# L11a456

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a456 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^3 w^2-u v^3 w+u v^2 w^3-4 u v^2 w^2+5 u v^2 w-u v^2-2 u v w^3+5 u v w^2-4 u v w+u v+u w^3-2 u w^2+u w-v^3 w^2+2 v^3 w-v^3-v^2 w^3+4 v^2 w^2-5 v^2 w+2 v^2+v w^3-5 v w^2+4 v w-v+w^2-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $q^6-3 q^5- q^{-5} +7 q^4+3 q^{-4} -11 q^3-7 q^{-3} +17 q^2+13 q^{-2} -18 q-16 q^{-1} +19$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^4 a^{-4} -a^4 z^2+2 z^2 a^{-4} + a^{-4} z^{-2} -a^4+2 a^{-4} -z^6 a^{-2} +2 a^2 z^4-3 z^4 a^{-2} +3 a^2 z^2-6 z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-5 a^{-2} -z^6-z^4+z^2+ z^{-2} +2$ (db) Kauffman polynomial $z^6 a^{-6} -3 z^4 a^{-6} +2 z^2 a^{-6} +3 z^7 a^{-5} +a^5 z^5-8 z^5 a^{-5} -2 a^5 z^3+5 z^3 a^{-5} +a^5 z+5 z^8 a^{-4} +3 a^4 z^6-14 z^6 a^{-4} -5 a^4 z^4+16 z^4 a^{-4} +3 a^4 z^2-13 z^2 a^{-4} - a^{-4} z^{-2} -a^4+6 a^{-4} +4 z^9 a^{-3} +5 a^3 z^7-5 z^7 a^{-3} -6 a^3 z^5-3 z^5 a^{-3} +a^3 z^3+6 z^3 a^{-3} +a^3 z-5 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +6 a^2 z^8+12 z^8 a^{-2} -7 a^2 z^6-41 z^6 a^{-2} +3 a^2 z^4+55 z^4 a^{-2} -a^2 z^2-39 z^2 a^{-2} -2 a^{-2} z^{-2} +13 a^{-2} +4 a z^9+8 z^9 a^{-1} +2 a z^7-11 z^7 a^{-1} -12 a z^5+10 a z^3+8 z^3 a^{-1} -3 a z-8 z a^{-1} +2 a^{-1} z^{-1} +z^{10}+13 z^8-36 z^6+44 z^4-28 z^2- z^{-2} +9$ (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 \
-5-4-3-2-10123456χ
13           11
11          2 -2
9         51 4
7        73  -4
5       104   6
3      87    -1
1     1110     1
-1    811      3
-3   58       -3
-5  28        6
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=0$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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