# L10a167

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a167 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(1) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-2 t(2) t(4)^2 t(3)^2+t(4)^2 t(3)^2+t(1) t(4) t(3)^2+t(2) t(4) t(3)^2-t(4) t(3)^2+t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(4)^2 t(3)+t(1) t(3)+t(2) t(3)-2 t(1) t(4) t(3)+t(1) t(2) t(4) t(3)-2 t(2) t(4) t(3)+t(4) t(3)-t(3)-2 t(1)+t(1) t(2)-t(2)+t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $7 q^{9/2}-10 q^{7/2}+8 q^{5/2}-\frac{1}{q^{5/2}}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{15/2}+3 q^{13/2}-6 q^{11/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-9 z^3 a^{-1} +10 z^3 a^{-3} -3 z^3 a^{-5} +4 a z-14 z a^{-1} +13 z a^{-3} -3 z a^{-5} +4 a z^{-1} -11 a^{-1} z^{-1} +10 a^{-3} z^{-1} -3 a^{-5} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -4 z^8 a^{-4} -z^8-a z^7-6 z^7 a^{-3} -7 z^7 a^{-5} +13 z^6 a^{-2} +4 z^6 a^{-4} -7 z^6 a^{-6} +2 z^6+6 a z^5+14 z^5 a^{-1} +27 z^5 a^{-3} +13 z^5 a^{-5} -6 z^5 a^{-7} +6 z^4 a^{-2} +11 z^4 a^{-4} +8 z^4 a^{-6} -3 z^4 a^{-8} +6 z^4-13 a z^3-30 z^3 a^{-1} -30 z^3 a^{-3} -6 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} -33 z^2 a^{-2} -16 z^2 a^{-4} -17 z^2+13 a z+28 z a^{-1} +21 z a^{-3} +3 z a^{-5} -3 z a^{-7} +24 a^{-2} +11 a^{-4} - a^{-6} +13-6 a z^{-1} -14 a^{-1} z^{-1} -12 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3}$ (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 \
-4-3-2-10123456χ
16          11
14         31-2
12        3  3
10       43  -1
8      63   3
6     57    2
4    43     1
2   48      4
0  11       0
-2  4        4
-411         0
-61          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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