L10n1

(Knotscape image)

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

Visit L10n1'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 X₃₈₄₉ X_11,19,12,18 X_17,5,18,20 X_19,13,20,12 X_9,16,10,17 X_13,2,14,3
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 $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 v u 4 -2 v u 3 -2 u 2 + 2 u -1$ (db)
Jones polynomial	$q^{3/2}-2 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + z 5 a 5 + 5 z 3 a 5 + 7 z a 5 + 4 a 5 z -1 - z 7 a 3 -6 z 5 a 3 -12 z 3 a 3 -11 z a 3 -4 a 3 z -1 + z 5 a + 4 z 3 a + 3 z a + a z -1$ (db)
Kauffman polynomial	$- z 2 a 8 + a 8 -2 z 3 a 7 + 3 z a 7 - a 7 z -1 - z 6 a 6 + 4 z 4 a 6 -8 z 2 a 6 + 4 a 6 -2 z 7 a 5 + 10 z 5 a 5 -19 z 3 a 5 + 15 z a 5 -4 a 5 z -1 - z 8 a 4 + 2 z 6 a 4 + 6 z 4 a 4 -15 z 2 a 4 + 7 a 4 -4 z 7 a 3 + 19 z 5 a 3 -27 z 3 a 3 + 16 z a 3 -4 a 3 z -1 - z 8 a 2 + 2 z 6 a 2 + 6 z 4 a 2 -11 z 2 a 2 + 4 a 2 -2 z 7 a + 9 z 5 a -10 z 3 a + 4 z a - a z -1 - z 6 + 4 z 4 -3 z 2 + 1$ (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 L10n1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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