# L10a50

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a50 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{6 u v^2-7 u v+2 u+2 v^3-7 v^2+6 v}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{1}{q^{23/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^{11} z-a^{11} z^{-1} -a^9 z^3+2 a^9 z+2 a^9 z^{-1} -3 a^7 z^3-2 a^7 z-3 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^3$ (db) Kauffman polynomial $-z^7 a^{13}+4 z^5 a^{13}-4 z^3 a^{13}+z a^{13}-3 z^8 a^{12}+14 z^6 a^{12}-19 z^4 a^{12}+6 z^2 a^{12}+2 a^{12}-2 z^9 a^{11}+5 z^7 a^{11}+3 z^5 a^{11}-9 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -8 z^8 a^{10}+30 z^6 a^{10}-29 z^4 a^{10}+2 z^2 a^{10}+5 a^{10}-2 z^9 a^9-z^7 a^9+17 z^5 a^9-15 z^3 a^9+4 z a^9-2 a^9 z^{-1} -5 z^8 a^8+9 z^6 a^8+z^4 a^8-4 z^2 a^8+3 a^8-7 z^7 a^7+12 z^5 a^7-3 z^3 a^7-7 z^6 a^6+8 z^4 a^6-a^6-6 z^5 a^5+6 z^3 a^5-3 z a^5+a^5 z^{-1} -3 z^4 a^4-z^3 a^3$ (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         31-2
-6        3  3
-8       43  -1
-10      63   3
-12     45    1
-14    55     0
-16   24      2
-18  25       -3
-20 12        1
-22 2         -2
-241          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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