L10n89

(Knotscape image)

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

Visit L10n89's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_20,15,17,16 X_18,11,19,12 X_12,17,13,18 X_16,19,5,20 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 - w u 3 + u 3 + 2 v u 2 -2 v w u 2 + 2 w u 2 -3 u 2 -2 v u + 3 v w u -2 w u + 2 u + v - v w + w$ (db)
Jones polynomial	$q -2 -3 q -3 + 7 q -4 -7 q -5 + 9 q -6 -8 q -7 + 7 q -8 -4 q -9 + 2 q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 10 z -2 + 2 a 10 -5 z 2 a 8 -2 a 8 z -2 -8 a 8 + 3 z 4 a 6 + 8 z 2 a 6 + a 6 z -2 + 6 a 6 + z 4 a 4 + z 2 a 4$ (db)
Kauffman polynomial	$3 z 4 a 12 -5 z 2 a 12 + 2 a 12 + z 7 a 11 + z 5 a 11 -2 z 3 a 11 + z 8 a 10 + 2 z 6 a 10 -3 z 4 a 10 + a 10 z -2 -2 a 10 + 5 z 7 a 9 -3 z 5 a 9 -8 z 3 a 9 + 8 z a 9 -2 a 9 z -1 + z 8 a 8 + 8 z 6 a 8 -18 z 4 a 8 + 16 z 2 a 8 + 2 a 8 z -2 -9 a 8 + 4 z 7 a 7 - z 5 a 7 -8 z 3 a 7 + 8 z a 7 -2 a 7 z -1 + 6 z 6 a 6 -11 z 4 a 6 + 10 z 2 a 6 + a 6 z -2 -6 a 6 + 3 z 5 a 5 -2 z 3 a 5 + z 4 a 4 - z 2 a 4$ (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 =

-4 is the signature of L10n89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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