L11n432

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_14,4,15,3 X_12,14,7,13 X₂₇₃₈ X_22,10,13,9 X_6,22,1,21 X_15,20,16,21 X_5,17,6,16 X_11,19,12,18 X_17,11,18,10 X_19,5,20,4
Gauss code	{1, -4, 2, 11, -8, -6}, {4, -1, 5, 10, -9, -3}, {3, -2, -7, 8, -10, 9, -11, 7, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v 2 w 2 u 2 + v w 2 u 2 + v 2 w u 2 - v u - w u + u - v w 2 - v 2 w + v w$ (db)
Jones polynomial	$q 9 - q 8 + q 7 + q 5 + q 4 + q 2 - q + 1$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 6 a -4 + z 4 a -2 -5 z 4 a -4 + 4 z 2 a -2 -5 z 2 a -4 + z 2 a -8 + 2 a -2 -3 a -6 + a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$z 8 a -6 + z 8 a -8 + z 7 a -3 + z 7 a -5 + z 7 a -7 + z 7 a -9 + z 6 a -2 + z 6 a -4 -8 z 6 a -6 -7 z 6 a -8 + z 6 a -10 -5 z 5 a -3 -7 z 5 a -5 -7 z 5 a -7 -5 z 5 a -9 -5 z 4 a -2 -5 z 4 a -4 + 21 z 4 a -6 + 16 z 4 a -8 -5 z 4 a -10 + 5 z 3 a -3 + 13 z 3 a -5 + 13 z 3 a -7 + 5 z 3 a -9 + 6 z 2 a -2 + 3 z 2 a -4 -25 z 2 a -6 -17 z 2 a -8 + 5 z 2 a -10 -9 z a -5 -9 z a -7 -2 a -2 + 3 a -4 + 11 a -6 + 7 a -8 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 =

3 is the signature of L11n432. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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