L8a12

(Knotscape image)

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

Visit L8a12's page at the original Knot Atlas.

Planar diagram presentation	X_10,1,11,2 X_16,7,9,8 X_12,3,13,4 X_14,5,15,6 X_4,13,5,14 X_6,15,7,16 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -7, 3, -5, 4, -6, 2, -8}, {7, -1, 8, -3, 5, -4, 6, -2}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 3 - v 3 u 2 + v 2 u 2 - v u 2 - v 2 u + v u - u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{2}{q^{17/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z 3 a 9 + 3 z a 9 + a 9 z -1 - z 5 a 7 -4 z 3 a 7 -4 z a 7 - a 7 z -1 - z 5 a 5 -4 z 3 a 5 -3 z a 5$ (db)
Kauffman polynomial	$- z 3 a 13 + 2 z a 13 - z 4 a 12 + z 2 a 12 - z 5 a 11 + z 3 a 11 - z a 11 - z 6 a 10 + 2 z 4 a 10 -3 z 2 a 10 - z 7 a 9 + 4 z 5 a 9 -8 z 3 a 9 + 5 z a 9 - a 9 z -1 -2 z 6 a 8 + 6 z 4 a 8 -5 z 2 a 8 + a 8 - z 7 a 7 + 4 z 5 a 7 -6 z 3 a 7 + 5 z a 7 - a 7 z -1 - z 6 a 6 + 3 z 4 a 6 - z 2 a 6 - z 5 a 5 + 4 z 3 a 5 -3 z a 5$ (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 =

-5 is the signature of L8a12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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