L11n453

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_15,2,16,3 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_4,17,1,18
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 $u$ , $v$ , $w$ , ...)	$v 2 w u 2 - v w u 2 -2 v 2 w x u 2 + v w x u 2 - v 2 w u + v w u + v x u + v 2 w x u - x u + u + v - v x + x -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{3}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 + a 13 z -3 + z 5 a 11 + 4 z 3 a 11 + z a 11 -5 a 11 z -1 -3 a 11 z -3 - z 7 a 9 -4 z 5 a 9 + 11 z a 9 + 10 a 9 z -1 + 3 a 9 z -3 - z 7 a 7 -6 z 5 a 7 -12 z 3 a 7 -11 z a 7 -5 a 7 z -1 - a 7 z -3$ (db)
Kauffman polynomial	$- z 2 a 16 -3 z 3 a 15 + 3 z a 15 - z 6 a 14 + z 4 a 14 - z 2 a 14 -3 z 7 a 13 + 12 z 5 a 13 -18 z 3 a 13 + 12 z a 13 -5 a 13 z -1 + a 13 z -3 -3 z 8 a 12 + 12 z 6 a 12 -12 z 4 a 12 -3 z 2 a 12 -3 a 12 z -2 + 10 a 12 - z 9 a 11 + 14 z 5 a 11 -25 z 3 a 11 + 21 z a 11 -12 a 11 z -1 + 3 a 11 z -3 -4 z 8 a 10 + 16 z 6 a 10 -10 z 4 a 10 -15 z 2 a 10 -6 a 10 z -2 + 19 a 10 - z 9 a 9 + 2 z 7 a 9 + 8 z 5 a 9 -22 z 3 a 9 + 23 z a 9 -12 a 9 z -1 + 3 a 9 z -3 - z 8 a 8 + 3 z 6 a 8 + 3 z 4 a 8 -12 z 2 a 8 -3 a 8 z -2 + 10 a 8 - z 7 a 7 + 6 z 5 a 7 -12 z 3 a 7 + 11 z a 7 -5 a 7 z -1 + a 7 z -3$ (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 =

-7 is the signature of L11n453. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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