L8a15

(Knotscape image)

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

Visit L8a15's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u - v w u + 2 w u -2 u -2 v + 2 v w -2 w + 1$ (db)
Jones polynomial	$q -3 + 4 q -1 -4 q -2 + 6 q -3 -4 q -4 + 4 q -5 - q -6 + q -7$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 8 z -2 -2 a 6 z -2 -3 a 6 + 3 z 2 a 4 + a 4 z -2 + 3 a 4 - z 4 a 2 - z 2 a 2 + z 2$ (db)
Kauffman polynomial	$z 4 a 8 -3 z 2 a 8 - a 8 z -2 + 3 a 8 + z 5 a 7 -3 z a 7 + 2 a 7 z -1 + z 6 a 6 + 2 z 4 a 6 -6 z 2 a 6 -2 a 6 z -2 + 5 a 6 + z 7 a 5 + 2 z 3 a 5 -3 z a 5 + 2 a 5 z -1 + 4 z 6 a 4 -5 z 4 a 4 - a 4 z -2 + 3 a 4 + z 7 a 3 + 2 z 5 a 3 -3 z 3 a 3 + 3 z 6 a 2 -5 z 4 a 2 + 2 z 2 a 2 + 3 z 5 a -5 z 3 a + z 4 - z 2$ (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 =

-2 is the signature of L8a15. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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