L8a13

(Knotscape image)

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

Visit L8a13'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_6,13,7,14 X_14,5,15,6 X_4,15,5,16 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -7, 3, -6, 5, -4, 2, -8}, {7, -1, 8, -3, 4, -5, 6, -2}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 -2 v 2 u 2 + 2 v u 2 - u 2 - v 3 u + 2 v 2 u -2 v u - v 2$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 - z 3 a 7 - z a 7 - a 7 z -1 -2 z 3 a 5 -3 z a 5 - z 3 a 3 - z a 3$ (db)
Kauffman polynomial	$- z 5 a 11 + 4 z 3 a 11 -4 z a 11 - z 6 a 10 + 2 z 4 a 10 - z 7 a 9 + 2 z 5 a 9 -2 z 3 a 9 + 3 z a 9 - a 9 z -1 -3 z 6 a 8 + 5 z 4 a 8 - z 2 a 8 + a 8 - z 7 a 7 - z 3 a 7 + 3 z a 7 - a 7 z -1 -2 z 6 a 6 + z 4 a 6 -3 z 5 a 5 + 4 z 3 a 5 -3 z a 5 -2 z 4 a 4 + z 2 a 4 - z 3 a 3 + z a 3$ (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 L8a13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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