L11n452

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,20,12,21 X_13,19,14,22 X_21,18,22,9 X_17,13,18,12 X_8,16,5,15 X_14,8,15,7 X_19,17,20,16 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 8, -7}, {-9, 3, -5, 4}, {11, -2, -3, 6, -4, -8, 7, 9, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v w u 3 - w u 3 -2 v w u 2 + 2 w u 2 - v x u 2 + x u 2 + v w u - w u + 2 v x u -2 x u - v x + x$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}-q^{5/2}+2 q^{3/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + 2 a 5 z -1 + a 5 z -3 -3 z 3 a 3 -8 z a 3 -8 a 3 z -1 -3 a 3 z -3 + 2 z 5 a + 9 z 3 a + 15 z a + 11 a z -1 + 3 a z -3 - z 5 a -1 -7 z 3 a -1 -11 z a -1 -6 a -1 z -1 - a -1 z -3 + z 3 a -3 + 3 z a -3 + a -3 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -6 a 2 z 8 -4 z 8 - a 5 z 7 -3 a 3 z 7 -5 a z 7 -4 z 7 a -1 - z 7 a -3 + 7 a 4 z 6 + 23 a 2 z 6 - z 6 a -4 + 17 z 6 + 5 a 5 z 5 + 29 a 3 z 5 + 44 a z 5 + 27 z 5 a -1 + 7 z 5 a -3 -3 a 4 z 4 -12 a 2 z 4 + 7 z 4 a -2 + 5 z 4 a -4 -7 z 4 -10 a 5 z 3 -45 a 3 z 3 -75 a z 3 -51 z 3 a -1 -11 z 3 a -3 -7 a 4 z 2 -19 a 2 z 2 -15 z 2 a -2 -4 z 2 a -4 -23 z 2 + 10 a 5 z + 34 a 3 z + 50 a z + 35 z a -1 + 9 z a -3 + 9 a 4 + 21 a 2 + 6 a -2 + a -4 + 18-5 a 5 z -1 -14 a 3 z -1 -18 a z -1 -11 a -1 z -1 -2 a -3 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 =

1 is the signature of L11n452. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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