# L11a305

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a305 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^3 v^3-3 u^3 v^2+u^3 v-3 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2+u v^3-5 u v^2+8 u v-3 u+v^2-3 v+2}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $14 q^{9/2}-15 q^{7/2}+12 q^{5/2}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-6 q^{15/2}+10 q^{13/2}-13 q^{11/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-7} -3 z^3 a^{-7} -2 z a^{-7} +z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +3 z a^{-3} + a^{-3} z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1}$ (db) Kauffman polynomial $z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +3 z^2 a^{-10} +4 z^7 a^{-9} -5 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +4 z^8 a^{-8} -4 z^6 a^{-8} -z^2 a^{-8} +3 z^9 a^{-7} -3 z^7 a^{-7} +5 z^5 a^{-7} -7 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +5 z^8 a^{-6} -15 z^6 a^{-6} +19 z^4 a^{-6} -8 z^2 a^{-6} +6 z^9 a^{-5} -15 z^7 a^{-5} +17 z^5 a^{-5} -6 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +5 z^8 a^{-4} -21 z^6 a^{-4} +25 z^4 a^{-4} -7 z^2 a^{-4} - a^{-4} +3 z^9 a^{-3} -5 z^7 a^{-3} -4 z^5 a^{-3} +10 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -12 z^6 a^{-2} +9 z^4 a^{-2} -2 z^2 a^{-2} +3 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -2 z a^{-1} +z^6-3 z^4+z^2$ (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 \
-3-2-1012345678χ
20           11
18          2 -2
16         41 3
14        62  -4
12       74   3
10      87    -1
8     76     1
6    58      3
4   47       -3
2  26        4
0 13         -2
-2 2          2
-41           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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