# L11n455

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n455 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(3)^2 t(2)^2+t(3)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(3) t(2)^2+t(1) t(4) t(2)^2-t(1) t(3) t(4) t(2)^2-2 t(3)^2 t(2)+t(1) t(2)-t(1) t(3) t(2)+t(3) t(2)+t(3)^2 t(4) t(2)-2 t(1) t(4) t(2)+t(1) t(3) t(4) t(2)-t(3) t(4) t(2)+t(3)^2-t(3)-t(3)^2 t(4)+t(1) t(4)+2 t(3) t(4)-t(4)}{\sqrt{t(1)} t(2) t(3) \sqrt{t(4)}}$ (db) Jones polynomial $2 q^{9/2}-6 q^{7/2}+\frac{1}{q^{7/2}}+5 q^{5/2}-\frac{4}{q^{5/2}}-9 q^{3/2}+\frac{5}{q^{3/2}}-q^{11/2}+7 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-2 z^5 a^{-1} +3 a z^3-8 z^3 a^{-1} +3 z^3 a^{-3} -a^3 z+6 a z-12 z a^{-1} +8 z a^{-3} -z a^{-5} +3 a z^{-1} -8 a^{-1} z^{-1} +7 a^{-3} z^{-1} -2 a^{-5} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -6 z^8 a^{-2} -2 z^8 a^{-4} -4 z^8-5 a z^7-7 z^7 a^{-1} -3 z^7 a^{-3} -z^7 a^{-5} -2 a^2 z^6+19 z^6 a^{-2} +7 z^6 a^{-4} +10 z^6+17 a z^5+39 z^5 a^{-1} +27 z^5 a^{-3} +5 z^5 a^{-5} +2 a^2 z^4-6 z^4 a^{-2} -3 z^4 a^{-4} -z^4-4 a^3 z^3-24 a z^3-52 z^3 a^{-1} -42 z^3 a^{-3} -10 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-18 z^2 a^{-2} -9 z^2 a^{-4} -10 z^2+2 a^3 z+16 a z+31 z a^{-1} +27 z a^{-3} +10 z a^{-5} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3}$ (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 \
-3-2-10123456χ
12         11
10        1 -1
8       51 4
6      23  1
4     73   4
2   134    2
0   65     1
-2 124      3
-4 34       -1
-6 3        3
-81         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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