L11n433

(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_20,16,21,15 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 $u$ , $v$ , $w$ , ...)	$- v 2 w 2 u 3 + v w 2 u 3 + v 2 w u 3 + v u 2 - v w u 2 + w u 2 - v w 2 u - v 2 w u + v w u - v - w + 1$ (db)
Jones polynomial	$q 10 - q 9 + q 8 - q 7 + q 6 + q 5 + 2 q 3 - q 2 + q$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$- z 8 a -6 + z 6 a -4 -7 z 6 a -6 + z 6 a -8 + 6 z 4 a -4 -16 z 4 a -6 + 6 z 4 a -8 + 10 z 2 a -4 -17 z 2 a -6 + 8 z 2 a -8 - z 2 a -10 + 6 a -4 -9 a -6 + 3 a -8 + a -4 z -2 -2 a -6 z -2 + a -8 z -2$ (db)
Kauffman polynomial	$z 9 a -5 + z 9 a -7 + z 8 a -4 + 3 z 8 a -6 + 2 z 8 a -8 -6 z 7 a -5 -5 z 7 a -7 + z 7 a -9 -7 z 6 a -4 -21 z 6 a -6 -13 z 6 a -8 + z 6 a -10 + 8 z 5 a -5 + z 5 a -7 -6 z 5 a -9 + z 5 a -11 + 16 z 4 a -4 + 44 z 4 a -6 + 22 z 4 a -8 -5 z 4 a -10 + z 4 a -12 + 3 z 3 a -5 + 13 z 3 a -7 + 7 z 3 a -9 -3 z 3 a -11 -16 z 2 a -4 -35 z 2 a -6 -11 z 2 a -8 + 5 z 2 a -10 -3 z 2 a -12 -9 z a -5 -9 z a -7 + 7 a -4 + 11 a -6 + 3 a -8 -2 a -10 + 2 a -5 z -1 + 2 a -7 z -1 - a -4 z -2 -2 a -6 z -2 - a -8 z -2$ (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 =

5 is the signature of L11n433. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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