L10a46

(Knotscape image)

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Visit L10a46's page at the original Knot Atlas.

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 2 v u 4 -3 u 4 -3 v u 3 + 4 u 3 + 4 v u 2 -3 u 2 -3 v u + 2 u + 2 v -1$ (db)
Jones polynomial	$-q^{15/2}+3 q^{13/2}-6 q^{11/2}+9 q^{9/2}-9 q^{7/2}+10 q^{5/2}-9 q^{3/2}+6 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 5 z 5 a -3 - z 5 a -5 + a z 3 -9 z 3 a -1 + 10 z 3 a -3 -3 z 3 a -5 + 4 a z -14 z a -1 + 11 z a -3 -3 z a -5 + 4 a z -1 -8 a -1 z -1 + 5 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 -5 z 8 a -2 -4 z 8 a -4 - z 8 - a z 7 -7 z 7 a -3 -8 z 7 a -5 + 11 z 6 a -2 -9 z 6 a -6 + 2 z 6 + 6 a z 5 + 12 z 5 a -1 + 26 z 5 a -3 + 14 z 5 a -5 -6 z 5 a -7 + 9 z 4 a -2 + 22 z 4 a -4 + 15 z 4 a -6 -3 z 4 a -8 + 5 z 4 -13 a z 3 -24 z 3 a -1 -20 z 3 a -3 -4 z 3 a -5 + 4 z 3 a -7 - z 3 a -9 -28 z 2 a -2 -23 z 2 a -4 -9 z 2 a -6 -14 z 2 + 12 a z + 20 z a -1 + 9 z a -3 - z a -7 + 14 a -2 + 9 a -4 + 2 a -6 + 8-4 a z -1 -8 a -1 z -1 -5 a -3 z -1 - a -5 z -1$ (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 =

3 is the signature of L10a46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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