L8a4

(Knotscape image)

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

Visit L8a4's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + v -1$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+4 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 5 - a 5 z -1 + 2 z 3 a 3 + 4 z a 3 + 3 a 3 z -1 - z 5 a -3 z 3 a -4 z a -2 a z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$- a 3 z 7 - a z 7 -2 a 4 z 6 -5 a 2 z 6 -3 z 6 -2 a 5 z 5 -4 a 3 z 5 -5 a z 5 -3 z 5 a -1 - a 6 z 4 + 5 a 2 z 4 - z 4 a -2 + 3 z 4 + 3 a 5 z 3 + 9 a 3 z 3 + 11 a z 3 + 5 z 3 a -1 + 2 a 6 z 2 + 4 a 4 z 2 + 2 a 2 z 2 + z 2 a -2 + z 2 -2 a 5 z -7 a 3 z -7 a z -2 z a -1 - a 6 -3 a 4 -3 a 2 + a 5 z -1 + 3 a 3 z -1 + 2 a z -1$ (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 L8a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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