L10a12

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,8,15,7 X_18,11,19,12 X_20,16,5,15 X_16,20,17,19 X_12,17,13,18 X_8,14,9,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 10, -2, 4, -7, 8, -3, 5, -6, 7, -4, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + 3 u 3 + 6 v u 2 -9 u 2 -9 v u + 6 u + 3 v -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-7 q^{3/2}+10 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{13}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + 3 z 3 a 3 + z a 3 - a 3 z -1 - z 5 a + z a + 2 a z -1 + 2 z 3 a -1 - a -1 z -1 - z a -3$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -7 a 2 z 8 -4 z 8 -3 a 5 z 7 -8 a 3 z 7 -10 a z 7 -5 z 7 a -1 -2 a 6 z 6 + 2 a 4 z 6 + 10 a 2 z 6 -3 z 6 a -2 + 3 z 6 - a 7 z 5 + 3 a 5 z 5 + 20 a 3 z 5 + 26 a z 5 + 9 z 5 a -1 - z 5 a -3 + 3 a 6 z 4 - a 4 z 4 -7 a 2 z 4 + 5 z 4 a -2 + 2 z 4 + 3 a 7 z 3 + a 5 z 3 -20 a 3 z 3 -26 a z 3 -6 z 3 a -1 + 2 z 3 a -3 + 3 a 4 z 2 + 6 a 2 z 2 -2 z 2 a -2 + z 2 -3 a 7 z -2 a 5 z + 9 a 3 z + 13 a z + 4 z a -1 - z a -3 - a 6 -2 a 4 -3 a 2 -1 + a 7 z -1 + a 5 z -1 - a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L10a12. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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