L10n45

(Knotscape image)

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

Visit L10n45's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_5,13,6,12 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_14,20,15,19 X_17,2,18,3
Gauss code	{1, 10, -6, 7, -5, -3}, {3, -1, 2, 6, -8, 5, 4, -9, -7, 8, -10, -2, 9, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + v u + v -1$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 5 z -1 - z 3 a 3 -4 z a 3 -2 a 3 z -1 + z 5 a + 5 z 3 a + 6 z a + 2 a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 - a 3 z 7 -2 a z 7 - z 7 a -1 + 6 a 2 z 6 + 6 z 6 + 6 a 3 z 5 + 12 a z 5 + 6 z 5 a -1 -10 a 2 z 4 -10 z 4 -10 a 3 z 3 -20 a z 3 -10 z 3 a -1 + a 4 z 2 + 6 a 2 z 2 + 5 z 2 + a 5 z + 7 a 3 z + 11 a z + 5 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10n45. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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