# L10n30

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n30 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^5-3 t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $\frac{6}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{5/2}}+q^{3/2}-\frac{7}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{5}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+4 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-9 z a^3-4 a^3 z^{-1} +z^5 a+3 z^3 a+2 z a+a z^{-1}$ (db) Kauffman polynomial $-3 z^2 a^8+a^8-z^5 a^7-5 z^3 a^7+4 z a^7-a^7 z^{-1} -5 z^6 a^6+11 z^4 a^6-13 z^2 a^6+4 a^6-6 z^7 a^5+18 z^5 a^5-24 z^3 a^5+16 z a^5-4 a^5 z^{-1} -2 z^8 a^4-3 z^6 a^4+21 z^4 a^4-21 z^2 a^4+7 a^4-9 z^7 a^3+29 z^5 a^3-28 z^3 a^3+15 z a^3-4 a^3 z^{-1} -2 z^8 a^2+z^6 a^2+13 z^4 a^2-14 z^2 a^2+4 a^2-3 z^7 a+10 z^5 a-9 z^3 a+3 z a-a z^{-1} -z^6+3 z^4-3 z^2+1$ (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 \
-5-4-3-2-10123χ
4        1-1
2       2 2
0      31 -2
-2     42  2
-4    44   0
-6   43    1
-8  24     2
-10 34      -1
-12 3       3
-142        -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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