# L11n165

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n165 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^5-u^2 v^3+u^2 v^2+u v^6-u v^5+u v^3-u v+u+v^4-v^3+v}{u v^3}$ (db) Jones polynomial $\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{27/2}}-\frac{1}{q^{25/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{21/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{11/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-2 z a^{13}-2 a^{13} z^{-1} +z^5 a^{11}+7 z^3 a^{11}+11 z a^{11}+5 a^{11} z^{-1} -z^7 a^9-6 z^5 a^9-10 z^3 a^9-7 z a^9-3 a^9 z^{-1} -z^7 a^7-6 z^5 a^7-10 z^3 a^7-5 z a^7$ (db) Kauffman polynomial $a^{16} z^6-5 a^{16} z^4+6 a^{16} z^2-a^{16}+a^{15} z^7-5 a^{15} z^5+6 a^{15} z^3-2 a^{15} z+a^{14} z^6-5 a^{14} z^4+4 a^{14} z^2-3 a^{13} z^3+4 a^{13} z-2 a^{13} z^{-1} +a^{12} z^8-8 a^{12} z^6+19 a^{12} z^4-19 a^{12} z^2+5 a^{12}+a^{11} z^9-8 a^{11} z^7+22 a^{11} z^5-31 a^{11} z^3+21 a^{11} z-5 a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+24 a^{10} z^4-17 a^{10} z^2+5 a^{10}+a^9 z^9-6 a^9 z^7+11 a^9 z^5-12 a^9 z^3+10 a^9 z-3 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+5 a^8 z^4+a^7 z^7-6 a^7 z^5+10 a^7 z^3-5 a^7 z$ (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-6           11
-8          110
-10         1  1
-12       111  1
-14      121   0
-16     111    1
-18    132     0
-20   1        1
-22   11       0
-24 11         0
-26            0
-281           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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