L11n242

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_13,18,14,19 X_19,22,20,11 X_15,21,16,20 X_21,15,22,14 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 + v u 3 - v 3 u 2 + 6 v 2 u 2 -6 v u 2 + u 2 + v 3 u -6 v 2 u + 6 v u - u + v 2 - v$ (db)
Jones polynomial	$-2 q^{3/2}+4 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 -2 z 5 a 3 -7 z 3 a 3 -11 z a 3 -6 a 3 z -1 + 4 z 3 a + 8 z a + 5 a z -1 -2 z a -1 -2 a -1 z -1$ (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)

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

-1 is the signature of L11n242. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 0$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = 1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 2$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$