# L10a2

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a2 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-3 t(2)^3+3 t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $\frac{14}{q^{9/2}}-\frac{15}{q^{7/2}}+\frac{13}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-4 \sqrt{q}+\frac{6}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^7 \left(-z^3\right)-a^7 z+2 a^5 z^5+5 a^5 z^3+2 a^5 z-a^5 z^{-1} -a^3 z^7-4 a^3 z^5-5 a^3 z^3+3 a^3 z^{-1} +a z^5+2 a z^3-a z-2 a z^{-1}$ (db) Kauffman polynomial $a^{10} z^4+4 a^9 z^5-2 a^9 z^3+8 a^8 z^6-9 a^8 z^4+3 a^8 z^2+9 a^7 z^7-11 a^7 z^5+5 a^7 z^3-2 a^7 z+6 a^6 z^8-a^6 z^6-9 a^6 z^4+2 a^6 z^2+a^6+2 a^5 z^9+11 a^5 z^7-30 a^5 z^5+20 a^5 z^3-4 a^5 z-a^5 z^{-1} +11 a^4 z^8-23 a^4 z^6+12 a^4 z^4-4 a^4 z^2+3 a^4+2 a^3 z^9+6 a^3 z^7-27 a^3 z^5+22 a^3 z^3-a^3 z-3 a^3 z^{-1} +5 a^2 z^8-13 a^2 z^6+9 a^2 z^4-3 a^2 z^2+3 a^2+4 a z^7-12 a z^5+9 a z^3+a z-2 a z^{-1} +z^6-2 z^4$ (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 \
-7-6-5-4-3-2-10123χ
4          1-1
2         3 3
0        31 -2
-2       83  5
-4      75   -2
-6     86    2
-8    67     1
-10   58      -3
-12  36       3
-14 15        -4
-16 3         3
-181          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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