L9a23

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 4 - v 2 u 3 + 3 v u 3 -2 u 3 + 2 v 2 u 2 -5 v u 2 + 2 u 2 -2 v 2 u + 3 v u - u - v$ (db)
Jones polynomial	$q^{5/2}-3 q^{3/2}+5 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + 2 z a 5 + 2 a 5 z -1 - z 5 a 3 -3 z 3 a 3 -5 z a 3 -3 a 3 z -1 - z 5 a -2 z 3 a - z a + a z -1 + z 3 a -1 + z a -1$ (db)
Kauffman polynomial	$- a 4 z 8 - a 2 z 8 -3 a 5 z 7 -6 a 3 z 7 -3 a z 7 -2 a 6 z 6 -4 a 4 z 6 -6 a 2 z 6 -4 z 6 - a 7 z 5 + 8 a 5 z 5 + 13 a 3 z 5 + a z 5 -3 z 5 a -1 + 4 a 6 z 4 + 13 a 4 z 4 + 15 a 2 z 4 - z 4 a -2 + 5 z 4 + 3 a 7 z 3 -10 a 5 z 3 -15 a 3 z 3 + 2 a z 3 + 4 z 3 a -1 - a 6 z 2 -11 a 4 z 2 -13 a 2 z 2 + z 2 a -2 -2 z 2 -2 a 7 z + 8 a 5 z + 11 a 3 z - z a -1 + 3 a 4 + 3 a 2 + 1-2 a 5 z -1 -3 a 3 z -1 - a z -1$ (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 L9a23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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