L11n56

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_14,7,15,8 X_8,15,9,16 X_13,21,14,20 X_17,5,18,22 X_21,19,22,18 X_19,13,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + u 3 + 4 v u 2 -4 u 2 -4 v u + 4 u + v -2$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+5 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{6}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- a z 5 - z 5 a -1 + a 3 z 3 -3 a z 3 -2 z 3 a -1 + z 3 a -3 + 2 a 3 z -5 a z + z a -3 + 2 a 3 z -1 -3 a z -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 - a 2 z 8 -3 z 8 a -2 -4 z 8 + 2 a z 7 - z 7 a -1 -3 z 7 a -3 + 2 a 2 z 6 + 9 z 6 a -2 - z 6 a -4 + 12 z 6 -4 a 3 z 5 -5 a z 5 + 9 z 5 a -1 + 10 z 5 a -3 -2 a 4 z 4 -7 a 2 z 4 -5 z 4 a -2 + 3 z 4 a -4 -13 z 4 - a 5 z 3 + 6 a 3 z 3 + 9 a z 3 -5 z 3 a -1 -7 z 3 a -3 + a 4 z 2 + 7 a 2 z 2 + z 2 a -2 -2 z 2 a -4 + 9 z 2 + a 5 z -6 a 3 z -9 a z - z a -1 + z a -3 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11n56. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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