# L11a420

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a420 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^2 t(3)^4-t(2)^2 t(3)^4-t(1) t(2) t(3)^4-2 t(1) t(2)^2 t(3)^3+3 t(2)^2 t(3)^3-t(1) t(3)^3+3 t(1) t(2) t(3)^3-2 t(2) t(3)^3+2 t(1) t(2)^2 t(3)^2-3 t(2)^2 t(3)^2+3 t(1) t(3)^2-4 t(1) t(2) t(3)^2+4 t(2) t(3)^2-2 t(3)^2+t(2)^2 t(3)-3 t(1) t(3)+2 t(1) t(2) t(3)-3 t(2) t(3)+2 t(3)+t(1)+t(2)-1}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $q^7-3 q^6+7 q^5-10 q^4- q^{-4} +14 q^3+2 q^{-3} -14 q^2-5 q^{-2} +15 q+9 q^{-1} -11$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^8 a^{-2} -6 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-15 z^4 a^{-2} +4 z^4 a^{-4} +10 z^4-4 a^2 z^2-21 z^2 a^{-2} +6 z^2 a^{-4} +18 z^2-4 a^2-16 a^{-2} +5 a^{-4} +15-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2}$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10}+2 a z^9+6 z^9 a^{-1} +4 z^9 a^{-3} +2 a^2 z^8+9 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8+a^3 z^7-3 a z^7-12 z^7 a^{-1} -z^7 a^{-3} +7 z^7 a^{-5} -8 a^2 z^6-35 z^6 a^{-2} -12 z^6 a^{-4} +6 z^6 a^{-6} -25 z^6-5 a^3 z^5-12 a z^5-11 z^5 a^{-1} -15 z^5 a^{-3} -8 z^5 a^{-5} +3 z^5 a^{-7} +10 a^2 z^4+47 z^4 a^{-2} +11 z^4 a^{-4} -8 z^4 a^{-6} +z^4 a^{-8} +37 z^4+8 a^3 z^3+28 a z^3+39 z^3 a^{-1} +23 z^3 a^{-3} +2 z^3 a^{-5} -2 z^3 a^{-7} -7 a^2 z^2-39 z^2 a^{-2} -9 z^2 a^{-4} +6 z^2 a^{-6} -z^2 a^{-8} -30 z^2-5 a^3 z-21 a z-33 z a^{-1} -16 z a^{-3} +z a^{-5} +4 a^2+20 a^{-2} +6 a^{-4} -2 a^{-6} +17+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 z^{-2}$ (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 \
-5-4-3-2-10123456χ
15           11
13          31-2
11         4  4
9        63  -3
7       84   4
5      77    0
3     87     1
1    59      4
-1   46       -2
-3  15        4
-5 14         -3
-7 1          1
-91           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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