L10n3

(Knotscape image)

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

Visit L10n3's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,10,6,11 X₈₄₉₃ X_11,19,12,18 X_17,5,18,20 X_19,13,20,12 X_9,16,10,17 X_2,14,3,13
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -9, 4, -6, 8, 10, -2, 3, 9, -7, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$q^{7/2}-2 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 - a 3 z 3 + 4 a z 3 -2 z 3 a -1 -2 a 3 z + 6 a z -5 z a -1 + z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a 2 z 8 - z 8 -2 a 3 z 7 -4 a z 7 -2 z 7 a -1 - a 4 z 6 + a 2 z 6 - z 6 a -2 + z 6 + 8 a 3 z 5 + 16 a z 5 + 8 z 5 a -1 + 4 a 4 z 4 + 8 a 2 z 4 + 3 z 4 a -2 + 7 z 4 -8 a 3 z 3 -19 a z 3 -13 z 3 a -1 -2 z 3 a -3 -4 a 4 z 2 -11 a 2 z 2 -7 z 2 a -2 - z 2 a -4 -13 z 2 + 4 a 3 z + 11 a z + 9 z a -1 + 2 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 =

1 is the signature of L10n3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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