# L10a159

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a159 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(2)^2 t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+4 t(1) t(2) t(3)^2-t(2) t(3)^2-t(1)^2 t(3)+t(1) t(2)^2 t(3)-t(2)^2 t(3)+2 t(1) t(3)+t(1)^2 t(2) t(3)-4 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}}$ (db) Jones polynomial $q^{-13} -3 q^{-12} +6 q^{-11} -8 q^{-10} +11 q^{-9} -10 q^{-8} +10 q^{-7} -7 q^{-6} +5 q^{-5} -2 q^{-4} + q^{-3}$ (db) Signature -6 (db) HOMFLY-PT polynomial $z^2 a^{12}+a^{12} z^{-2} +2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-2 a^{10} z^{-2} -9 a^{10}+2 z^6 a^8+9 z^4 a^8+13 z^2 a^8+a^8 z^{-2} +7 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6$ (db) Kauffman polynomial $a^{16} z^4-a^{16} z^2+3 a^{15} z^5-3 a^{15} z^3+5 a^{14} z^6-7 a^{14} z^4+5 a^{14} z^2-2 a^{14}+5 a^{13} z^7-6 a^{13} z^5+3 a^{13} z^3+3 a^{12} z^8+a^{12} z^6-8 a^{12} z^4+5 a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^9+6 a^{11} z^7-17 a^{11} z^5+17 a^{11} z^3-9 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-16 a^{10} z^6+22 a^{10} z^4-23 a^{10} z^2-2 a^{10} z^{-2} +11 a^{10}+a^9 z^9+3 a^9 z^7-14 a^9 z^5+15 a^9 z^3-9 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+18 a^8 z^4-18 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^2$ (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
-5          11
-7         21-1
-9        3  3
-11       42  -2
-13      63   3
-15     44    0
-17    76     1
-19   36      3
-21  35       -2
-23 14        3
-25 2         -2
-271          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-6$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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