# L10n104

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n104 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1) t(2)-t(3) t(4)) (t(3) t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{17/2}}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -2 a^7 z-6 a^7 z^{-1} +a^5 z^3+3 a^5 z^{-3} +6 a^5 z+9 a^5 z^{-1} -a^3 z^3-a^3 z^{-3} -4 a^3 z-4 a^3 z^{-1}$ (db) Kauffman polynomial $a^9 z^7-7 a^9 z^5+15 a^9 z^3-a^9 z^{-3} -12 a^9 z+5 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+3 a^8 z^{-2} -10 a^8+a^7 z^7-8 a^7 z^5+21 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^6-8 a^6 z^4+20 a^6 z^2+6 a^6 z^{-2} -19 a^6-a^5 z^5+7 a^5 z^3-3 a^5 z^{-3} -15 a^5 z+12 a^5 z^{-1} -a^4 z^4+6 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^3-a^3 z^{-3} -4 a^3 z+5 a^3 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-2        11
-4      1 12
-6     14  3
-8      3  3
-10   13    2
-12    2    2
-14  1      1
-161        1
-181        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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