# L10n89

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n89 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) \left(t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $2 q^{-10} -4 q^{-9} +7 q^{-8} -8 q^{-7} +9 q^{-6} -7 q^{-5} +7 q^{-4} -3 q^{-3} + q^{-2}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{10} z^{-2} +2 a^{10}-5 z^2 a^8-2 a^8 z^{-2} -8 a^8+3 z^4 a^6+8 z^2 a^6+a^6 z^{-2} +6 a^6+z^4 a^4+z^2 a^4$ (db) Kauffman polynomial $3 a^{12} z^4-5 a^{12} z^2+2 a^{12}+a^{11} z^7+a^{11} z^5-2 a^{11} z^3+a^{10} z^8+2 a^{10} z^6-3 a^{10} z^4+a^{10} z^{-2} -2 a^{10}+5 a^9 z^7-3 a^9 z^5-8 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^8+8 a^8 z^6-18 a^8 z^4+16 a^8 z^2+2 a^8 z^{-2} -9 a^8+4 a^7 z^7-a^7 z^5-8 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +6 a^6 z^6-11 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -6 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^4 z^2$ (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 \
-8-7-6-5-4-3-2-10χ
-3        11
-5       31-2
-7      4  4
-9     33  0
-11    64   2
-13   45    1
-15  34     -1
-17 14      3
-1913       -2
-212        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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