L8a18

(Knotscape image)

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Visit L8a18's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 - v 2 w u 2 + v w u 2 - v 2 u + v u - v w u + w u - v - w + 1$ (db)
Jones polynomial	$q 8 -2 q 7 + 3 q 6 -3 q 5 + 4 q 4 -2 q 3 + 3 q 2 - q + 1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$- z 6 a -4 + z 4 a -2 -5 z 4 a -4 + z 4 a -6 + 4 z 2 a -2 -8 z 2 a -4 + 3 z 2 a -6 + 4 a -2 -6 a -4 + 2 a -6 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$z 7 a -3 + z 7 a -5 + z 6 a -2 + 4 z 6 a -4 + 3 z 6 a -6 -3 z 5 a -3 + 3 z 5 a -7 -5 z 4 a -2 -16 z 4 a -4 -8 z 4 a -6 + 3 z 4 a -8 - z 3 a -3 -7 z 3 a -5 -4 z 3 a -7 + 2 z 3 a -9 + 8 z 2 a -2 + 18 z 2 a -4 + 6 z 2 a -6 -3 z 2 a -8 + z 2 a -10 + 6 z a -3 + 6 z a -5 -5 a -2 -8 a -4 -3 a -6 + a -8 -2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

4 is the signature of L8a18. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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