L8a6

(Knotscape image)

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

Visit L8a6's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,5,7 X_14,10,15,9 X_10,14,11,13 X_8,16,9,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -7, 2, -8}, {7, -1, 3, -6, 4, -5, 8, -2, 5, -4, 6, -3}

A Braid Representative

A Morse Link Presentation

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

-1 is the signature of L8a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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