L11a452

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_22,18,13,17 X_16,8,17,7 X_12,14,5,13 X_8,21,9,22 X_20,9,21,10 X_18,11,19,12 X_10,19,11,20 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -6, 7, -9, 8, -5}, {5, -2, 11, -4, 3, -8, 9, -7, 6, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 3 - v u 3 - v 2 w u 3 + 2 v w u 3 - w u 3 + v 3 u 2 -2 v 2 u 2 + 2 v u 2 - v 3 w u 2 + 2 v 2 w u 2 -2 v w u 2 + 2 w u 2 - u 2 -2 v 3 u + 2 v 2 u -2 v u + v 3 w u -2 v 2 w u + 2 v w u - w u + u + v 3 -2 v 2 + v + v 2 w - v w$ (db)
Jones polynomial	$- q 6 + 3 q 5 -6 q 4 + 9 q 3 -10 q 2 + 12 q -11 + 10 q -1 -6 q -2 + 5 q -3 -2 q -4 + q -5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + z 6 -2 a 2 z 4 + 3 z 4 a -2 - z 4 a -4 + 3 z 4 + a 4 z 2 -6 a 2 z 2 + 3 z 2 a -2 -2 z 2 a -4 + 3 z 2 + 2 a 4 -5 a 2 + 2 a -2 - a -4 + 2 + a 4 z -2 -2 a 2 z -2 + z -2$ (db)
Kauffman polynomial	$a 2 z 10 + z 10 + 2 a 3 z 9 + 6 a z 9 + 4 z 9 a -1 + a 4 z 8 + a 2 z 8 + 7 z 8 a -2 + 7 z 8 -10 a 3 z 7 -24 a z 7 -5 z 7 a -1 + 9 z 7 a -3 -6 a 4 z 6 -24 a 2 z 6 -13 z 6 a -2 + 9 z 6 a -4 -40 z 6 + 14 a 3 z 5 + 24 a z 5 -12 z 5 a -1 -16 z 5 a -3 + 6 z 5 a -5 + 13 a 4 z 4 + 52 a 2 z 4 -14 z 4 a -4 + 3 z 4 a -6 + 56 z 4 -2 a 3 z 3 + a z 3 + 12 z 3 a -1 + 4 z 3 a -3 -4 z 3 a -5 + z 3 a -7 -13 a 4 z 2 -39 a 2 z 2 + z 2 a -2 + 7 z 2 a -4 -32 z 2 -5 a 3 z -8 a z -3 z a -1 + z a -3 + z a -5 + 6 a 4 + 13 a 2 - a -4 + 9 + 2 a 3 z -1 + 2 a z -1 - a 4 z -2 -2 a 2 z -2 - z -2$ (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 =

2 is the signature of L11a452. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{7}$
$r = 1$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$