L11a98

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_12,3,13,4 X_10,5,11,6 X_22,11,5,12 X_4,21,1,22 X_18,14,19,13 X_16,8,17,7 X_8,18,9,17 X_20,16,21,15 X_14,20,15,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 7, -11, 10, -8, 9, -7, 11, -10, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 5 v u 4 -7 u 4 -8 v u 3 + 9 u 3 + 9 v u 2 -8 u 2 -7 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-15 q^{5/2}+19 q^{3/2}-22 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- a z 7 - z 7 a -1 + a 3 z 5 -4 a z 5 -3 z 5 a -1 + z 5 a -3 + 3 a 3 z 3 -7 a z 3 -3 z 3 a -1 + 2 z 3 a -3 + 3 a 3 z -5 a z -2 z a -1 + 2 z a -3 + a 3 z -1 -2 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -12 a z 9 -8 z 9 a -1 -3 a 4 z 8 -5 a 2 z 8 -14 z 8 a -2 -16 z 8 - a 5 z 7 + 11 a 3 z 7 + 28 a z 7 + z 7 a -1 -15 z 7 a -3 + 11 a 4 z 6 + 30 a 2 z 6 + 18 z 6 a -2 -10 z 6 a -4 + 47 z 6 + 4 a 5 z 5 -7 a 3 z 5 -14 a z 5 + 22 z 5 a -1 + 21 z 5 a -3 -4 z 5 a -5 -13 a 4 z 4 -34 a 2 z 4 + 2 z 4 a -2 + 9 z 4 a -4 - z 4 a -6 -29 z 4 -5 a 5 z 3 + 2 a z 3 -13 z 3 a -1 -10 z 3 a -3 + 5 a 4 z 2 + 12 a 2 z 2 -10 z 2 a -2 -5 z 2 a -4 + 2 z 2 + 2 a 5 z -2 a 3 z -2 a z + 4 z a -1 + 2 z a -3 - a 2 + 5 a -2 + 2 a -4 + 3 + a 3 z -1 -2 a -1 z -1 - a -3 z -1$ (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 L11a98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$