L9n5

(Knotscape image)

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

Visit L9n5's page at the original Knot Atlas.

L9n5 is $9^2_{44}$ in the Rolfsen table of links.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_11,18,12,5 X_17,10,18,11 X_2,14,3,13
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 $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$\frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 7 -3 z a 7 -2 a 7 z -1 + z 5 a 5 + 5 z 3 a 5 + 9 z a 5 + 5 a 5 z -1 -2 z 3 a 3 -6 z a 3 -3 a 3 z -1$ (db)
Kauffman polynomial	$a 10 z 4 -3 a 10 z 2 + a 10 + a 9 z 5 -2 a 9 z 3 + a 8 z 6 -2 a 8 z 4 + a 8 z 2 + a 7 z 7 -4 a 7 z 5 + 8 a 7 z 3 -5 a 7 z + 2 a 7 z -1 + 2 a 6 z 6 -7 a 6 z 4 + 12 a 6 z 2 -5 a 6 + a 5 z 7 -5 a 5 z 5 + 13 a 5 z 3 -13 a 5 z + 5 a 5 z -1 + a 4 z 6 -4 a 4 z 4 + 8 a 4 z 2 -5 a 4 + 3 a 3 z 3 -8 a 3 z + 3 a 3 z -1$ (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 =

-3 is the signature of L9n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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