L11a99

(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_18,16,19,15 X_16,8,17,7 X_8,18,9,17 X_20,14,21,13 X_14,20,15,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -9, 2, -4, 5, -3, 10, -11, 7, -8, 9, -7, 11, -10, 6, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 v u 3 + 3 u 3 + 10 v u 2 -10 u 2 -10 v u + 10 u + 3 v -4$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+8 q^{5/2}-11 q^{3/2}+15 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z 3 a 5 + z a 5 - z 5 a 3 - z 3 a 3 + 2 a 3 z -1 -2 z 5 a -4 z 3 a -5 z a -3 a z -1 - z 5 a -1 + 2 z a -1 + a -1 z -1 + z 3 a -3$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -10 a z 9 -6 z 9 a -1 -4 a 4 z 8 -2 a 2 z 8 -7 z 8 a -2 -5 z 8 -4 a 5 z 7 + 6 a 3 z 7 + 30 a z 7 + 16 z 7 a -1 -4 z 7 a -3 -3 a 6 z 6 + 2 a 4 z 6 + 10 a 2 z 6 + 21 z 6 a -2 - z 6 a -4 + 27 z 6 - a 7 z 5 + 5 a 5 z 5 -9 a 3 z 5 -38 a z 5 -13 z 5 a -1 + 10 z 5 a -3 + 6 a 6 z 4 + 5 a 4 z 4 -14 a 2 z 4 -16 z 4 a -2 + 2 z 4 a -4 -31 z 4 + 2 a 7 z 3 + 9 a 3 z 3 + 20 a z 3 + 6 z 3 a -1 -3 z 3 a -3 -3 a 6 z 2 -3 a 4 z 2 + 7 a 2 z 2 + 4 z 2 a -2 + 11 z 2 - a 7 z -4 a 3 z -7 a z -2 z a -1 -3 a 2 - a -2 -3 + 2 a 3 z -1 + 3 a z -1 + a -1 z -1$ (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 L11a99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$