# L11n254

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n254 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^4 v^4+u^3 v^2+u^2 v^3-u^2 v^2+u^2 v+u v^2+1}{u^2 v^2}$ (db) Jones polynomial $-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{27/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z^3 a^{13}-2 z a^{13}+z^7 a^{11}+8 z^5 a^{11}+19 z^3 a^{11}+14 z a^{11}+a^{11} z^{-1} -z^9 a^9-9 z^7 a^9-28 z^5 a^9-36 z^3 a^9-17 z a^9-a^9 z^{-1}$ (db) Kauffman polynomial $a^{17} z^3-2 a^{17} z+a^{16} z^4-a^{16} z^2+a^{15} z^5-2 a^{15} z^3+2 a^{15} z+a^{14} z^4+2 a^{13} z+a^{12} z^8-8 a^{12} z^6+19 a^{12} z^4-13 a^{12} z^2+a^{11} z^9-9 a^{11} z^7+27 a^{11} z^5-33 a^{11} z^3+15 a^{11} z-a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+19 a^{10} z^4-14 a^{10} z^2+a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-36 a^9 z^3+17 a^9 z-a^9 z^{-1}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-8          11
-10          11
-12        1  1
-14      1    1
-16     111   -1
-18    11     0
-20   111     -1
-22  11       0
-24  1        1
-2611         0
-281          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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