# L10a102

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a102 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1) t(2)+1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+2 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}-z^7 a^9-5 z^5 a^9-7 z^3 a^9-2 z a^9+a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-6 z a^7-a^7 z^{-1}$ (db) Kauffman polynomial $a^{17} z^3-a^{17} z+2 a^{16} z^4-a^{16} z^2+3 a^{15} z^5-2 a^{15} z^3+a^{15} z+4 a^{14} z^6-7 a^{14} z^4+6 a^{14} z^2+3 a^{13} z^7-4 a^{13} z^5+a^{13} z+2 a^{12} z^8-3 a^{12} z^6-a^{12} z^2+a^{11} z^9-2 a^{11} z^7+4 a^{11} z^5-11 a^{11} z^3+5 a^{11} z+3 a^{10} z^8-11 a^{10} z^6+12 a^{10} z^4-6 a^{10} z^2+a^9 z^9-4 a^9 z^7+5 a^9 z^5-3 a^9 z^3+a^9 z^{-1} +a^8 z^8-4 a^8 z^6+3 a^8 z^4+2 a^8 z^2-a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-6 a^7 z+a^7 z^{-1}$ (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
-6          11
-8         110
-10        2  2
-12       21  -1
-14      42   2
-16     22    0
-18    44     0
-20   22      0
-22  24       -2
-24 13        2
-26 1         -1
-281          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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