# L11a461

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a461 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3-t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+4 t(1) t(3)^2 t(2)^2-4 t(3)^2 t(2)^2-2 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)-3 t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+2 t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $q^{10}-3 q^9+7 q^8-10 q^7+14 q^6-15 q^5+16 q^4-13 q^3+10 q^2-6 q+4- q^{-1}$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^4 a^{-8} +3 z^2 a^{-8} + a^{-8} z^{-2} +2 a^{-8} -2 z^6 a^{-6} -8 z^4 a^{-6} -9 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +8 z^4 a^{-4} +5 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} +2 a^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -2 z^3 a^{-11} +6 z^6 a^{-10} -8 z^4 a^{-10} +6 z^2 a^{-10} -2 a^{-10} +7 z^7 a^{-9} -8 z^5 a^{-9} +2 z^3 a^{-9} +z a^{-9} +7 z^8 a^{-8} -11 z^6 a^{-8} +8 z^4 a^{-8} -6 z^2 a^{-8} - a^{-8} z^{-2} +3 a^{-8} +5 z^9 a^{-7} -6 z^7 a^{-7} -4 z^5 a^{-7} +9 z^3 a^{-7} -8 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +5 z^8 a^{-6} -29 z^6 a^{-6} +37 z^4 a^{-6} -24 z^2 a^{-6} -2 a^{-6} z^{-2} +9 a^{-6} +10 z^9 a^{-5} -33 z^7 a^{-5} +29 z^5 a^{-5} -z^3 a^{-5} -8 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +2 z^8 a^{-4} -28 z^6 a^{-4} +36 z^4 a^{-4} -13 z^2 a^{-4} - a^{-4} z^{-2} +3 a^{-4} +5 z^9 a^{-3} -19 z^7 a^{-3} +19 z^5 a^{-3} -5 z^3 a^{-3} +z a^{-3} +4 z^8 a^{-2} -16 z^6 a^{-2} +16 z^4 a^{-2} -2 z^2 a^{-2} -2 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +z^3 a^{-1}$ (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 \
-3-2-1012345678χ
21           11
19          31-2
17         4  4
15        63  -3
13       84   4
11      87    -1
9     87     1
7    58      3
5   58       -3
3  37        4
1 13         -2
-1 3          3
-31           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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