# L11a439

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a439 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1)^2 (t(3)-1)^2 (t(2) t(3)+1)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $q^{10}-4 q^9+9 q^8-14 q^7+18 q^6-20 q^5+21 q^4-16 q^3+13 q^2-7 q+4- q^{-1}$ (db) Signature 4 (db) HOMFLY-PT polynomial $z^4 a^{-8} +2 z^2 a^{-8} + a^{-8} -2 z^6 a^{-6} -7 z^4 a^{-6} -7 z^2 a^{-6} + a^{-6} z^{-2} -2 a^{-6} +z^8 a^{-4} +5 z^6 a^{-4} +9 z^4 a^{-4} +6 z^2 a^{-4} -2 a^{-4} z^{-2} - a^{-4} -z^6 a^{-2} -3 z^4 a^{-2} -z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2}$ (db) Kauffman polynomial $z^4 a^{-12} +4 z^5 a^{-11} -z^3 a^{-11} +9 z^6 a^{-10} -8 z^4 a^{-10} +3 z^2 a^{-10} - a^{-10} +13 z^7 a^{-9} -18 z^5 a^{-9} +9 z^3 a^{-9} -2 z a^{-9} +12 z^8 a^{-8} -15 z^6 a^{-8} +4 z^2 a^{-8} - a^{-8} +7 z^9 a^{-7} +z^7 a^{-7} -27 z^5 a^{-7} +21 z^3 a^{-7} -6 z a^{-7} +2 z^{10} a^{-6} +15 z^8 a^{-6} -49 z^6 a^{-6} +38 z^4 a^{-6} -9 z^2 a^{-6} + a^{-6} z^{-2} - a^{-6} +12 z^9 a^{-5} -28 z^7 a^{-5} +8 z^5 a^{-5} +10 z^3 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +2 z^{10} a^{-4} +7 z^8 a^{-4} -40 z^6 a^{-4} +46 z^4 a^{-4} -15 z^2 a^{-4} +2 a^{-4} z^{-2} -2 a^{-4} +5 z^9 a^{-3} -15 z^7 a^{-3} +10 z^5 a^{-3} +z^3 a^{-3} +2 z a^{-3} -2 a^{-3} z^{-1} +4 z^8 a^{-2} -15 z^6 a^{-2} +17 z^4 a^{-2} -5 z^2 a^{-2} + a^{-2} z^{-2} -2 a^{-2} +z^7 a^{-1} -3 z^5 a^{-1} +2 z^3 a^{-1}$ (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          3 -3
17         61 5
15        83  -5
13       106   4
11      1210    -2
9     98     1
7    712      5
5   69       -3
3  39        6
1 14         -3
-1 3          3
-31           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$ ${\mathbb Z}^{12}$ $r=4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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