L10n83

(Knotscape image)

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

Visit L10n83's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,16,6,17 X₈₄₉₃ X_17,20,18,15 X_19,10,20,11 X_9,18,10,19 X_15,14,16,5 X_2,12,3,11
Gauss code	{1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 -2 v u 2 + 2 v w u 2 - w u 2 + u 2 + v u - v w u + 2 w u -2 u - w + 1$ (db)
Jones polynomial	$1-2 q -1 + 5 q -2 -4 q -3 + 6 q -4 -5 q -5 + 5 q -6 -3 q -7 + q -8$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z 4 a 6 + 2 z 2 a 6 + a 6 z -2 + a 6 - z 6 a 4 -4 z 4 a 4 -5 z 2 a 4 -2 a 4 z -2 -4 a 4 + z 4 a 2 + 3 z 2 a 2 + a 2 z -2 + 3 a 2$ (db)
Kauffman polynomial	$z 2 a 10 - a 10 + 3 z 3 a 9 - z a 9 + 2 z 6 a 8 -5 z 4 a 8 + 9 z 2 a 8 -3 a 8 + 3 z 7 a 7 -9 z 5 a 7 + 10 z 3 a 7 -3 z a 7 + z 8 a 6 + 3 z 6 a 6 -18 z 4 a 6 + 20 z 2 a 6 + a 6 z -2 -7 a 6 + 5 z 7 a 5 -15 z 5 a 5 + 9 z 3 a 5 + z a 5 -2 a 5 z -1 + z 8 a 4 + 2 z 6 a 4 -17 z 4 a 4 + 18 z 2 a 4 + 2 a 4 z -2 -8 a 4 + 2 z 7 a 3 -6 z 5 a 3 + 2 z 3 a 3 + 3 z a 3 -2 a 3 z -1 + z 6 a 2 -4 z 4 a 2 + 6 z 2 a 2 + a 2 z -2 -4 a 2$ (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 =

-4 is the signature of L10n83. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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