# L11n458

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n458 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2-2 t(1) t(4) t(3)^2+2 t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+3 t(1) t(4) t(3)-3 t(1) t(2) t(4) t(3)+3 t(2) t(4) t(3)-3 t(4) t(3)+t(2) t(4)^2-t(4)^2-t(1) t(2)+t(2)+t(1) t(2) t(4)-2 t(2) t(4)+2 t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $11 q^{9/2}-11 q^{7/2}+5 q^{5/2}-3 q^{3/2}+q^{21/2}-4 q^{19/2}+7 q^{17/2}-11 q^{15/2}+12 q^{13/2}-15 q^{11/2}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-9} - a^{-9} z^{-3} - a^{-9} z^{-1} -z^5 a^{-7} +3 a^{-7} z^{-3} +4 z a^{-7} +6 a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -3 a^{-5} z^{-3} -10 z a^{-5} -9 a^{-5} z^{-1} +3 z^3 a^{-3} + a^{-3} z^{-3} +6 z a^{-3} +4 a^{-3} z^{-1}$ (db) Kauffman polynomial $z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -11 z^5 a^{-11} +8 z^3 a^{-11} -2 z a^{-11} +5 z^8 a^{-10} -11 z^6 a^{-10} +2 z^4 a^{-10} +2 z^2 a^{-10} +2 z^9 a^{-9} +7 z^7 a^{-9} -33 z^5 a^{-9} +32 z^3 a^{-9} - a^{-9} z^{-3} -16 z a^{-9} +5 a^{-9} z^{-1} +11 z^8 a^{-8} -25 z^6 a^{-8} +9 z^4 a^{-8} +10 z^2 a^{-8} +3 a^{-8} z^{-2} -10 a^{-8} +2 z^9 a^{-7} +10 z^7 a^{-7} -41 z^5 a^{-7} +52 z^3 a^{-7} -3 a^{-7} z^{-3} -31 z a^{-7} +12 a^{-7} z^{-1} +6 z^8 a^{-6} -10 z^6 a^{-6} +2 z^4 a^{-6} +18 z^2 a^{-6} +6 a^{-6} z^{-2} -19 a^{-6} +7 z^7 a^{-5} -19 z^5 a^{-5} +34 z^3 a^{-5} -3 a^{-5} z^{-3} -27 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -3 z^4 a^{-4} +9 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +6 z^3 a^{-3} - a^{-3} z^{-3} -10 z a^{-3} +5 a^{-3} z^{-1}$ (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 \
0123456789χ
22         1-1
20        3 3
18       41 -3
16      73  4
14     76   -1
12    85    3
10   59     4
8  66      0
6 17       6
424        -2
23         3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=9$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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