# L10a23

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a23 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) \left(v^2+1\right) \left(v^2-3 v+1\right)}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-2 a^7 z-a^7 z^{-1} +2 a^5 z^5+6 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3-2 a^3 z+a z^5+2 a z^3-a z-a z^{-1}$ (db) Kauffman polynomial $a^{10} z^4-a^{10} z^2+3 a^9 z^5-2 a^9 z^3+6 a^8 z^6-8 a^8 z^4+6 a^8 z^2-2 a^8+7 a^7 z^7-10 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +5 a^6 z^8-2 a^6 z^6-9 a^6 z^4+12 a^6 z^2-5 a^6+2 a^5 z^9+7 a^5 z^7-23 a^5 z^5+20 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +10 a^4 z^8-23 a^4 z^6+12 a^4 z^4+3 a^4 z^2-3 a^4+2 a^3 z^9+4 a^3 z^7-22 a^3 z^5+18 a^3 z^3-4 a^3 z+5 a^2 z^8-14 a^2 z^6+10 a^2 z^4-2 a^2 z^2+a^2+4 a z^7-12 a z^5+8 a z^3+a z-a z^{-1} +z^6-2 z^4$ (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 \
-7-6-5-4-3-2-10123χ
4          1-1
2         3 3
0        31 -2
-2       73  4
-4      75   -2
-6     75    2
-8    57     2
-10   57      -2
-12  25       3
-14 15        -4
-16 2         2
-181          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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