# L11a533

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a533 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 t(2) t(1)-2 t(2) t(3) t(1)+2 t(3) t(1)-2 t(2) t(4) t(1)+2 t(2) t(3) t(4) t(1)-3 t(3) t(4) t(1)+3 t(4) t(1)-3 t(1)-3 t(2)+3 t(2) t(3)-2 t(3)+2 t(2) t(4)-3 t(2) t(3) t(4)+3 t(3) t(4)-2 t(4)+2}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db) Jones polynomial $q^{9/2}-4 q^{7/2}+7 q^{5/2}-9 q^{3/2}+11 \sqrt{q}-\frac{14}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^{-1} +a^7 z^{-3} -3 z a^5-5 a^5 z^{-1} -3 a^5 z^{-3} +3 z^3 a^3+6 z a^3+7 a^3 z^{-1} +3 a^3 z^{-3} -z^5 a-z^3 a-3 z a-3 a z^{-1} -a z^{-3} -z^5 a^{-1} -z^3 a^{-1} +z^3 a^{-3}$ (db) Kauffman polynomial $a^7 z^5-4 a^7 z^3+a^7 z^{-3} +6 a^7 z-4 a^7 z^{-1} +a^6 z^6-6 a^6 z^2-3 a^6 z^{-2} +8 a^6+a^5 z^7+3 a^5 z^5-12 a^5 z^3+3 a^5 z^{-3} +14 a^5 z-9 a^5 z^{-1} +a^4 z^8+2 a^4 z^6+z^6 a^{-4} -2 z^4 a^{-4} -12 a^4 z^2-6 a^4 z^{-2} +15 a^4+a^3 z^9+a^3 z^7+4 z^7 a^{-3} +3 a^3 z^5-11 z^5 a^{-3} -12 a^3 z^3+4 z^3 a^{-3} +3 a^3 z^{-3} +14 a^3 z-9 a^3 z^{-1} +a^2 z^{10}-a^2 z^8+6 z^8 a^{-2} +5 a^2 z^6-19 z^6 a^{-2} -3 a^2 z^4+13 z^4 a^{-2} -6 a^2 z^2-3 a^2 z^{-2} +8 a^2+5 a z^9+4 z^9 a^{-1} -14 a z^7-10 z^7 a^{-1} +17 a z^5+5 z^5 a^{-1} -8 a z^3+a z^{-3} +6 a z-4 a z^{-1} +z^{10}+4 z^8-16 z^6+12 z^4$ (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
10           1-1
8          3 3
6         41 -3
4        53  2
2       64   -2
0      85    3
-2     710     3
-4    44      0
-6   17       6
-8  44        0
-10 15         4
-12            0
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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