# L11a537

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a537 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-t(1) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-2 t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2-t(1) t(3)^2-t(2) t(3)^2+2 t(1) t(4) t(3)^2-t(1) t(2) t(4) t(3)^2+3 t(2) t(4) t(3)^2-2 t(4) t(3)^2+t(3)^2+2 t(1) t(4)^2 t(3)-2 t(1) t(2) t(4)^2 t(3)+3 t(2) t(4)^2 t(3)-t(4)^2 t(3)+3 t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-4 t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-4 t(2) t(4) t(3)+2 t(4) t(3)-2 t(3)-t(1) t(4)^2+t(1) t(2) t(4)^2-t(2) t(4)^2-2 t(1)+t(1) t(2)-t(2)+3 t(1) t(4)-2 t(1) t(2) t(4)+2 t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+10 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-8 a^3 z+14 a z-9 z a^{-1} +2 z a^{-3} +2 a^5 z^{-1} -8 a^3 z^{-1} +11 a z^{-1} -6 a^{-1} z^{-1} + a^{-3} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3}$ (db) Kauffman polynomial $z^4 a^{-6} +a^5 z^7-5 a^5 z^5+4 z^5 a^{-5} +10 a^5 z^3-z^3 a^{-5} -a^5 z^{-3} -10 a^5 z+5 a^5 z^{-1} +2 a^4 z^8-6 a^4 z^6+9 z^6 a^{-4} +3 a^4 z^4-8 z^4 a^{-4} +7 a^4 z^2+3 z^2 a^{-4} +3 a^4 z^{-2} -9 a^4- a^{-4} +2 a^3 z^9+a^3 z^7+13 z^7 a^{-3} -24 a^3 z^5-19 z^5 a^{-3} +45 a^3 z^3+13 z^3 a^{-3} -3 a^3 z^{-3} -35 a^3 z-6 z a^{-3} +14 a^3 z^{-1} +2 a^{-3} z^{-1} +a^2 z^{10}+7 a^2 z^8+11 z^8 a^{-2} -23 a^2 z^6-10 z^6 a^{-2} +6 a^2 z^4-10 z^4 a^{-2} +24 a^2 z^2+14 z^2 a^{-2} +6 a^2 z^{-2} -21 a^2-6 a^{-2} +7 a z^9+5 z^9 a^{-1} -a z^7+12 z^7 a^{-1} -48 a z^5-52 z^5 a^{-1} +74 a z^3+53 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -49 a z-30 z a^{-1} +18 a z^{-1} +11 a^{-1} z^{-1} +z^{10}+16 z^8-36 z^6+2 z^4+28 z^2+3 z^{-2} -18$ (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         61 -5
6        83  5
4       107   -3
2      117    4
0     913     4
-2    88      0
-4   512       7
-6  25        -3
-8 16         5
-10 1          -1
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{11}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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