L10n91

(Knotscape image)

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

Visit L10n91's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,20,14,17 X_7,18,8,19 X_17,10,18,11 X_9,15,10,14 X_15,9,16,8 X_19,16,20,5 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v w u 3 + 2 u 3 + 2 v w u 2 - u 2 + v w u -2 u -2 v w + 1$ (db)
Jones polynomial	$1- q -1 + q -2 - q -3 + q -5 + q -6 + 2 q -7 - q -8 + q -9$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 10 z -2 -2 a 8 z -2 -4 a 8 + z 4 a 6 + 6 z 2 a 6 + a 6 z -2 + 6 a 6 - z 6 a 4 -5 z 4 a 4 -6 z 2 a 4 -4 a 4 + z 4 a 2 + 4 z 2 a 2 + 2 a 2$ (db)
Kauffman polynomial	$z 6 a 10 -5 z 4 a 10 + 6 z 2 a 10 + a 10 z -2 -4 a 10 + z 7 a 9 -4 z 5 a 9 + 4 z a 9 -2 a 9 z -1 + z 8 a 8 -5 z 6 a 8 + 3 z 4 a 8 + 6 z 2 a 8 + 2 a 8 z -2 -5 a 8 + 2 z 7 a 7 -12 z 5 a 7 + 16 z 3 a 7 -4 z a 7 -2 a 7 z -1 + z 8 a 6 -5 z 6 a 6 + 2 z 4 a 6 + 6 z 2 a 6 + a 6 z -2 -2 a 6 + 2 z 7 a 5 -13 z 5 a 5 + 22 z 3 a 5 -12 z a 5 + 2 z 6 a 4 -11 z 4 a 4 + 12 z 2 a 4 -2 a 4 + z 7 a 3 -5 z 5 a 3 + 6 z 3 a 3 -4 z a 3 + z 6 a 2 -5 z 4 a 2 + 6 z 2 a 2 -2 a 2$ (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 L10n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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