L10n52

(Knotscape image)

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

Visit L10n52's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_9,19,10,18 X_14,6,15,5 X_16,12,17,11 X_3,10,4,11 X_12,7,13,8 X_20,15,7,16 X_6,14,1,13 X_4,19,5,20 X_17,2,18,3
Gauss code	{1, 10, -5, -9, 3, -8}, {6, -1, -2, 5, 4, -6, 8, -3, 7, -4, -10, 2, 9, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 4 - u 4 -3 v u 3 + 2 u 3 - v 2 u 2 + 5 v u 2 - u 2 + 2 v 2 u -3 v u - v 2 + v$ (db)
Jones polynomial	$-2 q^{3/2}+4 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 + a 5 z -1 - z 5 a 3 -3 z 3 a 3 -5 z a 3 -2 a 3 z -1 + 3 z 3 a + 5 z a + 2 a z -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 4 z 8 -2 a 2 z 8 -4 a 5 z 7 -7 a 3 z 7 -3 a z 7 -3 a 6 z 6 + 2 a 4 z 6 + 4 a 2 z 6 - z 6 - a 7 z 5 + 11 a 5 z 5 + 21 a 3 z 5 + 9 a z 5 + 7 a 6 z 4 + 2 a 4 z 4 -7 a 2 z 4 -2 z 4 + 2 a 7 z 3 -9 a 5 z 3 -26 a 3 z 3 -18 a z 3 -3 z 3 a -1 -2 a 6 z 2 - a 4 z 2 + 4 a 2 z 2 + 3 z 2 + 4 a 5 z + 12 a 3 z + 12 a z + 4 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 =

-1 is the signature of L10n52. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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