L9n28

(Knotscape image)

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

Visit L9n28's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_18,10,15,9 X₈₄₉₃ X_5,17,6,16 X_17,5,18,14 X_10,16,11,15 X_2,12,3,11
Gauss code	{1, -9, 5, -3}, {8, 6, -7, -4}, {-6, -1, 2, -5, 4, -8, 9, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 -2 v u 2 + 2 v w u 2 - w u 2 + u 2 + v u - v w u + 2 w u -2 u - w + 1$ (db)
Jones polynomial	$3 q 5 -4 q 4 + 6 q 3 -5 q 2 + 6 q -4 + 3 q -1 - q -2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z 6 a -2 + 4 z 4 a -2 - z 4 a -4 - z 4 + 5 z 2 a -2 -3 z 2 a -4 -2 z 2 + 3 a -2 -4 a -4 + a -6 + a -2 z -2 -2 a -4 z -2 + a -6 z -2$ (db)
Kauffman polynomial	$2 z 7 a -1 + 2 z 7 a -3 + 8 z 6 a -2 + 5 z 6 a -4 + 3 z 6 + a z 5 - z 5 a -1 + z 5 a -3 + 3 z 5 a -5 -21 z 4 a -2 -13 z 4 a -4 -8 z 4 -2 a z 3 -7 z 3 a -1 -8 z 3 a -3 -3 z 3 a -5 + 17 z 2 a -2 + 18 z 2 a -4 + 6 z 2 a -6 + 5 z 2 + a z + 3 z a -1 + 7 z a -3 + 5 z a -5 -7 a -2 -10 a -4 -5 a -6 -1-2 a -3 z -1 -2 a -5 z -1 + a -2 z -2 + 2 a -4 z -2 + a -6 z -2$ (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 =

2 is the signature of L9n28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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