L10n48

(Knotscape image)

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

Visit L10n48's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_3,10,4,11 X_15,5,16,4 X_11,16,12,17 X_14,20,15,19 X_2,18,3,17
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 $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v u 2 + v 2 u -5 v u + u + 2 v -1$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + a 5 z -1 - z 5 a 3 -4 z 3 a 3 -6 z a 3 -2 a 3 z -1 + 2 z 3 a + 4 z a + 2 a z -1 - z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 -2 a 5 z 7 -3 a 3 z 7 - a z 7 -2 a 6 z 6 + 2 a 4 z 6 + 4 a 2 z 6 - a 7 z 5 + 6 a 5 z 5 + 11 a 3 z 5 + 4 a z 5 + 6 a 6 z 4 - a 4 z 4 -9 a 2 z 4 -2 z 4 + 3 a 7 z 3 -5 a 5 z 3 -18 a 3 z 3 -11 a z 3 - z 3 a -1 -3 a 6 z 2 + 5 a 2 z 2 + 2 z 2 - a 7 z + 3 a 5 z + 11 a 3 z + 9 a z + 2 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10n48. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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