# L10a169

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a169 at Knotilus! Compare L10n107.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) (w-1)^2 (x-1)^2}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $-q^{7/2}+5 q^{5/2}-11 q^{3/2}+15 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-3 a^3 z^3+3 a z^3-z^3 a^{-1} -a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3}$ (db) Kauffman polynomial $a^7 z^5+5 a^6 z^6-4 a^6 z^4+11 a^5 z^7-17 a^5 z^5+8 a^5 z^3-a^5 z^{-3} +a^5 z^{-1} +11 a^4 z^8-12 a^4 z^6+3 a^4 z^{-2} -2 a^4+4 a^3 z^9+19 a^3 z^7-48 a^3 z^5+z^5 a^{-3} +24 a^3 z^3-3 a^3 z^{-3} +a^3 z+22 a^2 z^8-34 a^2 z^6+5 z^6 a^{-2} +8 a^2 z^4-4 z^4 a^{-2} +6 a^2 z^{-2} -3 a^2+4 a z^9+19 a z^7+11 z^7 a^{-1} -48 a z^5-17 z^5 a^{-1} +24 a z^3+8 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} +a z+ a^{-1} z^{-1} +11 z^8-12 z^6+3 z^{-2} -2$ (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 \
-6-5-4-3-2-101234χ
8          11
6         4 -4
4        71 6
2       84  -4
0      147   7
-2     1012    2
-4    1210     2
-6   714      7
-8  48       -4
-10 17        6
-12 4         -4
-141          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{12}$ $r=-1$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{10}$ $r=0$ ${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$ ${\mathbb Z}^{14}$ $r=1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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