L10n21

(Knotscape image)

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

Visit L10n21's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_9,16,10,17 X_17,5,18,20 X_13,19,14,18 X_19,15,20,14 X_15,8,16,9 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 8, -4, -2, 10, -3, -6, 7, -8, 4, -5, 6, -7, 5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+3 q^{11/2}-4 q^{9/2}+5 q^{7/2}-6 q^{5/2}+5 q^{3/2}-5 \sqrt{q}+\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -3 -2 z 3 a -1 + 3 z 3 a -3 - z 3 a -5 + a z -3 z a -1 + 3 z a -3 - z a -5 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$- z 8 a -2 - z 8 a -4 - z 7 a -1 -4 z 7 a -3 -3 z 7 a -5 + 2 z 6 a -2 - z 6 a -4 -3 z 6 a -6 + 3 z 5 a -1 + 12 z 5 a -3 + 8 z 5 a -5 - z 5 a -7 -3 z 4 a -2 + 7 z 4 a -4 + 8 z 4 a -6 -2 z 4 - a z 3 -9 z 3 a -1 -15 z 3 a -3 -5 z 3 a -5 + 2 z 3 a -7 - z 2 a -2 -5 z 2 a -4 -3 z 2 a -6 + z 2 + 2 a z + 6 z a -1 + 6 z a -3 + 2 z a -5 + 1- a z -1 - a -1 z -1$ (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 L10n21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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