# L10a158

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a158 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(w-1) \left(u^2 v^2+u^2 v w-u^2 v-u^2 w+u v^2 w-u v^2+u v w^2-3 u v w+u v-u w^2+u w-v^2 w-v w^2+v w+w^2\right)}{u v w^{3/2}}$ (db) Jones polynomial $-q^5+3 q^4-5 q^3+9 q^2-10 q+12-10 q^{-1} +9 q^{-2} -5 q^{-3} +3 q^{-4} - q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^4 \left(-z^2\right)-z^2 a^{-4} +a^2 z^4+z^4 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +a^2+ a^{-2} +2 z^4+z^2-2 z^{-2} -2$ (db) Kauffman polynomial $2 a z^9+2 z^9 a^{-1} +4 a^2 z^8+4 z^8 a^{-2} +8 z^8+4 a^3 z^7-a z^7-z^7 a^{-1} +4 z^7 a^{-3} +3 a^4 z^6-9 a^2 z^6-9 z^6 a^{-2} +3 z^6 a^{-4} -24 z^6+a^5 z^5-8 a^3 z^5-2 a z^5-2 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -7 a^4 z^4+9 a^2 z^4+9 z^4 a^{-2} -7 z^4 a^{-4} +32 z^4-2 a^5 z^3+3 a^3 z^3+3 a z^3+3 z^3 a^{-1} +3 z^3 a^{-3} -2 z^3 a^{-5} +3 a^4 z^2-3 a^2 z^2-3 z^2 a^{-2} +3 z^2 a^{-4} -12 z^2+2 a z+2 z a^{-1} -2 a^2-2 a^{-2} -3-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        31 -2
5       62  4
3      54   -1
1     75    2
-1    57     2
-3   45      -1
-5  26       4
-7 13        -2
-9 2         2
-111          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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