L9a32

(Knotscape image)

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

Visit L9a32's page at the original Knot Atlas.

A traditional symbol of the Christian Trinity (a triquetra interlaced with a circle, or "Trinity knot")

Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_18,13,7,14 X_14,9,15,10 X_10,17,11,18 X_16,5,17,6 X₂₇₃₈ X_4,11,5,12 X_6,15,1,16
Gauss code	{1, -7, 2, -8, 6, -9}, {7, -1, 4, -5, 8, -2, 3, -4, 9, -6, 5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 4 -3 v u 3 + 3 u 3 -3 v 2 u 2 + 5 v u 2 -3 u 2 + 3 v 2 u -3 v u - v 2$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{8}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 3 a 9 z -1 -3 z 3 a 7 -4 z a 7 -2 a 7 z -1 -3 z 3 a 5 -3 z a 5 - z 3 a 3$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 10 z 5 a 11 -9 z 3 a 11 + 2 z a 11 - a 11 z -1 -2 z 8 a 10 + z 6 a 10 + 12 z 4 a 10 -14 z 2 a 10 + 3 a 10 -9 z 7 a 9 + 26 z 5 a 9 -22 z 3 a 9 + 11 z a 9 -3 a 9 z -1 -2 z 8 a 8 -5 z 6 a 8 + 21 z 4 a 8 -14 z 2 a 8 + 3 a 8 -6 z 7 a 7 + 10 z 5 a 7 -6 z 3 a 7 + 6 z a 7 -2 a 7 z -1 -7 z 6 a 6 + 9 z 4 a 6 -3 z 2 a 6 -6 z 5 a 5 + 6 z 3 a 5 -3 z a 5 -3 z 4 a 4 - z 3 a 3$ (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 L9a32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$