L8a21

(Knotscape image)

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

Visit L8a21's page at the original Knot Atlas.

L8a21 is a closed four-link chain.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u - v w u + w u - v x u - w x u + 2 x u - u - v + 2 v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 11 z -3 -4 a 9 z -1 -3 a 9 z -3 + 6 z a 7 + 8 a 7 z -1 + 3 a 7 z -3 -3 z 3 a 5 -6 z a 5 -4 a 5 z -1 - a 5 z -3 - z 3 a 3$ (db)
Kauffman polynomial	$- z 5 a 11 + 4 z 3 a 11 -6 z a 11 + 4 a 11 z -1 - a 11 z -3 - z 6 a 10 + 6 z 2 a 10 + 3 a 10 z -2 -8 a 10 - z 7 a 9 -2 z 5 a 9 + 11 z 3 a 9 -14 z a 9 + 9 a 9 z -1 -3 a 9 z -3 -5 z 6 a 8 + 5 z 4 a 8 + 12 z 2 a 8 + 6 a 8 z -2 -15 a 8 - z 7 a 7 -7 z 5 a 7 + 17 z 3 a 7 -14 z a 7 + 9 a 7 z -1 -3 a 7 z -3 -4 z 6 a 6 + 2 z 4 a 6 + 6 z 2 a 6 + 3 a 6 z -2 -8 a 6 -6 z 5 a 5 + 9 z 3 a 5 -6 z a 5 + 4 a 5 z -1 - a 5 z -3 -3 z 4 a 4 - z 3 a 3$ (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 L8a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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