# L10a142

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a142 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3 w^2+u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^2-2 u v w+u v+u w-u+v^3 w^3-v^3 w^2-v^2 w^3+2 v^2 w^2-v^2 w-v w^2+2 v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial $q^{-3} -2 q^{-4} +4 q^{-5} -4 q^{-6} +7 q^{-7} -7 q^{-8} +7 q^{-9} -5 q^{-10} +4 q^{-11} -2 q^{-12} + q^{-13}$ (db) Signature -6 (db) HOMFLY-PT polynomial $a^{12} z^2+a^{12} z^{-2} +3 a^{12}-3 a^{10} z^4-12 a^{10} z^2-2 a^{10} z^{-2} -11 a^{10}+2 a^8 z^6+10 a^8 z^4+15 a^8 z^2+a^8 z^{-2} +8 a^8+a^6 z^6+4 a^6 z^4+3 a^6 z^2$ (db) Kauffman polynomial $a^{16} z^4-2 a^{16} z^2+a^{16}+2 a^{15} z^5-3 a^{15} z^3+2 a^{14} z^6-a^{14} z^4-2 a^{14} z^2+2 a^{13} z^7-2 a^{13} z^5+a^{13} z^3+2 a^{12} z^8-5 a^{12} z^6+9 a^{12} z^4-8 a^{12} z^2-a^{12} z^{-2} +5 a^{12}+a^{11} z^9-8 a^{11} z^5+18 a^{11} z^3-11 a^{11} z+2 a^{11} z^{-1} +5 a^{10} z^8-22 a^{10} z^6+38 a^{10} z^4-31 a^{10} z^2-2 a^{10} z^{-2} +13 a^{10}+a^9 z^9-11 a^9 z^5+18 a^9 z^3-11 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-14 a^8 z^6+23 a^8 z^4-20 a^8 z^2-a^8 z^{-2} +8 a^8+2 a^7 z^7-7 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+3 a^6 z^2$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-5          11
-7         21-1
-9        2  2
-11       22  0
-13      52   3
-15     33    0
-17    44     0
-19   13      2
-21  34       -1
-23 13        2
-25 1         -1
-271          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-7$ $i=-5$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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