L11a52

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_14,6,15,5 X_10,4,11,3 X_20,12,21,11 X_22,14,5,13 X_12,22,13,21 X_16,9,17,10 X_2,16,3,15 X_8,17,9,18
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 6, -8, 7, -4, 10, -9, 11, -2, 3, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 4 v u 4 -4 u 4 -4 v u 3 + 4 u 3 + 4 v u 2 -4 u 2 -4 v u + 4 u + 2 v -2$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-11 q^{7/2}+12 q^{5/2}-12 q^{3/2}+10 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -1 + z 7 a -3 - a z 5 + 4 z 5 a -1 + 4 z 5 a -3 - z 5 a -5 -3 a z 3 + 3 z 3 a -1 + 5 z 3 a -3 -3 z 3 a -5 -3 z a -1 + 5 z a -3 -2 z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -3 a z 9 -8 z 9 a -1 -5 z 9 a -3 - a 2 z 8 -8 z 8 a -4 + 7 z 8 + 17 a z 7 + 36 z 7 a -1 + 9 z 7 a -3 -10 z 7 a -5 + 5 a 2 z 6 + 18 z 6 a -2 + 16 z 6 a -4 -9 z 6 a -6 -2 z 6 -31 a z 5 -48 z 5 a -1 + 9 z 5 a -3 + 20 z 5 a -5 -6 z 5 a -7 -7 a 2 z 4 -17 z 4 a -2 + z 4 a -4 + 13 z 4 a -6 -3 z 4 a -8 -9 z 4 + 18 a z 3 + 16 z 3 a -1 -16 z 3 a -3 -9 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 + 2 a 2 z 2 -2 z 2 a -2 -8 z 2 a -4 -5 z 2 a -6 + 3 z 2 + a z + 7 z a -1 + 9 z a -3 + 2 z a -5 - z a -7 + 3 a -2 + 3 a -4 + a -6 + 2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L11a52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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