L9a29

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,11,17,12 X_14,6,15,5 X_4,16,5,15 X_18,13,7,14 X_12,17,13,18
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -9, 8, -6, 7, -5, 9, -8}

A Braid Representative

A Morse Link Presentation

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

-5 is the signature of L9a29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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