L10a18

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,10,13,9 X_18,13,19,14 X_16,7,17,8 X_8,17,9,18 X_20,15,5,16 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, 4, -8, 7, -5, 6, -4, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 3 v u 4 -3 u 4 -4 v u 3 + 4 u 3 + 4 v u 2 -4 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{3/2}-3 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{10}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -3 z a 7 -2 a 7 z -1 + 2 z 5 a 5 + 8 z 3 a 5 + 11 z a 5 + 5 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -10 z 3 a 3 -10 z a 3 -3 a 3 z -1 + z 5 a + 3 z 3 a + 2 z a$ (db)
Kauffman polynomial	$- z 4 a 10 + 2 z 2 a 10 - a 10 -2 z 5 a 9 + 2 z 3 a 9 -3 z 6 a 8 + 2 z 4 a 8 -4 z 7 a 7 + 6 z 5 a 7 -8 z 3 a 7 + 5 z a 7 -2 a 7 z -1 -3 z 8 a 6 + z 6 a 6 + 6 z 4 a 6 -11 z 2 a 6 + 5 a 6 - z 9 a 5 -7 z 7 a 5 + 24 z 5 a 5 -29 z 3 a 5 + 17 z a 5 -5 a 5 z -1 -6 z 8 a 4 + 11 z 6 a 4 + 2 z 4 a 4 -11 z 2 a 4 + 5 a 4 - z 9 a 3 -6 z 7 a 3 + 26 z 5 a 3 -29 z 3 a 3 + 16 z a 3 -3 a 3 z -1 -3 z 8 a 2 + 6 z 6 a 2 + 2 z 4 a 2 -4 z 2 a 2 -3 z 7 a + 10 z 5 a -10 z 3 a + 4 z a - z 6 + 3 z 4 -2 z 2$ (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 =

-3 is the signature of L10a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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