L8a17

(Knotscape image)

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

Visit L8a17's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 - v w u 2 + v 2 u -2 v u - v 2 w u + 2 v w u - w u + u + v - v w + w$ (db)
Jones polynomial	$q -2 -2 q -3 + 4 q -4 -4 q -5 + 6 q -6 -4 q -7 + 4 q -8 -2 q -9 + q -10$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a 10 z -2 + a 10 -3 z 2 a 8 -2 a 8 z -2 -6 a 8 + 2 z 4 a 6 + 6 z 2 a 6 + a 6 z -2 + 5 a 6 + z 4 a 4 + 2 z 2 a 4$ (db)
Kauffman polynomial	$z 4 a 12 -2 z 2 a 12 + a 12 + 2 z 5 a 11 -3 z 3 a 11 + 2 z 6 a 10 -2 z 4 a 10 + z 2 a 10 + a 10 z -2 -3 a 10 + z 7 a 9 + 2 z 5 a 9 -6 z 3 a 9 + 6 z a 9 -2 a 9 z -1 + 5 z 6 a 8 -12 z 4 a 8 + 15 z 2 a 8 + 2 a 8 z -2 -8 a 8 + z 7 a 7 + 2 z 5 a 7 -6 z 3 a 7 + 6 z a 7 -2 a 7 z -1 + 3 z 6 a 6 -8 z 4 a 6 + 10 z 2 a 6 + a 6 z -2 -5 a 6 + 2 z 5 a 5 -3 z 3 a 5 + z 4 a 4 -2 z 2 a 4$ (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 =

-4 is the signature of L8a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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