# L11n242

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n242 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+t(2)^2 t(1)-4 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{9}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{11}{q^{5/2}}-2 q^{3/2}+\frac{10}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+4 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -2 a^3 z^5-7 a^3 z^3-11 a^3 z-6 a^3 z^{-1} +4 a z^3+8 a z+5 a z^{-1} -2 z a^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $-a^5 z^9-a^3 z^9-3 a^6 z^8-7 a^4 z^8-4 a^2 z^8-3 a^7 z^7-8 a^5 z^7-9 a^3 z^7-4 a z^7-a^8 z^6+4 a^6 z^6+15 a^4 z^6+9 a^2 z^6-z^6+9 a^7 z^5+33 a^5 z^5+36 a^3 z^5+12 a z^5+3 a^8 z^4+8 a^6 z^4-3 a^4 z^4-10 a^2 z^4-2 z^4-8 a^7 z^3-35 a^5 z^3-50 a^3 z^3-26 a z^3-3 z^3 a^{-1} -3 a^8 z^2-9 a^6 z^2-7 a^4 z^2+z^2+4 a^7 z+18 a^5 z+29 a^3 z+20 a z+5 z a^{-1} +a^8+3 a^6+3 a^4+a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -6 a^3 z^{-1} -5 a z^{-1} -2 a^{-1} z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
4         22
2        2 -2
0       62 4
-2      64  -2
-4     54   1
-6    56    1
-8   45     -1
-10  25      3
-12 14       -3
-14 2        2
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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