L11n459

L11n459

(Knotscape image)

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

Link L11n459.

A graph, L11n459.

A part of a knot and a part of a graph.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,21,8,20 X_19,5,20,10 X_13,19,14,22 X_21,11,22,18 X_17,15,18,14 X_9,17,10,16 X_15,9,16,8 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {11, -2, -5, 7, -9, 8, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 w u 2 - v w u 2 -2 v 2 x u 2 + v x u 2 - v 2 w u + v w u + w u + v 2 x u + v x u - x u + v w -2 w - v x + x$ (db)
Jones polynomial	$q^{15/2}-2 q^{13/2}+q^{11/2}-2 q^{9/2}-q^{5/2}-2 q^{3/2}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + a z 3 -5 z 3 a -1 + z 3 a -3 - z 3 a -5 + 3 a z -8 z a -1 + 6 z a -3 -2 z a -5 + z a -7 + 3 a z -1 -8 a -1 z -1 + 7 a -3 z -1 -2 a -5 z -1 + a z -3 -3 a -1 z -3 + 3 a -3 z -3 - a -5 z -3$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -3 z 8 a -2 -3 z 8 a -4 - z 8 a -6 - z 8 - a z 7 + 4 z 7 a -1 + 3 z 7 a -3 -4 z 7 a -5 -2 z 7 a -7 + 17 z 6 a -2 + 18 z 6 a -4 + 4 z 6 a -6 - z 6 a -8 + 4 z 6 + 6 a z 5 + 3 z 5 a -1 + 12 z 5 a -3 + 25 z 5 a -5 + 10 z 5 a -7 -18 z 4 a -2 -23 z 4 a -4 + 4 z 4 a -8 + z 4 -11 a z 3 -19 z 3 a -1 -35 z 3 a -3 -39 z 3 a -5 -12 z 3 a -7 -10 z 2 a -2 -2 z 2 a -6 -2 z 2 a -8 -10 z 2 + 10 a z + 23 z a -1 + 27 z a -3 + 20 z a -5 + 6 z a -7 + 19 a -2 + 10 a -4 + 10-5 a z -1 -12 a -1 z -1 -12 a -3 z -1 -5 a -5 z -1 -6 a -2 z -2 -3 a -4 z -2 -3 z -2 + a z -3 + 3 a -1 z -3 + 3 a -3 z -3 + a -5 z -3$ (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 L11n459. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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