# L11a511

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a511 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-2 t(1) t(2) t(3)^3+t(2) t(3)^3+2 t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-2 t(1) t(3)^2-3 t(1)^2 t(2) t(3)^2+5 t(1) t(2) t(3)^2-3 t(2) t(3)^2+t(3)^2-t(1)^2 t(3)-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-2 t(2)^2 t(3)+2 t(1) t(3)+3 t(1)^2 t(2) t(3)-5 t(1) t(2) t(3)+3 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+2 t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $-q^8+3 q^7-6 q^6+11 q^5-14 q^4+17 q^3-16 q^2+15 q-10+7 q^{-1} -3 q^{-2} + q^{-3}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} +z^6 a^{-4} +3 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} +4 a^{-4} +z^6 a^{-2} +z^4 a^{-2} +a^2 z^2-3 z^2 a^{-2} -2 a^{-2} z^{-2} +a^2-5 a^{-2} -2 z^4-3 z^2+ z^{-2} +1$ (db) Kauffman polynomial $z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +2 z^2 a^{-8} +5 z^7 a^{-7} -10 z^5 a^{-7} +6 z^3 a^{-7} -z a^{-7} +6 z^8 a^{-6} -14 z^6 a^{-6} +17 z^4 a^{-6} -9 z^2 a^{-6} +2 a^{-6} +4 z^9 a^{-5} -4 z^7 a^{-5} -2 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +z^{10} a^{-4} +10 z^8 a^{-4} -35 z^6 a^{-4} +53 z^4 a^{-4} -35 z^2 a^{-4} - a^{-4} z^{-2} +10 a^{-4} +7 z^9 a^{-3} -13 z^7 a^{-3} +7 z^5 a^{-3} +5 z^3 a^{-3} -8 z a^{-3} +2 a^{-3} z^{-1} +z^{10} a^{-2} +8 z^8 a^{-2} +a^2 z^6-27 z^6 a^{-2} -3 a^2 z^4+35 z^4 a^{-2} +3 a^2 z^2-29 z^2 a^{-2} -2 a^{-2} z^{-2} -a^2+12 a^{-2} +3 z^9 a^{-1} +3 a z^7-z^7 a^{-1} -8 a z^5-10 z^5 a^{-1} +5 a z^3+9 z^3 a^{-1} -6 z a^{-1} +2 a^{-1} z^{-1} +4 z^8-8 z^6+2 z^4-2 z^2- z^{-2} +4$ (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 \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        72  5
9       85   -3
7      96    3
5     89     1
3    78      -1
1   49       5
-1  36        -3
-3 15         4
-5 2          -2
-71           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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