# L10a67

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a67 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^2 t(2)^4-4 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-5 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-4 t(2)+1}{t(1) t(2)^2}$ (db) Jones polynomial $-\frac{7}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{21/2}}+\frac{2}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{6}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{7}{q^{11/2}}-\frac{1}{\sqrt{q}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $z^3 a^9+3 z a^9+a^9 z^{-1} -2 z^5 a^7-8 z^3 a^7-8 z a^7-2 a^7 z^{-1} +z^7 a^5+5 z^5 a^5+8 z^3 a^5+6 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-4 z a^3-a^3 z^{-1}$ (db) Kauffman polynomial $a^{13} z^3-a^{13} z+2 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-2 a^{11} z^3+a^{11} z+4 a^{10} z^6-6 a^{10} z^4+4 a^{10} z^2+4 a^9 z^7-8 a^9 z^5+6 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-6 a^8 z^6+a^8 z^4+a^7 z^9+3 a^7 z^7-20 a^7 z^5+24 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +5 a^6 z^8-19 a^6 z^6+20 a^6 z^4-8 a^6 z^2+a^6+a^5 z^9-14 a^5 z^5+23 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4-3 a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-5 a^3 z+a^3 z^{-1}$ (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 \
-8-7-6-5-4-3-2-1012χ
0          11
-2         1 -1
-4        31 2
-6       32  -1
-8      42   2
-10     33    0
-12    44     0
-14   23      1
-16  24       -2
-18 13        2
-20 1         -1
-221          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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