# L10n103

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n103 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z^{-3} -3 a^7 z^{-3} -3 a^7 z^{-1} -a^5 z^3+3 a^5 z^{-3} +2 a^5 z+6 a^5 z^{-1} +a^3 z^5+2 a^3 z^3-a^3 z^{-3} -a^3 z-3 a^3 z^{-1} -a z^3-a z$ (db) Kauffman polynomial $a^9 z^3-a^9 z^{-3} -3 a^9 z+3 a^9 z^{-1} +a^8 z^4+2 a^8 z^2+3 a^8 z^{-2} -6 a^8+a^7 z^7-4 a^7 z^5+12 a^7 z^3-3 a^7 z^{-3} -9 a^7 z+6 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+a^6 z^4+9 a^6 z^2+6 a^6 z^{-2} -11 a^6+4 a^5 z^7-12 a^5 z^5+16 a^5 z^3-3 a^5 z^{-3} -11 a^5 z+6 a^5 z^{-1} +a^4 z^8+a^4 z^6-7 a^4 z^4+8 a^4 z^2+3 a^4 z^{-2} -6 a^4+3 a^3 z^7-7 a^3 z^5+3 a^3 z^3-a^3 z^{-3} -4 a^3 z+3 a^3 z^{-1} +3 a^2 z^6-7 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-6-5-4-3-2-1012χ
2        11
0       2 -2
-2      31 2
-4     34  1
-6    41   3
-8  113    3
-10  64     2
-12 16      5
-14         0
-161        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{6}$ ${\mathbb Z}^{6}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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