L9a20

(Knotscape image)

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

Visit L9a20's page at the original Knot Atlas.

Planar diagram presentation	X₈₁₉₂ X_16,11,17,12 X_10,4,11,3 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_14,10,15,9 X_18,14,7,13 X_4,18,5,17
Gauss code	{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + v u 4 + 3 v 2 u 3 -4 v u 3 + u 3 -3 v 2 u 2 + 7 v u 2 -3 u 2 + v 2 u -4 v u + 3 u + v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 4 a z 5 - z 5 a -1 -2 a 3 z 3 + 4 a z 3 -2 z 3 a -1 - a z + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-3 a 2 z 8 -3 z 8 -7 a 3 z 7 -13 a z 7 -6 z 7 a -1 -7 a 4 z 6 -6 a 2 z 6 -4 z 6 a -2 -3 z 6 -4 a 5 z 5 + 8 a 3 z 5 + 25 a z 5 + 12 z 5 a -1 - z 5 a -3 - a 6 z 4 + 8 a 4 z 4 + 16 a 2 z 4 + 7 z 4 a -2 + 14 z 4 + 3 a 5 z 3 -2 a 3 z 3 -12 a z 3 -6 z 3 a -1 + z 3 a -3 -2 a 4 z 2 -6 a 2 z 2 -2 z 2 a -2 -6 z 2 - a 3 z - a z - a 2 + a 3 z -1 + a z -1$ (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 L9a20. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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