L9a24

(Knotscape image)

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,11,17,12 X_18,13,7,14 X_12,17,13,18 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -8, 2, -7, 3, -9}, {8, -1, 9, -2, 4, -6, 5, -3, 7, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^2 v^3-u^2 v^2+u^2 v-u^2+u v^4-3 u v^3+3 u v^2-3 u v+u-v^4+v^3-v^2+v}{u v^2}$ (db)
Jones polynomial	$-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -2 z a 7 - a 7 z -1 + z 5 a 5 + 3 z 3 a 5 + 4 z a 5 + 3 a 5 z -1 + z 5 a 3 + 2 z 3 a 3 - z a 3 -2 a 3 z -1 - z 3 a -2 z a$ (db)
Kauffman polynomial	$- z 4 a 10 + 2 z 2 a 10 -2 z 5 a 9 + 3 z 3 a 9 -3 z 6 a 8 + 6 z 4 a 8 -5 z 2 a 8 + a 8 -3 z 7 a 7 + 8 z 5 a 7 -13 z 3 a 7 + 7 z a 7 - a 7 z -1 - z 8 a 6 -2 z 6 a 6 + 8 z 4 a 6 -11 z 2 a 6 + 3 a 6 -5 z 7 a 5 + 15 z 5 a 5 -22 z 3 a 5 + 14 z a 5 -3 a 5 z -1 - z 8 a 4 - z 6 a 4 + 6 z 4 a 4 -6 z 2 a 4 + 3 a 4 -2 z 7 a 3 + 4 z 5 a 3 -3 z 3 a 3 + 5 z a 3 -2 a 3 z -1 -2 z 6 a 2 + 5 z 4 a 2 -2 z 2 a 2 - z 5 a + 3 z 3 a -2 z a$ (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 L9a24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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