# L10a1

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a1 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^4-3 v^3+5 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $-\frac{9}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{17}{q^{5/2}}-8 q^{3/2}+\frac{17}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+12 \sqrt{q}-\frac{17}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-4 a^3 z^3+6 a z^3-2 z^3 a^{-1} +a z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1}$ (db) Kauffman polynomial $-2 a^3 z^9-2 a z^9-7 a^4 z^8-13 a^2 z^8-6 z^8-9 a^5 z^7-17 a^3 z^7-15 a z^7-7 z^7 a^{-1} -5 a^6 z^6+3 a^4 z^6+15 a^2 z^6-4 z^6 a^{-2} +3 z^6-a^7 z^5+15 a^5 z^5+40 a^3 z^5+37 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+10 a^4 z^4+5 a^2 z^4+6 z^4 a^{-2} +7 z^4-6 a^5 z^3-23 a^3 z^3-26 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} -5 z^2+a^3 z+3 a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a z^{-1}$ (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 \
-6-5-4-3-2-101234χ
8          11
6         3 -3
4        51 4
2       73  -4
0      105   5
-2     99    0
-4    88     0
-6   59      4
-8  48       -4
-10 15        4
-12 4         -4
-141          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{10}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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