# L11n397

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n397 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (w-1)^2 (w+1) \left(v+w^2\right)}{\sqrt{u} \sqrt{v} w^{5/2}}$ (db) Jones polynomial $-q^5+2 q^4-2 q^3+q^2-q+1+2 q^{-1} - q^{-2} +3 q^{-3} - q^{-4} + q^{-5}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z^6-2 a^2 z^4+6 z^4+a^4 z^2-9 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +11 z^2+3 a^4-11 a^2-3 a^{-2} +11+2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} +4 z^{-2}$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-5 a^3 z^7-4 a z^7+2 z^7 a^{-1} +z^7 a^{-3} -7 a^4 z^6-27 a^2 z^6+2 z^6 a^{-4} -22 z^6+3 a^3 z^5-7 a z^5-15 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} +17 a^4 z^4+56 a^2 z^4-7 z^4 a^{-4} +46 z^4+11 a^3 z^3+32 a z^3+27 z^3 a^{-1} +3 z^3 a^{-3} -3 z^3 a^{-5} -19 a^4 z^2-49 a^2 z^2-5 z^2 a^{-2} +3 z^2 a^{-4} -38 z^2-15 a^3 z-29 a z-17 z a^{-1} -2 z a^{-3} +z a^{-5} +10 a^4+22 a^2+4 a^{-2} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 z^{-2}$ (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 \
-6-5-4-3-2-1012345χ
11           1-1
9          1 1
7         11 0
5       221  -1
3      121   0
1     242    0
-1    124     3
-3   12       1
-5  211       2
-7 13         2
-9            0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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