# L10a171

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a171 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v w^2 x-u v w^2+u v w x^2-3 u v w x+2 u v w+u v x-u v-u w^2 x-u w x^2+3 u w x-u w+u x^2-2 u x+u+v w^2 x^2-2 v w^2 x+v w^2-v w x^2+3 v w x-v w-v x-w^2 x^2+w^2 x+2 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9+a^9 z^{-1} +a^9 z^{-3} -3 z^3 a^7-6 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +2 z^5 a^5+6 z^3 a^5+10 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} +z^5 a^3-4 z a^3-4 a^3 z^{-1} -a^3 z^{-3} -z^3 a-z a$ (db) Kauffman polynomial $-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-10 z^3 a^9+11 z a^9-5 a^9 z^{-1} +a^9 z^{-3} -4 z^8 a^8+2 z^6 a^8+9 z^4 a^8-14 z^2 a^8-3 a^8 z^{-2} +10 a^8-z^9 a^7-13 z^7 a^7+42 z^5 a^7-54 z^3 a^7+33 z a^7-12 a^7 z^{-1} +3 a^7 z^{-3} -8 z^8 a^6+10 z^6 a^6+8 z^4 a^6-26 z^2 a^6-6 a^6 z^{-2} +19 a^6-z^9 a^5-13 z^7 a^5+42 z^5 a^5-54 z^3 a^5+33 z a^5-12 a^5 z^{-1} +3 a^5 z^{-3} -4 z^8 a^4+2 z^6 a^4+9 z^4 a^4-14 z^2 a^4-3 a^4 z^{-2} +10 a^4-5 z^7 a^3+10 z^5 a^3-10 z^3 a^3+11 z a^3-5 a^3 z^{-1} +a^3 z^{-3} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-z a$ (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         2 -2
-2        51 4
-4       64  -2
-6      83   5
-8     68    2
-10    86     2
-12   38      5
-14  46       -2
-16 15        4
-18 2         -2
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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