L9n11

(Knotscape image)

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

Visit L9n11's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_15,18,16,5 X_11,16,12,17 X_17,12,18,13 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -7, 9, -2, -5, 6, 7, -3, -4, 5, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 - u 2 - v u + 2 u + v -1$ (db)
Jones polynomial	$-\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z 3 a 5 - z a 5 + a 5 z -1 + z 5 a 3 + 3 z 3 a 3 + z a 3 - a 3 z -1 - z 3 a -2 z a$ (db)
Kauffman polynomial	$- z 2 a 8 -3 z 3 a 7 + z a 7 - z 6 a 6 + z 4 a 6 -2 z 2 a 6 - z 7 a 5 + 2 z 5 a 5 -4 z 3 a 5 + 4 z a 5 - a 5 z -1 -3 z 6 a 4 + 7 z 4 a 4 -4 z 2 a 4 + a 4 - z 7 a 3 + z 5 a 3 + 2 z 3 a 3 + z a 3 - a 3 z -1 -2 z 6 a 2 + 6 z 4 a 2 -3 z 2 a 2 - z 5 a + 3 z 3 a -2 z a$ (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 L9n11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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