# L11a52

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a52 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-11 q^{7/2}+12 q^{5/2}-12 q^{3/2}+10 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-5} -3 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +5 z a^{-3} +3 a^{-3} z^{-1} +z^7 a^{-1} -a z^5+4 z^5 a^{-1} -3 a z^3+3 z^3 a^{-1} -3 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-9} +3 z^4 a^{-8} +6 z^5 a^{-7} -4 z^3 a^{-7} +z a^{-7} +9 z^6 a^{-6} -13 z^4 a^{-6} +5 z^2 a^{-6} - a^{-6} +10 z^7 a^{-5} -20 z^5 a^{-5} +9 z^3 a^{-5} -2 z a^{-5} + a^{-5} z^{-1} +8 z^8 a^{-4} -16 z^6 a^{-4} -z^4 a^{-4} +8 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -9 z^7 a^{-3} -9 z^5 a^{-3} +16 z^3 a^{-3} -9 z a^{-3} +3 a^{-3} z^{-1} +2 z^{10} a^{-2} +a^2 z^8-5 a^2 z^6-18 z^6 a^{-2} +7 a^2 z^4+17 z^4 a^{-2} -2 a^2 z^2+2 z^2 a^{-2} -3 a^{-2} +3 a z^9+8 z^9 a^{-1} -17 a z^7-36 z^7 a^{-1} +31 a z^5+48 z^5 a^{-1} -18 a z^3-16 z^3 a^{-1} -a z-7 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}-7 z^8+2 z^6+9 z^4-3 z^2-2$ (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 \
-5-4-3-2-10123456χ
16           11
14          2 -2
12         41 3
10        52  -3
8       64   2
6      65    -1
4     66     0
2    68      2
0   24       -2
-2  26        4
-4 12         -1
-6 2          2
-81           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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