L9n15

(Knotscape image)

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

Visit L9n15's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_16,11,17,12 X_3,10,4,11 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_9,14,10,15 X_13,18,14,7 X_17,4,18,5
Gauss code	{1, -4, -3, 9, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 7, 4, -2, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 - v u 2 -1$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 11 - a 11 z -1 + z 5 a 9 + 6 z 3 a 9 + 9 z a 9 + 3 a 9 z -1 - z 7 a 7 -7 z 5 a 7 -15 z 3 a 7 -11 z a 7 -2 a 7 z -1$ (db)
Kauffman polynomial	$a 12 + z a 11 - a 11 z -1 - z 6 a 10 + 6 z 4 a 10 -9 z 2 a 10 + 3 a 10 - z 7 a 9 + 7 z 5 a 9 -15 z 3 a 9 + 12 z a 9 -3 a 9 z -1 - z 6 a 8 + 6 z 4 a 8 -9 z 2 a 8 + 3 a 8 - z 7 a 7 + 7 z 5 a 7 -15 z 3 a 7 + 11 z a 7 -2 a 7 z -1$ (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 =

-7 is the signature of L9n15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$	$i = -4$
$r = -6$		${\mathbb Z}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$	${\mathbb Z}$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$