# L10n101

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n101 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(4)-1) \left(-t(4)^2+t(1) t(4)+t(2) t(4)-t(1) t(3) t(4)+t(1) t(2) t(3) t(4)-t(2) t(3) t(4)-t(4)+t(1) t(2) t(3)\right)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}}$ (db) Jones polynomial $2 q^{9/2}-4 q^{7/2}+2 q^{5/2}-4 q^{3/2}-q^{17/2}-q^{13/2}-3 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $- a^{-9} z^{-3} - a^{-9} z^{-1} +3 a^{-7} z^{-3} +2 z a^{-7} +5 a^{-7} z^{-1} -3 a^{-5} z^{-3} -3 z a^{-5} -6 a^{-5} z^{-1} -z^5 a^{-3} -3 z^3 a^{-3} + a^{-3} z^{-3} -z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^7 a^{-9} -7 z^5 a^{-9} +15 z^3 a^{-9} - a^{-9} z^{-3} -14 z a^{-9} +6 a^{-9} z^{-1} +z^6 a^{-8} -9 z^4 a^{-8} +19 z^2 a^{-8} +3 a^{-8} z^{-2} -13 a^{-8} +z^7 a^{-7} -10 z^5 a^{-7} +28 z^3 a^{-7} -3 a^{-7} z^{-3} -27 z a^{-7} +14 a^{-7} z^{-1} +2 z^6 a^{-6} -14 z^4 a^{-6} +33 z^2 a^{-6} +6 a^{-6} z^{-2} -24 a^{-6} +z^7 a^{-5} -4 z^5 a^{-5} +10 z^3 a^{-5} -3 a^{-5} z^{-3} -16 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -10 z^4 a^{-4} +14 z^2 a^{-4} +3 a^{-4} z^{-2} -11 a^{-4} +z^7 a^{-3} -6 z^3 a^{-3} - a^{-3} z^{-3} +3 a^{-3} z^{-1} +2 z^6 a^{-2} -5 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-1} - a^{-1} z^{-1}$ (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 \
-2-1012345678χ
18          11
16          11
14        1  1
12      4    4
10     351   1
8    2      2
6    31     2
4  42       2
2 13        2
0 1         -1
-21          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}^{5}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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