# L11n34

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n34 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{4}{q^{7/2}}-11 q^{5/2}+\frac{7}{q^{5/2}}+14 q^{3/2}-\frac{12}{q^{3/2}}-15 \sqrt{q}+\frac{14}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -a^3 z+4 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1} -a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $6 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +a^4 z^6+3 z^6 a^{-4} -2 a^4 z^4+3 z^4 a^{-4} +a^4 z^2-3 z^2 a^{-4} + a^{-4} +4 a^3 z^7+8 z^7 a^{-3} -11 a^3 z^5-16 z^5 a^{-3} +9 a^3 z^3+24 z^3 a^{-3} -a^3 z-15 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +5 a^2 z^8+7 z^8 a^{-2} -10 a^2 z^6-10 z^6 a^{-2} +2 a^2 z^4+8 z^4 a^{-2} +a^2 z^2-6 z^2 a^{-2} +a^2+3 a^{-2} +2 a z^9+2 z^9 a^{-1} +9 a z^7+13 z^7 a^{-1} -35 a z^5-40 z^5 a^{-1} +27 a z^3+36 z^3 a^{-1} -5 a z-14 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +12 z^8-24 z^6+9 z^4-3 z^2+2$ (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 \
-5-4-3-2-101234χ
10         33
8        4 -4
6       73 4
4      74  -3
2     87   1
0    89    1
-2   46     -2
-4  38      5
-6 14       -3
-8 3        3
-101         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$

### Computer Talk

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