# L10a120

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a120 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 \left(-v^2\right)-u^3 v^3+u^3 v^2-u^3 v-u^2 v^4+u^2 v^3-u^2 v^2+u^2 v-u^2-u v^3+u v^2-u v-v^2}{u^2 v^2}$ (db) Jones polynomial $-\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{3}{q^{19/2}}-\frac{4}{q^{17/2}}+\frac{4}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{11} z^3+3 a^{11} z+a^{11} z^{-1} -a^9 z^5-4 a^9 z^3-4 a^9 z-a^9 z^{-1} -a^7 z^5-3 a^7 z^3-a^7 z-a^5 z^5-4 a^5 z^3-3 a^5 z$ (db) Kauffman polynomial $a^{15} z^5-4 a^{15} z^3+3 a^{15} z+a^{14} z^6-3 a^{14} z^4+a^{14} z^2+a^{13} z^7-3 a^{13} z^5+2 a^{13} z^3-a^{13} z+a^{12} z^8-4 a^{12} z^6+6 a^{12} z^4-3 a^{12} z^2+a^{11} z^9-6 a^{11} z^7+15 a^{11} z^5-15 a^{11} z^3+7 a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-10 a^{10} z^6+18 a^{10} z^4-8 a^{10} z^2+a^{10}+a^9 z^9-6 a^9 z^7+15 a^9 z^5-15 a^9 z^3+7 a^9 z-a^9 z^{-1} +a^8 z^8-4 a^8 z^6+6 a^8 z^4-3 a^8 z^2+a^7 z^7-3 a^7 z^5+2 a^7 z^3-a^7 z+a^6 z^6-3 a^6 z^4+a^6 z^2+a^5 z^5-4 a^5 z^3+3 a^5 z$ (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
-4          11
-6         110
-8        1  1
-10       21  -1
-12      21   1
-14     22    0
-16    22     0
-18   12      1
-20  12       -1
-22  1        1
-2411         0
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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