L9n23

(Knotscape image)

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

Visit L9n23's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 w u 2 + v w u 2 + v 2 w u - u - v + 1$ (db)
Jones polynomial	$1- q -1 + 2 q -2 - q -3 + 3 q -4 - q -5 + 2 q -6 - q -7$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$- a 8 + z 4 a 6 + 4 z 2 a 6 + a 6 z -2 + 3 a 6 - z 6 a 4 -5 z 4 a 4 -7 z 2 a 4 -2 a 4 z -2 -5 a 4 + z 4 a 2 + 4 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z a 9 + 2 z 2 a 8 -2 a 8 + z 5 a 7 -3 z 3 a 7 + 3 z a 7 + 2 z 6 a 6 -9 z 4 a 6 + 13 z 2 a 6 + a 6 z -2 -8 a 6 + z 7 a 5 -3 z 5 a 5 - z 3 a 5 + 5 z a 5 -2 a 5 z -1 + 3 z 6 a 4 -14 z 4 a 4 + 18 z 2 a 4 + 2 a 4 z -2 -9 a 4 + z 7 a 3 -4 z 5 a 3 + 2 z 3 a 3 + 3 z a 3 -2 a 3 z -1 + z 6 a 2 -5 z 4 a 2 + 7 z 2 a 2 + a 2 z -2 -4 a 2$ (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 =

-4 is the signature of L9n23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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