L10n28

(Knotscape image)

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

Visit L10n28's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_11,19,12,18 X_15,20,16,5 X_19,16,20,17 X_17,13,18,12 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, 10, -2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 - u 3 + 2 v u 2 + u 2 + v u + 2 u - v -1$ (db)
Jones polynomial	$-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{2}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z -1 -2 z a 7 -2 a 7 z -1 + a 5 z -1 + z 5 a 3 + 4 z 3 a 3 + 3 z a 3 + a 3 z -1 - z 3 a -3 z a - a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 6 z 5 a 9 -10 z 3 a 9 + 6 z a 9 - a 9 z -1 - z 8 a 8 + 6 z 6 a 8 -9 z 4 a 8 + 4 z 2 a 8 - a 8 -2 z 7 a 7 + 14 z 5 a 7 -26 z 3 a 7 + 14 z a 7 -2 a 7 z -1 - z 8 a 6 + 7 z 6 a 6 -13 z 4 a 6 + 8 z 2 a 6 -3 a 6 - z 7 a 5 + 7 z 5 a 5 -13 z 3 a 5 + 8 z a 5 - a 5 z -1 - z 4 a 4 + 4 z 2 a 4 -2 a 4 -2 z 5 a 3 + 7 z 3 a 3 -4 z a 3 + a 3 z -1 - z 6 a 2 + 3 z 4 a 2 - a 2 - z 5 a + 4 z 3 a -4 z a + a z -1$ (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 =

-3 is the signature of L10n28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$	$i = 0$
$r = -8$		${\mathbb Z}$
$r = -7$		${\mathbb Z}_2$	${\mathbb Z}$
$r = -6$		${\mathbb Z}$
$r = -5$		${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = 1$	${\mathbb Z}$
$r = 2$	${\mathbb Z}_2$	${\mathbb Z}$