# L10a117

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a117 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^4 \left(-v^3\right)-u^3 v^4+2 u^3 v^3-2 u^3 v^2-2 u^2 v^3+3 u^2 v^2-2 u^2 v-2 u v^2+2 u v-u-v}{u^2 v^2}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-7 z^3 a^9-3 z a^9-a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-6 z a^7$ (db) Kauffman polynomial $-z^3 a^{17}+z a^{17}-2 z^4 a^{16}+z^2 a^{16}-3 z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-3 z^6 a^{14}+3 z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+6 z^5 a^{13}-5 z^3 a^{13}+2 z a^{13}-2 z^8 a^{12}+4 z^6 a^{12}-2 z^4 a^{12}+2 z^2 a^{12}-z^9 a^{11}+2 z^7 a^{11}-2 z^5 a^{11}+7 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} -3 z^8 a^{10}+11 z^6 a^{10}-10 z^4 a^{10}+3 z^2 a^{10}-a^{10}-z^9 a^9+4 z^7 a^9-5 z^5 a^9+5 z^3 a^9-4 z a^9+a^9 z^{-1} -z^8 a^8+4 z^6 a^8-3 z^4 a^8-z^2 a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+6 z a^7$ (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
-6          11
-8         110
-10        2  2
-12       21  -1
-14      32   1
-16     32    -1
-18    33     0
-20   23      1
-22  23       -1
-24  2        2
-2612         -1
-281          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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