# L10a149

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a149 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(3)-1)^2 (2 t(3) t(2)-t(2)-t(3)+2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^8+5 q^7-9 q^6+13 q^5-15 q^4+17 q^3-14 q^2+12 q-6+3 q^{-1} - q^{-2}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-6} + a^{-6} z^{-2} +2 a^{-6} +z^6 a^{-4} +z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -7 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} +6 a^{-2} -z^4-2 z^2-1$ (db) Kauffman polynomial $z^5 a^{-9} +5 z^6 a^{-8} -6 z^4 a^{-8} + a^{-8} +9 z^7 a^{-7} -15 z^5 a^{-7} +4 z^3 a^{-7} +7 z^8 a^{-6} -5 z^6 a^{-6} -8 z^4 a^{-6} +5 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +2 z^9 a^{-5} +13 z^7 a^{-5} -28 z^5 a^{-5} +9 z^3 a^{-5} +5 z a^{-5} -2 a^{-5} z^{-1} +11 z^8 a^{-4} -12 z^6 a^{-4} -9 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -10 a^{-4} +2 z^9 a^{-3} +8 z^7 a^{-3} -16 z^5 a^{-3} +4 z^3 a^{-3} +7 z a^{-3} -2 a^{-3} z^{-1} +4 z^8 a^{-2} +z^6 a^{-2} -13 z^4 a^{-2} +18 z^2 a^{-2} + a^{-2} z^{-2} -9 a^{-2} +4 z^7 a^{-1} +a z^5-3 z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +a z+3 z a^{-1} +3 z^6-6 z^4+5 z^2-2$ (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 \
-3-2-101234567χ
17          1-1
15         4 4
13        51 -4
11       84  4
9      97   -2
7     86    2
5    69     3
3   68      -2
1  28       6
-1 14        -3
-3 2         2
-51          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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