# L10a75

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a75 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 u^2 v^2-2 u^2 v-2 u v^2+5 u v-2 u-2 v+3}{u v}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{5}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $z^3 a^{11}+2 z a^{11}-z^5 a^9-3 z^3 a^9-2 z a^9-z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $a^{15} z^5-3 a^{15} z^3+a^{15} z+2 a^{14} z^6-6 a^{14} z^4+3 a^{14} z^2+2 a^{13} z^7-5 a^{13} z^5+2 a^{13} z^3+2 a^{12} z^8-7 a^{12} z^6+10 a^{12} z^4-5 a^{12} z^2+a^{11} z^9-3 a^{11} z^7+5 a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^8-12 a^{10} z^6+20 a^{10} z^4-8 a^{10} z^2+a^9 z^9-4 a^9 z^7+9 a^9 z^5-6 a^9 z^3+a^9 z+a^8 z^8-2 a^8 z^6+2 a^8 z^4-a^8 z^2+a^7 z^7-a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 z^{-1}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-4          11
-6         110
-8        2  2
-10       21  -1
-12      42   2
-14     22    0
-16    44     0
-18   23      1
-20  13       -2
-22 12        1
-24 1         -1
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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