L10n113

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_20,15,17,16 X_16,19,13,20 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, 8, -9}, {-10, 3, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u v w + v w + u v x w - x w - v y w + y w + u x - u v x + u v y - u x y + x y - y$ (db)
Jones polynomial	$q 6 - q 5 + 5 q 4 - q 3 + 5 q 2 + 5 + q -1 + q -3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2 z 4 a -2 - z 4 + a 2 z 2 + 9 z 2 a -2 -3 z 2 a -4 -7 z 2 + 3 a 2 + 16 a -2 -8 a -4 + a -6 -12 + 3 a 2 z -2 + 15 a -2 z -2 -9 a -4 z -2 + 2 a -6 z -2 -11 z -2 + a 2 z -4 + 6 a -2 z -4 -4 a -4 z -4 + a -6 z -4 -4 z -4$ (db)
Kauffman polynomial	$z 7 a -3 + z 7 a -5 + a 2 z 6 + 5 z 6 a -2 + 6 z 6 a -4 + z 6 a -6 + z 6 + a z 5 + 5 z 5 a -1 + 3 z 5 a -3 - z 5 a -5 -6 a 2 z 4 -25 z 4 a -2 -25 z 4 a -4 -5 z 4 a -6 -11 z 4 -10 a z 3 -30 z 3 a -1 -30 z 3 a -3 -10 z 3 a -5 + 10 a 2 z 2 + 40 z 2 a -2 + 30 z 2 a -4 + 10 z 2 a -6 + 30 z 2 + 20 a z + 55 z a -1 + 55 z a -3 + 20 z a -5 -10 a 2 -31 a -2 -25 a -4 -10 a -6 -25-15 a z -1 -41 a -1 z -1 -41 a -3 z -1 -15 a -5 z -1 + 5 a 2 z -2 + 18 a -2 z -2 + 14 a -4 z -2 + 5 a -6 z -2 + 14 z -2 + 4 a z -3 + 12 a -1 z -3 + 12 a -3 z -3 + 4 a -5 z -3 - a 2 z -4 -6 a -2 z -4 -4 a -4 z -4 - a -6 z -4 -4 z -4$ (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 =

0 is the signature of L10n113. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -1$	$i = 1$	$i = 3$
$r = -4$		${\mathbb Z}$	${\mathbb Z}$
$r = -3$
$r = -2$		${\mathbb Z}$
$r = -1$		${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{6}$	${\mathbb Z}^{10}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{4}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 4$	${\mathbb Z}^{5}$	${\mathbb Z}^{5}$
$r = 5$	${\mathbb Z}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$