L11n90

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_17,22,18,5 X_14,9,15,10 X_10,20,11,19 X_21,9,22,8 X_18,14,19,13 X_20,15,21,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 u 3 -2 v u 2 + 5 u 2 + 5 v u -2 u -2 v$ (db)
Jones polynomial	$-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 -3 z a 7 -2 a 7 z -1 + 2 z 3 a 5 + 2 z a 5 + a 5 z -1 + 2 z 3 a 3 + 2 z a 3 + a 3 z -1 -3 z a - a z -1$ (db)
Kauffman polynomial	$- z 7 a 9 + 5 z 5 a 9 -8 z 3 a 9 + 5 z a 9 - a 9 z -1 -2 z 8 a 8 + 9 z 6 a 8 -11 z 4 a 8 + 4 z 2 a 8 - a 8 - z 9 a 7 + 15 z 5 a 7 -25 z 3 a 7 + 13 z a 7 -2 a 7 z -1 -5 z 8 a 6 + 21 z 6 a 6 -25 z 4 a 6 + 13 z 2 a 6 -3 a 6 - z 9 a 5 - z 7 a 5 + 15 z 5 a 5 -19 z 3 a 5 + 9 z a 5 - a 5 z -1 -3 z 8 a 4 + 12 z 6 a 4 -17 z 4 a 4 + 12 z 2 a 4 -2 a 4 -2 z 7 a 3 + 5 z 5 a 3 -2 z 3 a 3 -3 z a 3 + a 3 z -1 -3 z 4 a 2 + 3 z 2 a 2 - a 2 -4 z a + a z -1$ (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 L11n90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -1$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$