# L11a474

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a474 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1) \left(t(2)^2 t(3)^3-t(2) t(3)^3-t(2)^2 t(3)^2+t(2) t(3)^2-t(3)^2+t(2)^2 t(3)-t(2) t(3)+t(3)+t(2)-1\right)}{\sqrt{t(1)} t(2) t(3)^2}$ (db) Jones polynomial $-q^9+3 q^8-6 q^7+9 q^6-11 q^5+13 q^4-11 q^3+11 q^2-7 q+5-2 q^{-1} + q^{-2}$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-6} -4 z^4 a^{-6} -5 z^2 a^{-6} -3 a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +14 z^4 a^{-4} +17 z^2 a^{-4} + a^{-4} z^{-2} +9 a^{-4} -2 z^6 a^{-2} -10 z^4 a^{-2} -16 z^2 a^{-2} -2 a^{-2} z^{-2} -10 a^{-2} +z^4+4 z^2+ z^{-2} +4$ (db) Kauffman polynomial $z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +2 z^8 a^{-2} +10 z^8 a^{-4} +9 z^8 a^{-6} +z^8-10 z^7 a^{-1} -28 z^7 a^{-3} -8 z^7 a^{-5} +10 z^7 a^{-7} -29 z^6 a^{-2} -54 z^6 a^{-4} -22 z^6 a^{-6} +9 z^6 a^{-8} -6 z^6+15 z^5 a^{-1} +27 z^5 a^{-3} -13 z^5 a^{-5} -19 z^5 a^{-7} +6 z^5 a^{-9} +61 z^4 a^{-2} +76 z^4 a^{-4} +12 z^4 a^{-6} -13 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-5 z^3 a^{-1} +3 z^3 a^{-3} +19 z^3 a^{-5} +6 z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -47 z^2 a^{-2} -44 z^2 a^{-4} -5 z^2 a^{-6} +5 z^2 a^{-8} -13 z^2-4 z a^{-1} -9 z a^{-3} -6 z a^{-5} +z a^{-9} +15 a^{-2} +13 a^{-4} +2 a^{-6} - a^{-8} +6+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-4-3-2-101234567χ
19           1-1
17          2 2
15         41 -3
13        52  3
11       64   -2
9      75    2
7     68     2
5    55      0
3   48       4
1  13        -2
-1 14         3
-3 1          -1
-51           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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