# L11n252

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n252 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2)^2 t(1)^4-t(2)^3 t(1)^3+t(2)^2 t(1)^3-t(1)^3-t(2)^2 t(1)^2-t(2)^4 t(1)+t(2)^2 t(1)-t(2) t(1)-t(2)^2}{t(1)^2 t(2)^2}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}-\frac{1}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^{11} z+a^{11} z^{-1} -a^9 z^{-1} -a^7 z^5-4 a^7 z^3-3 a^7 z-a^5 z^5-4 a^5 z^3-3 a^5 z$ (db) Kauffman polynomial $-z^7 a^{13}+7 z^5 a^{13}-14 z^3 a^{13}+8 z a^{13}+z^4 a^{12}-2 z^2 a^{12}+z^5 a^{11}-3 z^3 a^{11}+a^{11} z^{-1} -z^2 a^{10}-a^{10}-z^5 a^9+2 z^3 a^9-2 z a^9+a^9 z^{-1} -z^6 a^8+2 z^4 a^8-z^7 a^7+4 z^5 a^7-5 z^3 a^7+3 z a^7-z^6 a^6+3 z^4 a^6-z^2 a^6-z^5 a^5+4 z^3 a^5-3 z a^5$ (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         110
-8        1  1
-10       11  0
-12     121   0
-14     11    0
-16   122     -1
-18    1      1
-20  11       0
-221          1
-241          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $i=-2$ $r=-10$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-9$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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