L9a13

(Knotscape image)

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

Visit L9a13's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_18,13,5,14 X_14,7,15,8 X_8,17,9,18 X_16,9,17,10 X_10,15,11,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 6, -7, 9, -2, 3, -4, 7, -6, 5, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 -4 v u 2 + 5 u 2 + 5 v u -4 u -2 v$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 3 z a 9 + 2 a 9 z -1 -2 z 3 a 7 - z a 7 -3 z 3 a 5 -4 z a 5 - a 5 z -1 - z 3 a 3$ (db)
Kauffman polynomial	$- z 6 a 12 + 4 z 4 a 12 -5 z 2 a 12 + 2 a 12 -2 z 7 a 11 + 7 z 5 a 11 -7 z 3 a 11 + 2 z a 11 - a 11 z -1 - z 8 a 10 -2 z 6 a 10 + 15 z 4 a 10 -17 z 2 a 10 + 5 a 10 -6 z 7 a 9 + 16 z 5 a 9 -12 z 3 a 9 + 6 z a 9 -2 a 9 z -1 - z 8 a 8 -7 z 6 a 8 + 21 z 4 a 8 -13 z 2 a 8 + 3 a 8 -4 z 7 a 7 + 3 z 5 a 7 + 3 z 3 a 7 -6 z 6 a 6 + 7 z 4 a 6 - z 2 a 6 - a 6 -6 z 5 a 5 + 7 z 3 a 5 -4 z a 5 + a 5 z -1 -3 z 4 a 4 - z 3 a 3$ (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 L9a13. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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