L9n4

(Knotscape image)

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

Visit L9n4's page at the original Knot Atlas.

L9n4 is $9^2_{43}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_11,18,12,5 X_17,10,18,11 X_13,2,14,3
Gauss code	{1, 9, -5, -3}, {-4, -1, 2, 5, -6, 8, -7, 4, -9, -2, 3, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u v^5-1}{\sqrt{u} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 11 -2 a 11 z -1 + z 5 a 9 + 6 z 3 a 9 + 10 z a 9 + 5 a 9 z -1 - z 7 a 7 -7 z 5 a 7 -15 z 3 a 7 -11 z a 7 -3 a 7 z -1$ (db)
Kauffman polynomial	$- a 14 - z 3 a 11 + 4 z a 11 -2 a 11 z -1 - z 6 a 10 + 6 z 4 a 10 -10 z 2 a 10 + 5 a 10 - z 7 a 9 + 7 z 5 a 9 -16 z 3 a 9 + 15 z a 9 -5 a 9 z -1 - z 6 a 8 + 6 z 4 a 8 -10 z 2 a 8 + 5 a 8 - z 7 a 7 + 7 z 5 a 7 -15 z 3 a 7 + 11 z a 7 -3 a 7 z -1$ (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 L9n4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$	$i = -4$
$r = -7$	${\mathbb Z}$	${\mathbb Z}$
$r = -6$	${\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}^{2}$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$