L9n12

(Knotscape image)

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

Visit L9n12's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_7,15,8,14 X_15,9,16,8 X_13,5,14,18 X_17,13,18,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, -4, 5, 9, -2, -3, 7, -6, 4, -5, 3, -7, 6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 - v u 3 + u 3 + v u 2 - u 2 - v$ (db)
Jones polynomial	$q^{11/2}-q^{9/2}+q^{7/2}-q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 5 a -1 + a z 3 -6 z 3 a -1 + 4 a z -9 z a -1 + 2 z a -3 + z a -5 + 3 a z -1 -5 a -1 z -1 + 2 a -3 z -1$ (db)
Kauffman polynomial	$- a z 7 - z 7 a -1 - z 6 a -2 - z 6 + 7 a z 5 + 8 z 5 a -1 - z 5 a -5 + 7 z 4 a -2 - z 4 a -4 - z 4 a -6 + 7 z 4 -15 a z 3 -19 z 3 a -1 - z 3 a -3 + 3 z 3 a -5 -13 z 2 a -2 + 2 z 2 a -4 + 3 z 2 a -6 -12 z 2 + 12 a z + 16 z a -1 + 3 z a -3 - z a -5 + 5 a -2 - a -6 + 5-3 a z -1 -5 a -1 z -1 -2 a -3 z -1$ (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 =

1 is the signature of L9n12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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