L11n454

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_19,22,20,15 X_21,14,22,11 X_13,20,14,21 X_9,18,10,19 X_11,10,12,5 X_17,1,18,4
Gauss code	{1, -4, -3, 11}, {-10, 2, -8, 7}, {-2, -1, 5, 3, -9, 10}, {4, -5, -11, 9, -6, 8, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v x u 2 -2 v w x u 2 + w x u 2 - x u 2 + 2 v w 2 u - w 2 u -2 v w u + 2 w u - v x u + 2 v w x u -2 w x u + 2 x u - v w 2 + w 2 + v w -2 w$ (db)
Jones polynomial	$-\frac{2}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 9 - z a 9 + a 9 z -1 + a 9 z -3 + z 5 a 7 + 2 z 3 a 7 - z a 7 -6 a 7 z -1 -3 a 7 z -3 + z 5 a 5 + 3 z 3 a 5 + 7 z a 5 + 9 a 5 z -1 + 3 a 5 z -3 -2 z 3 a 3 -5 z a 3 -4 a 3 z -1 - a 3 z -3$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 - z 2 a 12 -3 z 7 a 11 + 11 z 5 a 11 -10 z 3 a 11 + 3 z a 11 -3 z 8 a 10 + 9 z 6 a 10 -4 z 4 a 10 - z 2 a 10 - z 9 a 9 -3 z 7 a 9 + 21 z 5 a 9 -28 z 3 a 9 + 16 z a 9 -5 a 9 z -1 + a 9 z -3 -5 z 8 a 8 + 15 z 6 a 8 -10 z 4 a 8 -7 z 2 a 8 -3 a 8 z -2 + 10 a 8 - z 9 a 7 -2 z 7 a 7 + 16 z 5 a 7 -31 z 3 a 7 + 25 z a 7 -12 a 7 z -1 + 3 a 7 z -3 -2 z 8 a 6 + 4 z 6 a 6 - z 4 a 6 -15 z 2 a 6 -6 a 6 z -2 + 19 a 6 -2 z 7 a 5 + 6 z 5 a 5 -16 z 3 a 5 + 19 z a 5 -12 a 5 z -1 + 3 a 5 z -3 - z 6 a 4 + 2 z 4 a 4 -8 z 2 a 4 -3 a 4 z -2 + 10 a 4 -3 z 3 a 3 + 7 z a 3 -5 a 3 z -1 + a 3 z -3$ (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 =

-3 is the signature of L11n454. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}^{2}$	${\mathbb Z}^{2}$