# L10a115

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a115 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 \left(-v^2\right)-2 u^3 v^3+2 u^3 v^2-u^3 v-u^2 v^4+2 u^2 v^3-3 u^2 v^2+2 u^2 v-u^2-u v^3+2 u v^2-2 u v-v^2}{u^2 v^2}$ (db) Jones polynomial $-\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{5}{q^{19/2}}-\frac{6}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{7}{q^{13/2}}+\frac{5}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $z^3 a^{11}+3 z a^{11}+a^{11} z^{-1} -z^5 a^9-3 z^3 a^9-2 z a^9-a^9 z^{-1} -2 z^5 a^7-7 z^3 a^7-5 z a^7-z^5 a^5-3 z^3 a^5-z a^5$ (db) Kauffman polynomial $-z^5 a^{15}+4 z^3 a^{15}-4 z a^{15}-z^6 a^{14}+2 z^4 a^{14}-z^7 a^{13}+z^5 a^{13}+z^3 a^{13}-z^8 a^{12}+2 z^6 a^{12}-5 z^4 a^{12}+6 z^2 a^{12}-z^9 a^{11}+4 z^7 a^{11}-12 z^5 a^{11}+13 z^3 a^{11}-6 z a^{11}+a^{11} z^{-1} -3 z^8 a^{10}+10 z^6 a^{10}-16 z^4 a^{10}+7 z^2 a^{10}-a^{10}-z^9 a^9+2 z^7 a^9-3 z^5 a^9+2 z^3 a^9-4 z a^9+a^9 z^{-1} -2 z^8 a^8+5 z^6 a^8-4 z^4 a^8-3 z^7 a^7+10 z^5 a^7-11 z^3 a^7+5 z a^7-2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5$ (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         21-1
-8        2  2
-10       32  -1
-12      42   2
-14     33    0
-16    34     -1
-18   23      1
-20  13       -2
-22  2        2
-2411         0
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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