# L11a510

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a510 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^2 v^2 w^3-u^2 v^2 w^2-u^2 v w^3+2 u^2 v w^2-u^2 v w-u^2 w^2+2 u^2 w-u^2-2 u v^2 w^3+2 u v^2 w^2-u v^2 w+u v w^3-2 u v w^2+2 u v w-u v+u w^2-2 u w+2 u+v^2 w^3-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{u v w^{3/2}}$ (db) Jones polynomial $-q^9+3 q^8-5 q^7+8 q^6-10 q^5+11 q^4-10 q^3+10 q^2+ q^{-2} -6 q-2 q^{-1} +5$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-15 z^2 a^{-2} +13 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-8 a^{-2} +5 a^{-4} - a^{-6} +4-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2}$ (db) Kauffman polynomial $z^3 a^{-11} +3 z^4 a^{-10} -z^2 a^{-10} +5 z^5 a^{-9} -3 z^3 a^{-9} +7 z^6 a^{-8} -9 z^4 a^{-8} +2 z^2 a^{-8} +8 z^7 a^{-7} -16 z^5 a^{-7} +7 z^3 a^{-7} -z a^{-7} +7 z^8 a^{-6} -17 z^6 a^{-6} +9 z^4 a^{-6} -4 z^2 a^{-6} + a^{-6} +4 z^9 a^{-5} -6 z^7 a^{-5} -13 z^5 a^{-5} +17 z^3 a^{-5} -3 z a^{-5} +z^{10} a^{-4} +7 z^8 a^{-4} -42 z^6 a^{-4} +59 z^4 a^{-4} -32 z^2 a^{-4} - a^{-4} z^{-2} +8 a^{-4} +6 z^9 a^{-3} -24 z^7 a^{-3} +22 z^5 a^{-3} +4 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +z^8 a^{-2} -24 z^6 a^{-2} +51 z^4 a^{-2} -38 z^2 a^{-2} -2 a^{-2} z^{-2} +12 a^{-2} +2 z^9 a^{-1} -10 z^7 a^{-1} +14 z^5 a^{-1} -2 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +z^8-6 z^6+13 z^4-13 z^2- z^{-2} +6$ (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
19           1-1
17          2 2
15         31 -2
13        52  3
11       64   -2
9      54    1
7     67     1
5    44      0
3   37       4
1  23        -1
-1 14         3
-3 1          -1
-51           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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