L10a19

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,10,13,9 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X_20,13,5,14 X_14,19,15,20 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -6, 3, -2, 10, -3, 7, -8, 4, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 6 v u 2 -6 u 2 -6 v u + 6 u + 2 v -2$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 2 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 - z 5 a 3 - z 3 a 3 - z 5 a -2 z 3 a -3 z a - a z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$- a 5 z 9 - a 3 z 9 -2 a 6 z 8 -5 a 4 z 8 -3 a 2 z 8 -2 a 7 z 7 -3 a 5 z 7 -5 a 3 z 7 -4 a z 7 - a 8 z 6 + 2 a 6 z 6 + 7 a 4 z 6 -4 z 6 + 6 a 7 z 5 + 12 a 5 z 5 + 10 a 3 z 5 + a z 5 -3 z 5 a -1 + 4 a 8 z 4 + 8 a 6 z 4 + 4 a 4 z 4 + 4 a 2 z 4 - z 4 a -2 + 3 z 4 -5 a 7 z 3 -9 a 5 z 3 -3 a 3 z 3 + 5 a z 3 + 4 z 3 a -1 -5 a 8 z 2 -12 a 6 z 2 -8 a 4 z 2 -2 a 2 z 2 + z 2 a -2 + 2 a 7 z + 5 a 5 z -5 a z -2 z a -1 + 2 a 8 + 5 a 6 + 3 a 4 - a 2 - a 7 z -1 -2 a 5 z -1 + a z -1$ (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 L10a19. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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