# L10a164

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a164 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^2 t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3-t(1)^2 t(3)^2+2 t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2+2 t(1)^2 t(2) t(3)^2-4 t(1) t(2) t(3)^2+3 t(2) t(3)^2-t(3)^2+2 t(1)^2 t(3)+t(1)^2 t(2)^2 t(3)-2 t(1) t(2)^2 t(3)+t(2)^2 t(3)-2 t(1) t(3)-3 t(1)^2 t(2) t(3)+4 t(1) t(2) t(3)-2 t(2) t(3)-t(1)^2+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^{-2} -3 q^{-3} +7 q^{-4} -10 q^{-5} +13 q^{-6} -13 q^{-7} +13 q^{-8} -9 q^{-9} +7 q^{-10} -3 q^{-11} + q^{-12}$ (db) Signature -4 (db) HOMFLY-PT polynomial $a^{12} z^{-2} +a^{12}-4 a^{10} z^2-2 a^{10} z^{-2} -6 a^{10}+3 a^8 z^4+6 a^8 z^2+a^8 z^{-2} +4 a^8+3 a^6 z^4+5 a^6 z^2+a^6+a^4 z^4+a^4 z^2$ (db) Kauffman polynomial $z^6 a^{14}-3 z^4 a^{14}+3 z^2 a^{14}-a^{14}+3 z^7 a^{13}-8 z^5 a^{13}+5 z^3 a^{13}+4 z^8 a^{12}-9 z^6 a^{12}+4 z^4 a^{12}-3 z^2 a^{12}-a^{12} z^{-2} +5 a^{12}+2 z^9 a^{11}+3 z^7 a^{11}-18 z^5 a^{11}+17 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} +10 z^8 a^{10}-25 z^6 a^{10}+26 z^4 a^{10}-23 z^2 a^{10}-2 a^{10} z^{-2} +11 a^{10}+2 z^9 a^9+7 z^7 a^9-22 z^5 a^9+19 z^3 a^9-9 z a^9+2 a^9 z^{-1} +6 z^8 a^8-9 z^6 a^8+10 z^4 a^8-10 z^2 a^8-a^8 z^{-2} +5 a^8+7 z^7 a^7-9 z^5 a^7+5 z^3 a^7+6 z^6 a^6-8 z^4 a^6+6 z^2 a^6-a^6+3 z^5 a^5-2 z^3 a^5+z^4 a^4-z^2 a^4$ (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
-3          11
-5         31-2
-7        4  4
-9       63  -3
-11      74   3
-13     66    0
-15    77     0
-17   48      4
-19  35       -2
-21 15        4
-23 2         -2
-251          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=-5$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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