# L10a146

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a146 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^3 w+2 u v^2 w^2-3 u v^2 w+u v^2+u v w^3-3 u v w^2+2 u v w-u w^3+u w^2-v^3 w+v^3-2 v^2 w^2+3 v^2 w-v^2-v w^3+3 v w^2-2 v w-w^2}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $q^9-2 q^8+5 q^7-7 q^6+10 q^5-9 q^4+10 q^3-8 q^2+5 q+ q^{-1} -2$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} -2 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +z^2+ a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2}$ (db) Kauffman polynomial $z^6 a^{-10} -4 z^4 a^{-10} +4 z^2 a^{-10} +2 z^7 a^{-9} -6 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -11 z^6 a^{-8} +18 z^4 a^{-8} -19 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +z^9 a^{-7} +3 z^7 a^{-7} -15 z^5 a^{-7} +17 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +6 z^8 a^{-6} -18 z^6 a^{-6} +28 z^4 a^{-6} -30 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} +5 z^7 a^{-5} -17 z^5 a^{-5} +21 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -3 z^6 a^{-4} +3 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +4 z^7 a^{-3} -6 z^5 a^{-3} +6 z^3 a^{-3} +3 z^6 a^{-2} -2 z^4 a^{-2} +2 z^5 a^{-1} -2 z^3 a^{-1} +z^4-2 z^2+1$ (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 \
-2-1012345678χ
19          11
17         21-1
15        3  3
13       42  -2
11      63   3
9     45    1
7    65     1
5   24      2
3  36       -3
1 14        3
-1 1         -1
-31          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=8$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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