} +2 a^2 z^6-27 z^6 a^{-2} -3 a^2 z^4+29 z^4 a^{-2} -12 z^2 a^{-2} +a^2+3 a^{-2} +3 z^9 a^{-1} +3 a z^7+2 z^7 a^{-1} -2 a z^5-12 z^5 a^{-1} -3 a z^3+19 z^3 a^{-1} +3 a z-10 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +4 z^8-5 z^6+7 z^4-7 z^2+2$ (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 \
-4-3-2-101234567χ
16           1-1
14          3 3
12         51 -4
10        83  5
8       85   -3
6      108    2
4     88     0
2    710      -3
0   510       5
-2  15        -4
-4 15         4
-6 1          -1
-81           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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