# L11a43

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a43 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-3 q^{9/2}+\frac{2}{q^{9/2}}+5 q^{7/2}-\frac{5}{q^{7/2}}-9 q^{5/2}+\frac{8}{q^{5/2}}+11 q^{3/2}-\frac{10}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{12}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} +z a^{-5} -2 a^3 z^3-2 z^3 a^{-3} -3 a^3 z-2 a^3 z^{-1} -2 z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +a z^3+z^3 a^{-1} +a z+a z^{-1} +2 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^2 z^{10}-z^{10}-2 a^3 z^9-5 a z^9-3 z^9 a^{-1} -2 a^4 z^8-2 a^2 z^8-4 z^8 a^{-2} -4 z^8-a^5 z^7+4 a^3 z^7+12 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +8 a^4 z^6+16 a^2 z^6+3 z^6 a^{-2} -4 z^6 a^{-4} +15 z^6+5 a^5 z^5+8 a^3 z^5-3 z^5 a^{-5} -9 a^4 z^4-18 a^2 z^4+z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -12 z^4-8 a^5 z^3-20 a^3 z^3-13 a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +4 z^3 a^{-5} +4 a^4 z^2+8 a^2 z^2+z^2 a^{-6} +5 z^2+5 a^5 z+13 a^3 z+8 a z-3 z a^{-1} -5 z a^{-3} -2 z a^{-5} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
12           1-1
10          2 2
8         31 -2
6        62  4
4       53   -2
2      86    2
0     67     1
-2    46      -2
-4   46       2
-6  14        -3
-8 14         3
-10 1          -1
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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