# L11n169

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n169 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^2 v^6-u^2 v^4+u^2 v^3+2 u v^4-3 u v^3+2 u v^2+v^3-v^2+1}{u v^3}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}+\frac{1}{q^{29/2}}-\frac{2}{q^{27/2}}+\frac{3}{q^{25/2}}-\frac{4}{q^{23/2}}+\frac{4}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-2 z^3 a^{13}-5 z a^{13}-2 a^{13} z^{-1} +z^7 a^{11}+9 z^5 a^{11}+24 z^3 a^{11}+22 z a^{11}+5 a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-37 z^3 a^9-20 z a^9-3 a^9 z^{-1}$ (db) Kauffman polynomial $a^{18} z^4-2 a^{18} z^2+2 a^{17} z^5-4 a^{17} z^3+a^{17} z+2 a^{16} z^6-4 a^{16} z^4+3 a^{16} z^2-a^{16}+a^{15} z^7-a^{15} z^5+a^{15} z^3+2 a^{14} z^6-3 a^{14} z^4+3 a^{14} z^2+2 a^{13} z^5-3 a^{13} z^3+5 a^{13} z-2 a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-24 a^{12} z^2+5 a^{12}+a^{11} z^9-10 a^{11} z^7+33 a^{11} z^5-45 a^{11} z^3+26 a^{11} z-5 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+24 a^{10} z^4-22 a^{10} z^2+5 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+20 a^9 z-3 a^9 z^{-1}$ (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
-8           11
-10           11
-12         1  1
-14       2    2
-16      111   -1
-18     32     1
-20    221     -1
-22   22       0
-24  12        1
-26 12         -1
-28 1          1
-301           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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