# L11n42

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n42 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{3 t(1) t(2)^3-4 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-4 t(2)+3}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{13} z^{-1} +z^3 a^{11}+3 z a^{11}+2 a^{11} z^{-1} -z^5 a^9-3 z^3 a^9-2 z a^9-a^9 z^{-1} -z^5 a^7-2 z^3 a^7+z a^7+a^7 z^{-1} -z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $3 a^{14} z^4-8 a^{14} z^2+3 a^{14}+a^{13} z^7-a^{13} z^5-3 a^{13} z^3+3 a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-10 a^{12} z^6+24 a^{12} z^4-24 a^{12} z^2+7 a^{12}+a^{11} z^9-4 a^{11} z^7+10 a^{11} z^5-11 a^{11} z^3+8 a^{11} z-2 a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+25 a^{10} z^4-16 a^{10} z^2+4 a^{10}+a^9 z^9-4 a^9 z^7+9 a^9 z^5-7 a^9 z^3+4 a^9 z-a^9 z^{-1} +a^8 z^8-2 a^8 z^6+2 a^8 z^4-a^8 z^2+a^7 z^7-a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        110
-8       2  2
-10      21  -1
-12     42   2
-14    33    0
-16   33     0
-18  13      2
-20 23       -1
-22 1        1
-242         -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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