L6a1

(Knotscape image)

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

Visit L6a1's page at the original Knot Atlas.

A kolam with two cycles/components[1]

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u + 2 u + 2 v -1$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 -2 z a 3 - a 3 z -1 + z 3 a + z a - z a -1$ (db)
Kauffman polynomial	$- z 3 a 5 + 2 z a 5 - a 5 z -1 - z 4 a 4 + a 4 - z 5 a 3 + z a 3 - a 3 z -1 -3 z 4 a 2 + 3 z 2 a 2 - z 5 a -2 z 4 + 3 z 2 - z 3 a -1 + z a -1$ (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 L6a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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