# L11n37

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n37 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $q^{15/2}-3 q^{13/2}+6 q^{11/2}-8 q^{9/2}+9 q^{7/2}-10 q^{5/2}+8 q^{3/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $2 z^5 a^{-3} -3 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +a z-5 z a^{-1} +9 z a^{-3} -6 z a^{-5} +z a^{-7} +a z^{-1} -2 a^{-1} z^{-1} +3 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-3} -z^9 a^{-5} -3 z^8 a^{-2} -6 z^8 a^{-4} -3 z^8 a^{-6} -2 z^7 a^{-1} -5 z^7 a^{-3} -6 z^7 a^{-5} -3 z^7 a^{-7} +9 z^6 a^{-2} +15 z^6 a^{-4} +5 z^6 a^{-6} -z^6 a^{-8} +6 z^5 a^{-1} +24 z^5 a^{-3} +27 z^5 a^{-5} +9 z^5 a^{-7} -15 z^4 a^{-2} -9 z^4 a^{-4} +6 z^4 a^{-6} +3 z^4 a^{-8} -3 z^4-a z^3-16 z^3 a^{-1} -36 z^3 a^{-3} -28 z^3 a^{-5} -7 z^3 a^{-7} +7 z^2 a^{-2} +z^2 a^{-4} -7 z^2 a^{-6} -3 z^2 a^{-8} +2 z^2+2 a z+11 z a^{-1} +20 z a^{-3} +14 z a^{-5} +3 z a^{-7} -2 a^{-2} +2 a^{-6} + a^{-8} -a z^{-1} -2 a^{-1} z^{-1} -3 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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 \
-2-101234567χ
16         1-1
14        2 2
12       41 -3
10      42  2
8     54   -1
6    54    1
4   35     2
2  45      -1
0 15       4
-2 2        -2
-41         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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