# L10a87

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a87 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{4 u^2 v^2-4 u^2 v+u^2-4 u v^2+9 u v-4 u+v^2-4 v+4}{u v}$ (db) Jones polynomial $-\frac{6}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{6}{q^{21/2}}+\frac{9}{q^{19/2}}-\frac{11}{q^{17/2}}+\frac{12}{q^{15/2}}-\frac{11}{q^{13/2}}+\frac{8}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $z^3 a^{11}+z a^{11}-z^5 a^9-z^3 a^9+z a^9-2 z^5 a^7-5 z^3 a^7-2 z a^7+a^7 z^{-1} -z^5 a^5-3 z^3 a^5-3 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $-z^5 a^{15}+2 z^3 a^{15}-z a^{15}-3 z^6 a^{14}+6 z^4 a^{14}-3 z^2 a^{14}-4 z^7 a^{13}+6 z^5 a^{13}-z^3 a^{13}-z a^{13}-3 z^8 a^{12}+z^6 a^{12}+3 z^4 a^{12}-z^9 a^{11}-5 z^7 a^{11}+6 z^5 a^{11}+3 z^3 a^{11}-3 z a^{11}-5 z^8 a^{10}+3 z^6 a^{10}+4 z^4 a^{10}-3 z^2 a^{10}-z^9 a^9-4 z^7 a^9+4 z^5 a^9+2 z^3 a^9-z a^9-2 z^8 a^8-3 z^6 a^8+10 z^4 a^8-6 z^2 a^8-3 z^7 a^7+4 z^5 a^7-z^3 a^7-z a^7+a^7 z^{-1} -2 z^6 a^6+3 z^4 a^6-a^6-z^5 a^5+3 z^3 a^5-3 z a^5+a^5 z^{-1}$ (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         21-1
-8        4  4
-10       42  -2
-12      74   3
-14     54    -1
-16    67     -1
-18   46      2
-20  25       -3
-22 14        3
-24 2         -2
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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