# L10a121

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a121 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 \left(-v^2\right)-3 u^3 v^3+5 u^3 v^2-3 u^3 v-u^2 v^4+5 u^2 v^3-9 u^2 v^2+5 u^2 v-u^2-3 u v^3+5 u v^2-3 u v-v^2}{u^2 v^2}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{14}{q^{13/2}}+\frac{16}{q^{15/2}}-\frac{14}{q^{17/2}}+\frac{11}{q^{19/2}}-\frac{8}{q^{21/2}}+\frac{3}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{11} z^3+a^{11} z+a^{11} z^{-1} -a^9 z^5+2 a^9 z-a^9 z^{-1} -3 a^7 z^5-9 a^7 z^3-7 a^7 z-a^5 z^5-2 a^5 z^3-a^5 z$ (db) Kauffman polynomial $-z^5 a^{15}+2 z^3 a^{15}-z a^{15}-3 z^6 a^{14}+4 z^4 a^{14}-z^2 a^{14}-6 z^7 a^{13}+12 z^5 a^{13}-12 z^3 a^{13}+7 z a^{13}-5 z^8 a^{12}+3 z^6 a^{12}+6 z^4 a^{12}-5 z^2 a^{12}-2 z^9 a^{11}-8 z^7 a^{11}+18 z^5 a^{11}-9 z^3 a^{11}+z a^{11}+a^{11} z^{-1} -10 z^8 a^{10}+12 z^6 a^{10}+4 z^4 a^{10}-8 z^2 a^{10}-a^{10}-2 z^9 a^9-8 z^7 a^9+18 z^5 a^9-9 z^3 a^9+z a^9+a^9 z^{-1} -5 z^8 a^8+3 z^6 a^8+6 z^4 a^8-5 z^2 a^8-6 z^7 a^7+12 z^5 a^7-12 z^3 a^7+7 z a^7-3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z a^5$ (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         31-2
-8        5  5
-10       63  -3
-12      85   3
-14     86    -2
-16    68     -2
-18   58      3
-20  36       -3
-22  5        5
-2413         -2
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{3}$ $r=-8$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-6$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-5$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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