# L11a132

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a132 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 (u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-4 q^{9/2}+4 q^{7/2}-4 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{19/2}-2 q^{17/2}+2 q^{15/2}-3 q^{13/2}+4 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-1} -z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} +a z-z a^{-1} -z a^{-7} +z a^{-9} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^8 a^{-10} -6 z^6 a^{-10} +10 z^4 a^{-10} -4 z^2 a^{-10} +2 z^9 a^{-9} -13 z^7 a^{-9} +26 z^5 a^{-9} -17 z^3 a^{-9} +2 z a^{-9} +z^{10} a^{-8} -5 z^8 a^{-8} +6 z^6 a^{-8} -z^4 a^{-8} +3 z^9 a^{-7} -17 z^7 a^{-7} +29 z^5 a^{-7} -17 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} -5 z^8 a^{-6} +9 z^6 a^{-6} -10 z^4 a^{-6} +4 z^2 a^{-6} +z^9 a^{-5} -3 z^7 a^{-5} +z^5 a^{-5} +z^8 a^{-4} -2 z^6 a^{-4} +z^7 a^{-3} -z^5 a^{-3} +z^6 a^{-2} +z^5 a^{-1} +a z^3+z^3 a^{-1} -2 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^4-1$ (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 \
-2-10123456789χ
20           1-1
18          1 1
16         11 0
14        21  1
12       21   -1
10      22    0
8     22     0
6    22      0
4   12       1
2  22        0
0 13         2
-2            0
-41           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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