L9a49

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u 2 - v w u 2 + w u 2 - u 2 + 2 v 2 u -3 v u - v 2 w u + 3 v w u -2 w u + u - v 2 + v + v 2 w -2 v w$ (db)
Jones polynomial	$- q 5 + 3 q 4 -4 q 3 + 7 q 2 -7 q + 8-6 q -1 + 5 q -2 -2 q -3 + q -4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a 4 -2 z 2 a 2 + a 2 z -2 + z 4 - z 2 -2 z -2 -3 + z 4 a -2 + z 2 a -2 + a -2 z -2 + 2 a -2 - z 2 a -4$ (db)
Kauffman polynomial	$z 8 a -2 + z 8 + 2 a z 7 + 5 z 7 a -1 + 3 z 7 a -3 + 3 a 2 z 6 + 4 z 6 a -2 + 3 z 6 a -4 + 4 z 6 + 2 a 3 z 5 + 2 a z 5 -7 z 5 a -1 -6 z 5 a -3 + z 5 a -5 + a 4 z 4 -4 a 2 z 4 -13 z 4 a -2 -8 z 4 a -4 -10 z 4 -2 a 3 z 3 -7 a z 3 - z 3 a -1 + 2 z 3 a -3 -2 z 3 a -5 -2 a 4 z 2 + 4 a 2 z 2 + 11 z 2 a -2 + 5 z 2 a -4 + 12 z 2 + 6 a z + 6 z a -1 + a 4 -3 a 2 -5 a -2 -8-2 a z -1 -2 a -1 z -1 + a 2 z -2 + a -2 z -2 + 2 z -2$ (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 =

0 is the signature of L9a49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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