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

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