L10n29

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,14,8,15 X_15,20,16,5 X_9,17,10,16 X_19,9,20,8 X_13,19,14,18 X_17,13,18,12 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -5, 2, -10, 8, -7, 3, -4, 5, -8, 7, -6, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 3 v u 4 - u 4 -4 v u 3 + 4 u 3 + 4 v u 2 -4 u 2 - v u + 3 u -1$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-6 q^{3/2}+8 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 - z 5 a -1 -4 a 3 z 3 + 9 a z 3 -3 z 3 a -1 + a 5 z -7 a 3 z + 7 a z -3 z a -1 + 2 a 5 z -1 -4 a 3 z -1 + 3 a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 8 -2 z 8 -5 a 3 z 7 -9 a z 7 -4 z 7 a -1 -3 a 4 z 6 -3 a 2 z 6 -3 z 6 a -2 -3 z 6 + 13 a 3 z 5 + 21 a z 5 + 7 z 5 a -1 - z 5 a -3 + 3 a 4 z 4 + 11 a 2 z 4 + 6 z 4 a -2 + 14 z 4 -6 a 5 z 3 -23 a 3 z 3 -21 a z 3 -2 z 3 a -1 + 2 z 3 a -3 -4 a 4 z 2 -11 a 2 z 2 -3 z 2 a -2 -10 z 2 + 8 a 5 z + 17 a 3 z + 12 a z + 2 z a -1 - z a -3 + 2 a 4 + 3 a 2 + a -2 + 3-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L10n29. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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