L11n241

(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_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_18,14,19,13 X_22,20,11,19 X_20,15,21,16 X_14,21,15,22 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 $u$ , $v$ , $w$ , ...)	$- v 3 u 3 + 3 v 2 u 3 -3 v u 3 + u 3 + v 3 u 2 -2 v 2 u 2 + 2 v u 2 - u 2 - v 3 u + 2 v 2 u -2 v u + u + v 3 -3 v 2 + 3 v -1$ (db)
Jones polynomial	$2 q^{13/2}-5 q^{11/2}+7 q^{9/2}-9 q^{7/2}+9 q^{5/2}-9 q^{3/2}+7 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -8 z 3 a -1 + 10 z 3 a -3 -4 z 3 a -5 + 3 a z -10 z a -1 + 12 z a -3 -6 z a -5 + z a -7 + 2 a z -1 -5 a -1 z -1 + 6 a -3 z -1 -4 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -7 z 8 a -2 -5 z 8 a -4 -2 z 8 - a z 7 -3 z 7 a -1 -10 z 7 a -3 -8 z 7 a -5 + 23 z 6 a -2 + 10 z 6 a -4 -5 z 6 a -6 + 8 z 6 + 5 a z 5 + 26 z 5 a -1 + 47 z 5 a -3 + 25 z 5 a -5 - z 5 a -7 -16 z 4 a -2 + 2 z 4 a -4 + 9 z 4 a -6 -9 z 4 -9 a z 3 -41 z 3 a -1 -58 z 3 a -3 -31 z 3 a -5 -5 z 3 a -7 -8 z 2 a -4 -9 z 2 a -6 -3 z 2 a -8 + 2 z 2 + 7 a z + 24 z a -1 + 31 z a -3 + 18 z a -5 + 4 z a -7 + a -2 + 3 a -4 + 3 a -6 + a -8 + 1-2 a z -1 -5 a -1 z -1 -6 a -3 z -1 -4 a -5 z -1 - a -7 z -1$ (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 =

3 is the signature of L11n241. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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