L8a3

(Knotscape image)

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

Visit L8a3's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 3 -2 v u 2 + 4 u 2 + 4 v u -2 u - v$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 -2 a 5 z -1 + 2 z 3 a 3 + 3 z a 3 + 2 a 3 z -1 + z 3 a - z a - a z -1 - z a -1$ (db)
Kauffman polynomial	$- z 5 a 7 + 3 z 3 a 7 -3 z a 7 + a 7 z -1 -2 z 6 a 6 + 5 z 4 a 6 -2 z 2 a 6 - z 7 a 5 -2 z 5 a 5 + 11 z 3 a 5 -8 z a 5 + 2 a 5 z -1 -5 z 6 a 4 + 11 z 4 a 4 -6 z 2 a 4 + a 4 - z 7 a 3 -4 z 5 a 3 + 12 z 3 a 3 -9 z a 3 + 2 a 3 z -1 -3 z 6 a 2 + 4 z 4 a 2 -3 z 2 a 2 -3 z 5 a + 3 z 3 a -3 z a + a z -1 -2 z 4 + z 2 - z 3 a -1 + z a -1$ (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 L8a3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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