L7a1

(Knotscape image)

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

Visit L7a1's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_10,6,11,5 X₈₄₉₃ X_14,10,5,9 X_2,12,3,11
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 2 v u 2 -2 u 2 -2 v u + 2 u + v -1$ (db)
Jones polynomial	$-q^{9/2}+3 q^{7/2}-4 q^{5/2}+4 q^{3/2}-5 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{3}{q^{3/2}}+\frac{1}{q^{5/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 - a z 3 + 3 z 3 a -1 - z 3 a -3 - a z + 2 z a -1 - z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 6 a -2 -2 z 6 -3 a z 5 -7 z 5 a -1 -4 z 5 a -3 - a 2 z 4 - z 4 a -2 -3 z 4 a -4 + z 4 + 6 a z 3 + 12 z 3 a -1 + 5 z 3 a -3 - z 3 a -5 + a 2 z 2 + 3 z 2 a -2 + 2 z 2 a -4 + 2 z 2 -2 a z -4 z a -1 -2 z a -3 + 1- a z -1 - a -1 z -1$ (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 L7a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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