# L10a22

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a22 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $q^{17/2}-4 q^{15/2}+9 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +6 z^3 a^{-3} +3 z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -10 z^4 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +11 z^7 a^{-7} -15 z^5 a^{-7} +8 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +7 z^8 a^{-6} +z^6 a^{-6} -19 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +2 z^9 a^{-5} +16 z^7 a^{-5} -41 z^5 a^{-5} +26 z^3 a^{-5} -7 z a^{-5} +2 a^{-5} z^{-1} +12 z^8 a^{-4} -19 z^6 a^{-4} -4 z^4 a^{-4} +11 z^2 a^{-4} -3 a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} -33 z^5 a^{-3} +26 z^3 a^{-3} -6 z a^{-3} +5 z^8 a^{-2} -10 z^6 a^{-2} +2 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +9 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+z^2$ (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 \
-3-2-101234567χ
18          1-1
16         3 3
14        61 -5
12       63  3
10      96   -3
8     86    2
6    69     3
4   68      -2
2  38       5
0 14        -3
-2 3         3
-41          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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