# L10a107

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a107 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^2 t(1)^3-2 t(2) t(1)^3+2 t(2)^3 t(1)^2-7 t(2)^2 t(1)^2+7 t(2) t(1)^2-3 t(1)^2-3 t(2)^3 t(1)+7 t(2)^2 t(1)-7 t(2) t(1)+2 t(1)-2 t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $\frac{15}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{11}{q^{5/2}}-\frac{7}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{14}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9+a^9 z^{-1} -3 z^3 a^7-4 z a^7-a^7 z^{-1} +2 z^5 a^5+4 z^3 a^5+2 z a^5+z^5 a^3-2 z a^3-z^3 a-z a$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-12 a^9 z^5+12 a^9 z^3-8 a^9 z+a^9 z^{-1} +5 a^8 z^8-3 a^8 z^6-7 a^8 z^4+6 a^8 z^2-a^8+2 a^7 z^9+8 a^7 z^7-21 a^7 z^5+17 a^7 z^3-7 a^7 z+a^7 z^{-1} +10 a^6 z^8-15 a^6 z^6+5 a^6 z^4+a^6 z^2+2 a^5 z^9+7 a^5 z^7-17 a^5 z^5+10 a^5 z^3-a^5 z+5 a^4 z^8-6 a^4 z^6+3 a^4 z^4-2 a^4 z^2+5 a^3 z^7-8 a^3 z^5+5 a^3 z^3-2 a^3 z+3 a^2 z^6-5 a^2 z^4+2 a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        51 4
-4       73  -4
-6      74   3
-8     87    -1
-10    67     -1
-12   58      3
-14  36       -3
-16  5        5
-1813         -2
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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