# L10a94

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a94 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1) t(2)+1) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{4}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{11} z^3+2 a^{11} z-a^9 z^5-2 a^9 z^3+a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-4 a^7 z-a^7 z^{-1} -a^5 z^5-3 a^5 z^3-2 a^5 z$ (db) Kauffman polynomial $-z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-2 z^6 a^{14}+5 z^4 a^{14}-3 z^2 a^{14}-2 z^7 a^{13}+2 z^5 a^{13}+2 z^3 a^{13}-z a^{13}-2 z^8 a^{12}+3 z^6 a^{12}-4 z^4 a^{12}+3 z^2 a^{12}-z^9 a^{11}-3 z^5 a^{11}+6 z^3 a^{11}-3 z a^{11}-4 z^8 a^{10}+8 z^6 a^{10}-10 z^4 a^{10}+4 z^2 a^{10}-z^9 a^9-z^7 a^9+2 z^5 a^9-3 z^3 a^9+3 z a^9-a^9 z^{-1} -2 z^8 a^8+z^6 a^8+3 z^4 a^8-3 z^2 a^8+a^8-3 z^7 a^7+7 z^5 a^7-7 z^3 a^7+5 z a^7-a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-2 z a^5$ (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        3  3
-10       42  -2
-12      53   2
-14     44    0
-16    55     0
-18   24      2
-20  25       -3
-22 13        2
-24 1         -1
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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