# L10a78

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a78 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-4 t(1)^2 t(2)^3+4 t(1) t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-4 t(2)+1}{t(1) t(2)^2}$ (db) Jones polynomial $\frac{8}{q^{9/2}}-\frac{11}{q^{7/2}}-q^{5/2}+\frac{10}{q^{5/2}}+3 q^{3/2}-\frac{10}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{11/2}}-5 \sqrt{q}+\frac{8}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^5 z^5+3 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-10 a^3 z^3-10 a^3 z-3 a^3 z^{-1} +2 a z^5+7 a z^3-z^3 a^{-1} +6 a z-2 z a^{-1} +a z^{-1}$ (db) Kauffman polynomial $-z^3 a^9-3 z^4 a^8-6 z^5 a^7+5 z^3 a^7-2 z a^7-8 z^6 a^6+10 z^4 a^6-2 z^2 a^6-9 z^7 a^5+19 z^5 a^5-13 z^3 a^5+7 z a^5-2 a^5 z^{-1} -6 z^8 a^4+10 z^6 a^4+4 z^4 a^4-7 z^2 a^4+3 a^4-2 z^9 a^3-5 z^7 a^3+31 z^5 a^3-31 z^3 a^3+12 z a^3-3 a^3 z^{-1} -9 z^8 a^2+31 z^6 a^2-26 z^4 a^2+z^2 a^2+3 a^2-2 z^9 a+3 z^7 a+10 z^5 a-17 z^3 a+5 z a-a z^{-1} -3 z^8+13 z^6-17 z^4+6 z^2+1-z^7 a^{-1} +4 z^5 a^{-1} -5 z^3 a^{-1} +2 z a^{-1}$ (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 \
-6-5-4-3-2-101234χ
6          11
4         2 -2
2        31 2
0       52  -3
-2      53   2
-4     66    0
-6    54     1
-8   36      3
-10  35       -2
-12  3        3
-1413         -2
-161          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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