L10n53

(Knotscape image)

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

Visit L10n53's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_20,13,7,14 X_9,15,10,14 X_19,11,20,10 X_5,16,6,17 X_15,18,16,19 X₂₇₃₈ X_4,11,5,12 X_17,6,18,1
Gauss code	{1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 3 v u 2 - u 2 + 3 v 2 u -5 v u + 3 u - v 2 + 3 v -1$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$2 z a 7 + a 7 z -1 -3 z 3 a 5 -5 z a 5 - a 5 z -1 + z 5 a 3 + 2 z 3 a 3 + z a 3 - z 3 a - z a$ (db)
Kauffman polynomial	$-3 z 3 a 9 + 3 z a 9 - z 6 a 8 -2 z 4 a 8 + 2 z 2 a 8 -2 z 7 a 7 + 2 z 5 a 7 -3 z 3 a 7 - z a 7 + a 7 z -1 - z 8 a 6 -2 z 6 a 6 + 2 z 4 a 6 + z 2 a 6 - a 6 -5 z 7 a 5 + 8 z 5 a 5 -2 z 3 a 5 -3 z a 5 + a 5 z -1 - z 8 a 4 -4 z 6 a 4 + 11 z 4 a 4 -4 z 2 a 4 -3 z 7 a 3 + 5 z 5 a 3 -3 z 6 a 2 + 7 z 4 a 2 -3 z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L10n53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -6$	${\mathbb Z}^{2}$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = 1$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$