# L10a133

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a133 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-2 t(1) t(3)^2 t(2)+t(3)^2 t(2)-t(1) t(2)+3 t(1) t(3) t(2)-3 t(3) t(2)+2 t(2)+2 t(1)-2 t(1) t(3)+t(3)-2}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $q^{10}-3 q^9+5 q^8-7 q^7+9 q^6-8 q^5+8 q^4-5 q^3+4 q^2-q+1$ (db) Signature 4 (db) HOMFLY-PT polynomial $-z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -4 z^4 a^{-4} -3 z^4 a^{-6} +z^4 a^{-8} +4 z^2 a^{-2} -6 z^2 a^{-4} -z^2 a^{-6} +2 z^2 a^{-8} +4 a^{-2} -6 a^{-4} +2 a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} -z^2 a^{-12} +3 z^5 a^{-11} -4 z^3 a^{-11} +4 z^6 a^{-10} -5 z^4 a^{-10} +z^2 a^{-10} +4 z^7 a^{-9} -6 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -6 z^6 a^{-8} +9 z^4 a^{-8} -5 z^2 a^{-8} + a^{-8} +z^9 a^{-7} +z^7 a^{-7} -3 z^5 a^{-7} +2 z^3 a^{-7} +4 z^8 a^{-6} -11 z^6 a^{-6} +11 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +z^9 a^{-5} -2 z^7 a^{-5} +4 z^5 a^{-5} -9 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +z^8 a^{-4} -9 z^4 a^{-4} +13 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +z^7 a^{-3} -2 z^5 a^{-3} -3 z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +z^6 a^{-2} -5 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2}$ (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 \
-2-1012345678χ
21          11
19         2 -2
17        31 2
15       53  -2
13      42   2
11     45    1
9    44     0
7   25      3
5  23       -1
3 14        3
1           0
-11          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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