L10n6

(Knotscape image)

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

Visit L10n6's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_13,5,14,20 X_11,19,12,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -8, 9, -7, 4, 10, -2, 3, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+2 q^{11/2}-3 q^{9/2}+4 q^{7/2}-4 q^{5/2}+4 q^{3/2}-4 \sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -3 -2 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 + a z -5 z a -1 + 6 z a -3 -2 z a -5 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 a -4 - z 7 a -1 -3 z 7 a -3 -2 z 7 a -5 + 4 z 6 a -2 + 2 z 6 a -4 -2 z 6 a -6 + 5 z 5 a -1 + 12 z 5 a -3 + 6 z 5 a -5 - z 5 a -7 -6 z 4 a -2 + z 4 a -4 + 6 z 4 a -6 - z 4 - a z 3 -13 z 3 a -1 -19 z 3 a -3 -4 z 3 a -5 + 3 z 3 a -7 -2 z 2 a -2 -5 z 2 a -4 -3 z 2 a -6 + 3 a z + 11 z a -1 + 11 z a -3 + 2 z a -5 - z a -7 + 3 a -2 + 3 a -4 + a -6 + 2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L10n6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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