# L10a148

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a148 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^3 w^3-u v^3 w^2-u v^2 w^3+3 u v^2 w^2-u v^2 w-2 u v w^2+3 u v w-u v-u w+u-v^3 w^3+v^3 w^2+v^2 w^3-3 v^2 w^2+2 v^2 w+v w^2-3 v w+v+w-1}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $q^7-3 q^6+5 q^5-7 q^4+10 q^3+ q^{-3} -9 q^2-3 q^{-2} +10 q+5 q^{-1} -6$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-4} +4 z^4 a^{-4} +4 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} -z^8 a^{-2} -6 z^6 a^{-2} -12 z^4 a^{-2} -9 z^2 a^{-2} -2 a^{-2} z^{-2} -3 a^{-2} +z^6+4 z^4+4 z^2+ z^{-2} +2$ (db) Kauffman polynomial $2 z^9 a^{-1} +2 z^9 a^{-3} +8 z^8 a^{-2} +4 z^8 a^{-4} +4 z^8+3 a z^7-3 z^7 a^{-1} -2 z^7 a^{-3} +4 z^7 a^{-5} +a^2 z^6-30 z^6 a^{-2} -10 z^6 a^{-4} +4 z^6 a^{-6} -15 z^6-10 a z^5-5 z^5 a^{-1} -3 z^5 a^{-3} -5 z^5 a^{-5} +3 z^5 a^{-7} -3 a^2 z^4+41 z^4 a^{-2} +15 z^4 a^{-4} -5 z^4 a^{-6} +z^4 a^{-8} +17 z^4+6 a z^3+7 z^3 a^{-1} +7 z^3 a^{-3} +2 z^3 a^{-5} -4 z^3 a^{-7} +a^2 z^2-22 z^2 a^{-2} -10 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} -9 z^2-3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-2} - z^{-2}$ (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 \
-4-3-2-10123456χ
15          11
13         2 -2
11        31 2
9       53  -2
7      52   3
5     45    1
3    65     1
1   37      4
-1  23       -1
-3 13        2
-5 2         -2
-71          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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