# L11n359

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n359 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(1) t(3)^2+t(1) t(2) t(3)^2-t(2) t(3)^2-t(2)^2 t(3)-t(1) t(2) t(3)+t(2) t(3)+t(1)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^5+ q^{-5} +2 q^4- q^{-4} -2 q^3+2 q^{-3} +3 q^2-2 q+2$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^6-2 a^2 z^4+6 z^4+a^4 z^2-9 a^2 z^2-2 z^2 a^{-2} -z^2 a^{-4} +9 z^2+3 a^4-8 a^2- a^{-2} +6+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2}$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+3 a^2 z^8+2 z^8-6 a^3 z^7-6 a z^7+z^7 a^{-1} +z^7 a^{-3} -7 a^4 z^6-22 a^2 z^6+2 z^6 a^{-2} +2 z^6 a^{-4} -15 z^6+8 a^3 z^5+5 a z^5-6 z^5 a^{-1} -2 z^5 a^{-3} +z^5 a^{-5} +16 a^4 z^4+48 a^2 z^4-7 z^4 a^{-2} -7 z^4 a^{-4} +32 z^4+2 a^3 z^3+10 a z^3+9 z^3 a^{-1} -2 z^3 a^{-3} -3 z^3 a^{-5} -16 a^4 z^2-40 a^2 z^2+5 z^2 a^{-2} +5 z^2 a^{-4} -24 z^2-7 a^3 z-10 a z-3 z a^{-1} +z a^{-3} +z a^{-5} +7 a^4+14 a^2- a^{-2} - a^{-4} +8+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2}$ (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       131  1
3      122   1
1     132    0
-1    123     2
-3   121      0
-5  112       2
-7 12         1
-9            0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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