L10a34

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 7 v u 2 -7 u 2 -7 v u + 7 u + 2 v -2$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-7 q^{9/2}+9 q^{7/2}-12 q^{5/2}+12 q^{3/2}-11 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z 5 a -1 + z 5 a -3 -2 a z 3 + z 3 a -1 + z 3 a -3 - z 3 a -5 + a 3 z -3 a z + 2 z a -1 + a 3 z -1 -2 a z -1 + 2 a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -6 z 8 a -2 -4 z 8 a -4 -2 z 8 -2 a z 7 -4 z 7 a -1 -8 z 7 a -3 -6 z 7 a -5 -2 a 2 z 6 + 8 z 6 a -2 + 3 z 6 a -4 -4 z 6 a -6 - z 6 - a 3 z 5 - a z 5 + 5 z 5 a -1 + 18 z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + 4 a 2 z 4 -3 z 4 a -2 + 4 z 4 a -4 + 7 z 4 a -6 + 4 z 4 + 3 a 3 z 3 + 9 a z 3 + 2 z 3 a -1 -10 z 3 a -3 -5 z 3 a -5 + z 3 a -7 -2 a 2 z 2 - z 2 a -2 -2 z 2 a -4 - z 2 a -6 -2 z 2 -3 a 3 z -8 a z -6 z a -1 - z a -3 + 1 + a 3 z -1 + 2 a z -1 + 2 a -1 z -1 + a -3 z -1$ (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 L10a34. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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