L10n82

(Knotscape image)

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

Visit L10n82's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_11,20,12,13 X_7,18,8,19 X_9,16,10,17 X_17,8,18,9 X_13,10,14,11 X_19,12,20,5 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- w u 3 + v 2 u 2 - v u 2 - v 2 w u 2 + v w u 2 - v 2 u + v u + v 2 w u - v w u + v 3$ (db)
Jones polynomial	$q -2 + 3 q -1 -2 q -2 + 3 q -3 -2 q -4 + q -5 + q -6 + q -8$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 8 z -2 + a 8 - z 2 a 6 -2 a 6 z -2 -3 a 6 + z 2 a 4 + a 4 z -2 + a 4 - z 4 a 2 -2 z 2 a 2 + z 2 + 1$ (db)
Kauffman polynomial	$z 8 a 8 -8 z 6 a 8 + 21 z 4 a 8 -22 z 2 a 8 - a 8 z -2 + 8 a 8 + z 7 a 7 -8 z 5 a 7 + 17 z 3 a 7 -11 z a 7 + 2 a 7 z -1 + z 8 a 6 -9 z 6 a 6 + 25 z 4 a 6 -27 z 2 a 6 -2 a 6 z -2 + 13 a 6 + z 7 a 5 -8 z 5 a 5 + 19 z 3 a 5 -11 z a 5 + 2 a 5 z -1 + 3 z 4 a 4 -6 z 2 a 4 - a 4 z -2 + 5 a 4 + 2 z 5 a 3 -2 z 3 a 3 + z 6 a 2 -3 z 2 a 2 + 2 z 5 a -4 z 3 a + z 4 -2 z 2 + 1$ (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 =

-2 is the signature of L10n82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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