L11a50

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_12,6,13,5 X_10,4,11,3 X_22,14,5,13 X_14,22,15,21 X_20,12,21,11 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, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 v u 3 + 4 u 3 + 10 v u 2 -10 u 2 -10 v u + 10 u + 4 v -4$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-7 q^{9/2}+11 q^{7/2}-15 q^{5/2}+18 q^{3/2}-18 \sqrt{q}+\frac{15}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 + 2 z 5 a -1 + z 5 a -3 - a 3 z 3 + 3 z 3 a -1 + z 3 a -3 - z 3 a -5 -2 a z + z a -1 + 2 z a -3 - z a -5 + a 3 z -1 - a z -1 - a -1 z -1 + 2 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -5 a z 9 -10 z 9 a -1 -5 z 9 a -3 -6 a 2 z 8 -5 z 8 a -2 -7 z 8 a -4 -4 z 8 -4 a 3 z 7 + 10 a z 7 + 27 z 7 a -1 + 7 z 7 a -3 -6 z 7 a -5 - a 4 z 6 + 16 a 2 z 6 + 22 z 6 a -2 + 17 z 6 a -4 -3 z 6 a -6 + 19 z 6 + 11 a 3 z 5 -4 a z 5 -29 z 5 a -1 + z 5 a -3 + 14 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -10 a 2 z 4 -31 z 4 a -2 -19 z 4 a -4 + 5 z 4 a -6 -19 z 4 -6 a 3 z 3 + 3 a z 3 + 8 z 3 a -1 -15 z 3 a -3 -12 z 3 a -5 + 2 z 3 a -7 + 2 a 2 z 2 + 15 z 2 a -2 + 8 z 2 a -4 + 9 z 2 - a 3 z -3 a z + 4 z a -1 + 11 z a -3 + 5 z a -5 - a 2 -3 a -2 - a -4 -2 + a 3 z -1 + a z -1 - a -1 z -1 -2 a -3 z -1 - a -5 z -1$ (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 =

1 is the signature of L11a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$