L10n51

(Knotscape image)

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

Visit L10n51's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_11,19,12,18 X_5,12,6,13 X_17,4,18,5 X_14,7,15,8 X_16,14,17,13 X_20,15,7,16 X_6,19,1,20
Gauss code	{1, -2, 3, 6, -5, -10}, {7, -1, 2, -3, -4, 5, 8, -7, 9, -8, -6, 4, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-2 u^2 v^2+u^2 v+u v^4-3 u v^3+3 u v^2-3 u v+u+v^3-2 v^2}{u v^2}$ (db)
Jones polynomial	$-\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z + a 9 z -1 -2 a 7 z 3 -4 a 7 z -2 a 7 z -1 + a 5 z 5 + 3 a 5 z 3 + 4 a 5 z + 2 a 5 z -1 -2 a 3 z 3 -4 a 3 z - a 3 z -1$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -2 z a 11 -2 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -2 z 7 a 9 + 4 z 5 a 9 -2 z 3 a 9 + 3 z a 9 - a 9 z -1 - z 8 a 8 + 2 z 4 a 8 -4 z 7 a 7 + 11 z 5 a 7 -15 z 3 a 7 + 9 z a 7 -2 a 7 z -1 - z 8 a 6 + z 6 a 6 -3 z 4 a 6 + 2 z 2 a 6 - a 6 -2 z 7 a 5 + 6 z 5 a 5 -13 z 3 a 5 + 9 z a 5 -2 a 5 z -1 - z 6 a 4 -3 z 3 a 3 + 5 z a 3 - a 3 z -1$ (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 L10n51. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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