# L10n35

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n35 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-q^{19/2}+3 q^{17/2}-5 q^{15/2}+8 q^{13/2}-8 q^{11/2}+8 q^{9/2}-8 q^{7/2}+4 q^{5/2}-3 q^{3/2}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z a^{-9} - a^{-9} z^{-1} +3 z^3 a^{-7} +7 z a^{-7} +5 a^{-7} z^{-1} -2 z^5 a^{-5} -8 z^3 a^{-5} -13 z a^{-5} -8 a^{-5} z^{-1} +3 z^3 a^{-3} +7 z a^{-3} +4 a^{-3} z^{-1}$ (db) Kauffman polynomial $z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} -2 a^{-10} +3 z^7 a^{-9} -3 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} + a^{-9} z^{-1} +z^8 a^{-8} +8 z^6 a^{-8} -26 z^4 a^{-8} +24 z^2 a^{-8} -9 a^{-8} +7 z^7 a^{-7} -14 z^5 a^{-7} +12 z^3 a^{-7} -9 z a^{-7} +5 a^{-7} z^{-1} +z^8 a^{-6} +8 z^6 a^{-6} -25 z^4 a^{-6} +31 z^2 a^{-6} -14 a^{-6} +4 z^7 a^{-5} -10 z^5 a^{-5} +20 z^3 a^{-5} -19 z a^{-5} +8 a^{-5} z^{-1} +3 z^6 a^{-4} -6 z^4 a^{-4} +12 z^2 a^{-4} -8 a^{-4} +6 z^3 a^{-3} -10 z a^{-3} +4 a^{-3} z^{-1}$ (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 \
012345678χ
20        11
18       2 -2
16      31 2
14     52  -3
12    33   0
10   55    0
8  33     0
6 15      4
423       -1
23        3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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