L10n55

(Knotscape image)

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

Visit L10n55's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_11,17,12,16 X_8,9,1,10 X_17,9,18,20 X_3,12,4,13 X_7,14,8,15 X_13,6,14,7 X_5,18,6,19 X_19,4,20,5 X_15,2,16,3
Gauss code	{1, 10, -5, 9, -8, 7, -6, -3}, {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{u^3 (-v)+u^2 v^3-3 u^2 v^2+3 u^2 v-2 u^2-2 u v^3+3 u v^2-3 u v+u-v^2}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$\frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 9 z + a 9 z -1 -2 a 7 z 3 -3 a 7 z - a 7 z -1 + a 5 z 5 + 2 a 5 z 3 + a 5 z -2 a 3 z 3 -3 a 3 z$ (db)
Kauffman polynomial	$a 11 z 5 -3 a 11 z 3 + 2 a 11 z + 2 a 10 z 6 -4 a 10 z 4 + a 10 z 2 + 3 a 9 z 7 -8 a 9 z 5 + 9 a 9 z 3 -7 a 9 z + a 9 z -1 + a 8 z 8 + 3 a 8 z 6 -10 a 8 z 4 + 6 a 8 z 2 - a 8 + 5 a 7 z 7 -11 a 7 z 5 + 12 a 7 z 3 -7 a 7 z + a 7 z -1 + a 6 z 8 + 2 a 6 z 6 -4 a 6 z 4 + 3 a 6 z 2 + 2 a 5 z 7 -2 a 5 z 5 + 3 a 5 z 3 - a 5 z + a 4 z 6 + 2 a 4 z 4 -2 a 4 z 2 + 3 a 3 z 3 -3 a 3 z$ (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 L10n55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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