# L11n55

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n55 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(1) t(2)^3-3 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-6 t(2)-3 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $\frac{11}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{3/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} +a^5 z^5-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+a^3 z^3-a z^3-a z$ (db) Kauffman polynomial $-3 z^4 a^{10}+4 z^2 a^{10}-a^{10}-z^7 a^9-3 z^5 a^9+6 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+z^6 a^8-4 z^4 a^8+8 z^2 a^8-3 a^8-z^9 a^7-3 z^7 a^7+z^5 a^7+7 z^3 a^7-8 z a^7+3 a^7 z^{-1} -5 z^8 a^6+4 z^6 a^6+3 z^2 a^6-3 a^6-z^9 a^5-6 z^7 a^5+11 z^5 a^5-3 z^3 a^5-3 z a^5+2 a^5 z^{-1} -3 z^8 a^4+7 z^4 a^4-4 z^2 a^4-4 z^7 a^3+6 z^5 a^3-2 z^3 a^3+z a^3-3 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+2 z^3 a-z a$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        2 -2
-2       41 3
-4      63  -3
-6     63   3
-8    56    1
-10   66     0
-12  36      3
-14 25       -3
-16 3        3
-182         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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