# L11a535

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a535 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-2 t(1) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-2 t(2) t(4)^2 t(3)^2+2 t(4)^2 t(3)^2+t(1) t(4) t(3)^2-t(1) t(2) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(4)^2 t(3)-t(4)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-3 t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-3 t(2) t(4) t(3)+2 t(4) t(3)-t(3)-2 t(1)+2 t(1) t(2)-2 t(2)+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-4 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{5}{q^{7/2}}-11 q^{5/2}+\frac{6}{q^{5/2}}+13 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-15 \sqrt{q}+\frac{12}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+12 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-11 a^3 z+16 a z-6 z a^{-1} +3 a^5 z^{-1} -10 a^3 z^{-1} +11 a z^{-1} -4 a^{-1} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3}$ (db) Kauffman polynomial $-a^2 z^{10}-z^{10}-a^3 z^9-5 a z^9-4 z^9 a^{-1} -a^4 z^8+a^2 z^8-7 z^8 a^{-2} -5 z^8-a^5 z^7-a^3 z^7+12 a z^7+4 z^7 a^{-1} -8 z^7 a^{-3} +2 a^4 z^6-2 a^2 z^6+10 z^6 a^{-2} -7 z^6 a^{-4} +13 z^6+6 a^5 z^5+17 a^3 z^5-2 a z^5+9 z^5 a^{-3} -4 z^5 a^{-5} +6 a^4 z^4+19 a^2 z^4-4 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +z^4-13 a^5 z^3-32 a^3 z^3-16 a z^3+3 z^3 a^{-5} -17 a^4 z^2-33 a^2 z^2-16 z^2+13 a^5 z+28 a^3 z+21 a z+3 z a^{-1} -3 z a^{-3} +13 a^4+24 a^2- a^{-2} +11-6 a^5 z^{-1} -14 a^3 z^{-1} -12 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3}$ (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          3 3
8         41 -3
6        73  4
4       97   -2
2      64    2
0     710     3
-2    55      0
-4   28       6
-6  34        -1
-8 15         4
-10            0
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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