# L10a98

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a98 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+2}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-\frac{1}{q^{27/2}}+\frac{2}{q^{25/2}}-\frac{3}{q^{23/2}}+\frac{4}{q^{21/2}}-\frac{5}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{3}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $a^{11} z^5+4 a^{11} z^3+3 a^{11} z-a^9 z^7-5 a^9 z^5-6 a^9 z^3+a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-7 a^7 z-a^7 z^{-1}$ (db) Kauffman polynomial $-z^3 a^{17}+z a^{17}-2 z^4 a^{16}+2 z^2 a^{16}-2 z^5 a^{15}+z^3 a^{15}-2 z^6 a^{14}+2 z^4 a^{14}-z^2 a^{14}-2 z^7 a^{13}+4 z^5 a^{13}-3 z^3 a^{13}+z a^{13}-2 z^8 a^{12}+7 z^6 a^{12}-9 z^4 a^{12}+5 z^2 a^{12}-z^9 a^{11}+3 z^7 a^{11}-3 z^5 a^{11}+5 z^3 a^{11}-3 z a^{11}-3 z^8 a^{10}+13 z^6 a^{10}-15 z^4 a^{10}+5 z^2 a^{10}-z^9 a^9+4 z^7 a^9-3 z^5 a^9-z^3 a^9+2 z a^9-a^9 z^{-1} -z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+7 z a^7-a^7 z^{-1}$ (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       11  0
-14      32   1
-16     21    -1
-18    33     0
-20   12      1
-22  23       -1
-24 12        1
-26 1         -1
-281          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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