L9n16

(Knotscape image)

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

Visit L9n16's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_11,17,12,16 X_3,10,4,11 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_14,10,15,9 X_18,14,7,13 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 $u$ , $v$ , $w$ , ...)	$- v u 4 + v u 3 - u 3 - v u 2 - v 2 u + v u - v$ (db)
Jones polynomial	$\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 3 z a 5 + 2 a 5 z -1 - z 5 a 3 -5 z 3 a 3 -8 z a 3 -3 a 3 z -1 + z 3 a + 2 z a + a z -1$ (db)
Kauffman polynomial	$- z 5 a 7 + 4 z 3 a 7 -3 z a 7 - z 6 a 6 + 3 z 4 a 6 - z 2 a 6 - z 7 a 5 + 4 z 5 a 5 -6 z 3 a 5 + 6 z a 5 -2 a 5 z -1 -2 z 6 a 4 + 7 z 4 a 4 -8 z 2 a 4 + 3 a 4 - z 7 a 3 + 5 z 5 a 3 -12 z 3 a 3 + 11 z a 3 -3 a 3 z -1 - z 6 a 2 + 4 z 4 a 2 -8 z 2 a 2 + 3 a 2 -2 z 3 a + 2 z a - a z -1 - z 2 + 1$ (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 L9n16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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