# L11a429

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a429 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+2 t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-2 t(2)^2-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-5 t(1) t(2)+9 t(1) t(3) t(2)-9 t(3) t(2)+4 t(2)+2 t(1) t(3)^2-2 t(3)^2+2 t(1)-4 t(1) t(3)+5 t(3)-2}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $q^9-4 q^8+9 q^7-14 q^6+20 q^5-22 q^4+23 q^3-19 q^2- q^{-2} +15 q+4 q^{-1} -8$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} -2 z^4 a^{-6} -z^4+4 z^2 a^{-2} -4 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} -z^2+4 a^{-2} -6 a^{-4} +2 a^{-6} + a^{-2} z^{-2} -2 a^{-4} z^{-2} + a^{-6} z^{-2}$ (db) Kauffman polynomial $z^6 a^{-10} -2 z^4 a^{-10} +z^2 a^{-10} +4 z^7 a^{-9} -9 z^5 a^{-9} +5 z^3 a^{-9} +7 z^8 a^{-8} -16 z^6 a^{-8} +11 z^4 a^{-8} -4 z^2 a^{-8} + a^{-8} +6 z^9 a^{-7} -7 z^7 a^{-7} -4 z^5 a^{-7} +3 z^3 a^{-7} +2 z^{10} a^{-6} +12 z^8 a^{-6} -34 z^6 a^{-6} +23 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +12 z^9 a^{-5} -18 z^7 a^{-5} +6 z^5 a^{-5} -5 z^3 a^{-5} +6 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +13 z^8 a^{-4} -28 z^6 a^{-4} +12 z^4 a^{-4} +8 z^2 a^{-4} +2 a^{-4} z^{-2} -8 a^{-4} +6 z^9 a^{-3} -9 z^5 a^{-3} +z^3 a^{-3} +6 z a^{-3} -2 a^{-3} z^{-1} +8 z^8 a^{-2} -7 z^6 a^{-2} -4 z^4 a^{-2} +8 z^2 a^{-2} + a^{-2} z^{-2} -5 a^{-2} +7 z^7 a^{-1} +a z^5-9 z^5 a^{-1} -a z^3+3 z^3 a^{-1} +4 z^6-6 z^4+3 z^2$ (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 \
-3-2-1012345678χ
19           11
17          3 -3
15         61 5
13        94  -5
11       115   6
9      119    -2
7     1211     1
5    812      4
3   711       -4
1  310        7
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{12}$ $r=3$ ${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=4$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$ ${\mathbb Z}^{11}$ $r=5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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