L10a16

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 -3 v u 2 + 6 u 2 + 6 v u -3 u -2 v$ (db)
Jones polynomial	$-q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 -3 z a 7 -2 a 7 z -1 + 2 z 3 a 5 + 2 z a 5 + a 5 z -1 + 2 z 3 a 3 + z a 3 + a 3 z -1 + z 3 a - z a - a z -1 - z a -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -8 z 3 a 9 + 5 z a 9 - a 9 z -1 -2 z 8 a 8 + 9 z 6 a 8 -11 z 4 a 8 + 4 z 2 a 8 - a 8 - z 9 a 7 + 15 z 5 a 7 -26 z 3 a 7 + 13 z a 7 -2 a 7 z -1 -5 z 8 a 6 + 20 z 6 a 6 -23 z 4 a 6 + 11 z 2 a 6 -3 a 6 - z 9 a 5 -2 z 7 a 5 + 16 z 5 a 5 -18 z 3 a 5 + 8 z a 5 - a 5 z -1 -3 z 8 a 4 + 8 z 6 a 4 -8 z 4 a 4 + 7 z 2 a 4 -2 a 4 -3 z 7 a 3 + 3 z 5 a 3 + 4 z 3 a 3 -4 z a 3 + a 3 z -1 -3 z 6 a 2 + 2 z 4 a 2 + z 2 a 2 - a 2 -3 z 5 a + 3 z 3 a -3 z a + a z -1 -2 z 4 + z 2 - z 3 a -1 + z a -1$ (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 L10a16. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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