L9a47

(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_10,13,5,14 X_18,15,11,16 X_14,7,15,8 X_8,18,9,17 X_16,10,17,9 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 5, -6, 7, -3}, {9, -2, 3, -5, 4, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v u 2 - v w u 2 + w u 2 - u 2 + 2 v 2 u -4 v u - v 2 w u + 4 v w u -2 w u + u - v 2 + v + v 2 w -2 v w + w$ (db)
Jones polynomial	$q -3 + 6 q -1 -7 q -2 + 10 q -3 -8 q -4 + 8 q -5 -5 q -6 + 3 q -7 - q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$- a 8 + 3 z 2 a 6 + a 6 z -2 + 3 a 6 -2 z 4 a 4 -4 z 2 a 4 -2 a 4 z -2 -5 a 4 - z 4 a 2 + z 2 a 2 + a 2 z -2 + 3 a 2 + z 2$ (db)
Kauffman polynomial	$z 5 a 9 -2 z 3 a 9 + z a 9 + 3 z 6 a 8 -7 z 4 a 8 + 5 z 2 a 8 -2 a 8 + 3 z 7 a 7 -3 z 5 a 7 -4 z 3 a 7 + 3 z a 7 + z 8 a 6 + 8 z 6 a 6 -24 z 4 a 6 + 21 z 2 a 6 + a 6 z -2 -8 a 6 + 7 z 7 a 5 -10 z 5 a 5 + 5 z a 5 -2 a 5 z -1 + z 8 a 4 + 10 z 6 a 4 -26 z 4 a 4 + 24 z 2 a 4 + 2 a 4 z -2 -9 a 4 + 4 z 7 a 3 -3 z 5 a 3 - z 3 a 3 + 3 z a 3 -2 a 3 z -1 + 5 z 6 a 2 -8 z 4 a 2 + 7 z 2 a 2 + a 2 z -2 -4 a 2 + 3 z 5 a -3 z 3 a + z 4 - z 2$ (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 =

-2 is the signature of L9a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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