# L10n110

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n110 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(3)^2-t(1) t(2) t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-t(2) t(4)^2 t(3)+t(4)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+2 t(1) t(4) t(3)-2 t(1) t(2) t(4) t(3)+2 t(2) t(4) t(3)-2 t(4) t(3)+t(2) t(4)^2-t(4)^2+t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-6 q^{9/2}+6 q^{7/2}-10 q^{5/2}+7 q^{3/2}-\frac{3}{q^{3/2}}-q^{13/2}+3 q^{11/2}-8 \sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-5} - a^{-5} z^{-3} -z a^{-5} -2 a^{-5} z^{-1} +z^5 a^{-3} +3 z^3 a^{-3} +3 a^{-3} z^{-3} +7 z a^{-3} +7 a^{-3} z^{-1} -4 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-9 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 z^8 a^{-2} -2 z^8 a^{-4} -5 z^7 a^{-1} -9 z^7 a^{-3} -4 z^7 a^{-5} -z^6 a^{-2} -z^6 a^{-4} -3 z^6 a^{-6} -3 z^6+17 z^5 a^{-1} +27 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +11 z^4 a^{-2} +11 z^4 a^{-4} +6 z^4 a^{-6} +6 z^4-6 a z^3-33 z^3 a^{-1} -36 z^3 a^{-3} -7 z^3 a^{-5} +2 z^3 a^{-7} -23 z^2 a^{-2} -13 z^2 a^{-4} -z^2 a^{-6} -11 z^2+11 a z+27 z a^{-1} +25 z a^{-3} +8 z a^{-5} -z a^{-7} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3}$ (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 \
-2-10123456χ
14        11
12       2 -2
10      41 3
8     44  0
6    62   4
4   36    3
2  54     1
0 15      4
-223       -1
-43        3
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-2$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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