# L11a443

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a443 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3 w^2-u v^3 w+u v^2 w^3-3 u v^2 w^2+3 u v^2 w-u v^2-u v w^3+3 u v w^2-4 u v w+2 u v-u w^2+2 u w-2 v^3 w^2+v^3 w-2 v^2 w^3+4 v^2 w^2-3 v^2 w+v^2+v w^3-3 v w^2+3 v w-v+w^2-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $-q^8+3 q^7-5 q^6+10 q^5-13 q^4+15 q^3-14 q^2+13 q-9+6 q^{-1} -2 q^{-2} + q^{-3}$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-4 z^2 a^{-2} +4 z^2 a^{-4} -2 z^2 a^{-6} -4 z^2+2 a^2-5 a^{-2} +3 a^{-4} -2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -7 z^4 a^{-8} +4 z^2 a^{-8} +4 z^7 a^{-7} -7 z^5 a^{-7} +3 z^3 a^{-7} +4 z^8 a^{-6} -5 z^6 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +3 z^9 a^{-5} -3 z^7 a^{-5} +3 z^5 a^{-5} -z^3 a^{-5} +z a^{-5} +z^{10} a^{-4} +4 z^8 a^{-4} -8 z^6 a^{-4} +7 z^4 a^{-4} -4 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} -8 z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +3 z^8 a^{-2} +a^2 z^6-8 z^6 a^{-2} -4 a^2 z^4+9 z^4 a^{-2} +5 a^2 z^2-14 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+9 a^{-2} +2 z^9 a^{-1} +2 a z^7+z^7 a^{-1} -5 a z^5-12 z^5 a^{-1} +2 a z^3+13 z^3 a^{-1} +a z-8 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-7 z^6+5 z^4-4 z^2- z^{-2} +3$ (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 \
-4-3-2-101234567χ
17           1-1
15          2 2
13         31 -2
11        72  5
9       85   -3
7      75    2
5     78     1
3    67      -1
1   48       4
-1  25        -3
-3  4         4
-512          -1
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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