L10n20

(Knotscape image)

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

Visit L10n20's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_11,14,12,15 X₈₄₉₃ X_5,13,6,12 X_13,5,14,20 X_9,16,10,17 X_15,10,16,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -8, 9, -4, 6, -7, 4, -9, 8, 10, -2, 3, 7}

A Braid Representative

A Morse Link Presentation

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

-1 is the signature of L10n20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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