L9a12

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 2 u 4 + 2 v u 3 -2 u 3 -2 v u 2 + 2 u 2 + 2 v u -2 u - v$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{5}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{2}{q^{21/2}}+\frac{1}{q^{23/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 11 -2 a 11 z -1 + 3 z 3 a 9 + 9 z a 9 + 5 a 9 z -1 -2 z 5 a 7 -8 z 3 a 7 -9 z a 7 -3 a 7 z -1 - z 5 a 5 -3 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 4 a 14 + 2 z 2 a 14 - a 14 -2 z 5 a 13 + 3 z 3 a 13 -2 z 6 a 12 + z 4 a 12 + 2 z 2 a 12 -2 z 7 a 11 + 3 z 5 a 11 -5 z 3 a 11 + 5 z a 11 -2 a 11 z -1 - z 8 a 10 - z 6 a 10 + 6 z 4 a 10 -11 z 2 a 10 + 5 a 10 -5 z 7 a 9 + 17 z 5 a 9 -26 z 3 a 9 + 16 z a 9 -5 a 9 z -1 - z 8 a 8 - z 6 a 8 + 9 z 4 a 8 -12 z 2 a 8 + 5 a 8 -3 z 7 a 7 + 11 z 5 a 7 -15 z 3 a 7 + 10 z a 7 -3 a 7 z -1 -2 z 6 a 6 + 5 z 4 a 6 - z 2 a 6 - z 5 a 5 + 3 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L9a12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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