# L10a73

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a73 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 t(2) t(1)^2-2 t(1)^2+3 t(2)^2 t(1)-5 t(2) t(1)+3 t(1)-2 t(2)^2+3 t(2)}{t(1) t(2)}$ (db) Jones polynomial $\frac{5}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^{11} z-a^9 z^3-2 a^7 z^3-a^7 z+a^7 z^{-1} -2 a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z$ (db) Kauffman polynomial $a^{13} z^7-5 a^{13} z^5+7 a^{13} z^3-2 a^{13} z+2 a^{12} z^8-10 a^{12} z^6+15 a^{12} z^4-7 a^{12} z^2+a^{11} z^9-2 a^{11} z^7-4 a^{11} z^5+7 a^{11} z^3-a^{11} z+4 a^{10} z^8-15 a^{10} z^6+15 a^{10} z^4-5 a^{10} z^2+a^9 z^9-6 a^9 z^5+3 a^9 z^3+2 a^8 z^8-2 a^8 z^6-4 a^8 z^4+2 a^8 z^2+3 a^7 z^7-4 a^7 z^5-a^7 z^3+3 a^7 z-a^7 z^{-1} +3 a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+3 a^5 z^5-3 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^4-a^4 z^2+a^3 z^3-a^3 z$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-2          11
-4         21-1
-6        2  2
-8       32  -1
-10      42   2
-12     23    1
-14    44     0
-16   23      1
-18  13       -2
-20 12        1
-22 1         -1
-241          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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