# L10a165

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a165 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v w x-2 u v w-2 u v x+2 u v-2 u w x+2 u w+3 u x-2 u-2 v w x+3 v w+2 v x-2 v+2 w x-2 w-2 x+1}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x}}$ (db) Jones polynomial $-\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^{11} z^{-3} -4 a^9 z^{-1} -3 a^9 z^{-3} +6 z a^7+8 a^7 z^{-1} +3 a^7 z^{-3} -4 z^3 a^5-6 z a^5-4 a^5 z^{-1} -a^5 z^{-3} +z^5 a^3+z^3 a^3-z^3 a$ (db) Kauffman polynomial $-z^5 a^{11}+4 z^3 a^{11}-6 z a^{11}+4 a^{11} z^{-1} -a^{11} z^{-3} -z^6 a^{10}+6 z^2 a^{10}+3 a^{10} z^{-2} -8 a^{10}-z^7 a^9-3 z^5 a^9+12 z^3 a^9-14 z a^9+9 a^9 z^{-1} -3 a^9 z^{-3} -z^8 a^8-2 z^6 a^8+12 z^2 a^8+6 a^8 z^{-2} -15 a^8-z^9 a^7-6 z^5 a^7+13 z^3 a^7-14 z a^7+9 a^7 z^{-1} -3 a^7 z^{-3} -5 z^8 a^6+9 z^6 a^6-9 z^4 a^6+6 z^2 a^6+3 a^6 z^{-2} -8 a^6-z^9 a^5-5 z^7 a^5+10 z^5 a^5-z^3 a^5-6 z a^5+4 a^5 z^{-1} -a^5 z^{-3} -4 z^8 a^4+6 z^6 a^4-2 z^4 a^4-6 z^7 a^3+13 z^5 a^3-5 z^3 a^3-4 z^6 a^2+7 z^4 a^2-z^5 a+z^3 a$ (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         3 -3
-2        41 3
-4       54  -1
-6      63   3
-8     45    1
-10    76     1
-12   410      6
-14  11       0
-16  4        4
-1811         0
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{10}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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