L11n69

(Knotscape image)

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Visit L11n69's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_22,18,5,17 X_18,12,19,11 X_20,14,21,13 X_12,20,13,19 X_14,22,15,21 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 2 v u 4 -3 u 4 -3 v u 3 + 3 u 3 + 3 v u 2 -3 u 2 -3 v u + 2 u + v -1$ (db)
Jones polynomial	$q^{11/2}-3 q^{9/2}+5 q^{7/2}-7 q^{5/2}+9 q^{3/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{2}{q^{7/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + a z 5 -5 z 5 a -1 + z 5 a -3 + 2 a z 3 -8 z 3 a -1 + 3 z 3 a -3 + a 3 z -2 a z -3 z a -1 + 2 z a -3 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -3 z 8 a -2 -4 z 8 + 2 a z 7 -2 z 7 a -1 -4 z 7 a -3 + 2 a 2 z 6 + 5 z 6 a -2 -4 z 6 a -4 + 11 z 6 -3 a 3 z 5 -4 a z 5 + 8 z 5 a -1 + 6 z 5 a -3 -3 z 5 a -5 -2 a 2 z 4 -5 z 4 a -2 + 5 z 4 a -4 - z 4 a -6 -13 z 4 + 10 a 3 z 3 + 10 a z 3 -9 z 3 a -1 -5 z 3 a -3 + 4 z 3 a -5 + 6 a 2 z 2 + z 2 a -2 - z 2 a -4 + z 2 a -6 + 9 z 2 -8 a 3 z -9 a z + z a -1 + 2 z a -3 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11n69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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