# L10n106

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n106 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(2)-1) (t(3)-1) (t(4)-1) (t(1) t(3) t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z a^7+a^7 z^{-3} +z^5 a^5+4 z^3 a^5+3 z a^5-2 a^5 z^{-1} -3 a^5 z^{-3} -z^7 a^3-5 z^5 a^3-7 z^3 a^3-2 z a^3+4 a^3 z^{-1} +3 a^3 z^{-3} +z^5 a+3 z^3 a-2 a z^{-1} -a z^{-3}$ (db) Kauffman polynomial $-z^2 a^8-4 z^3 a^7+3 z a^7+2 a^7 z^{-1} -a^7 z^{-3} -2 z^6 a^6+4 z^4 a^6-3 z^2 a^6+3 a^6 z^{-2} -4 a^6-4 z^7 a^5+16 z^5 a^5-24 z^3 a^5+11 z a^5+3 a^5 z^{-1} -3 a^5 z^{-3} -2 z^8 a^4+4 z^6 a^4+3 z^4 a^4-4 z^2 a^4+6 a^4 z^{-2} -7 a^4-7 z^7 a^3+28 z^5 a^3-32 z^3 a^3+11 z a^3+3 a^3 z^{-1} -3 a^3 z^{-3} -2 z^8 a^2+5 z^6 a^2+2 z^4 a^2-3 z^2 a^2+3 a^2 z^{-2} -4 a^2-3 z^7 a+12 z^5 a-12 z^3 a+3 z a+2 a z^{-1} -a z^{-3} -z^6+3 z^4-z^2$ (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      11 0
-2     52  3
-4   123   2
-6   63    3
-8 125     4
-10 32      1
-12 3       3
-141        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{6}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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