L11n98

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_13,21,14,20 X_7,17,8,16 X_9,19,10,18 X_17,9,18,8 X_19,11,20,10 X_15,5,16,22 X_21,15,22,14 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 6, -5, 7, 11, -2, -3, 9, -8, 4, -6, 5, -7, 3, -9, 8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 7 - v u 5 + u 5 + 2 v u 4 -2 u 4 -2 v u 3 + 2 u 3 + v u 2 - u 2 - v$ (db)
Jones polynomial	$q^{17/2}-2 q^{15/2}+4 q^{13/2}-4 q^{11/2}+4 q^{9/2}-4 q^{7/2}+2 q^{5/2}-q^{3/2}-\sqrt{q}-\frac{1}{q^{3/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 7 a -3 + z 5 a -1 -8 z 5 a -3 + 6 z 3 a -1 -20 z 3 a -3 + 3 z 3 a -5 + z 3 a -7 + 10 z a -1 -20 z a -3 + 7 z a -5 + z a -7 + 4 a -1 z -1 -7 a -3 z -1 + 3 a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -1 - z 9 a -3 - z 8 a -2 - z 8 a -4 + 9 z 7 a -1 + 10 z 7 a -3 - z 7 a -7 + 9 z 6 a -2 + 9 z 6 a -4 -2 z 6 a -6 -2 z 6 a -8 -28 z 5 a -1 -33 z 5 a -3 -3 z 5 a -5 -2 z 5 a -9 -24 z 4 a -2 -26 z 4 a -4 + z 4 a -6 + 2 z 4 a -8 - z 4 a -10 + 37 z 3 a -1 + 46 z 3 a -3 + 6 z 3 a -5 + 3 z 3 a -9 + 23 z 2 a -2 + 24 z 2 a -4 + z 2 a -8 + 2 z 2 a -10 -20 z a -1 -29 z a -3 -8 z a -5 + z a -7 -7 a -2 -7 a -4 - a -10 + 4 a -1 z -1 + 7 a -3 z -1 + 3 a -5 z -1$ (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 =

3 is the signature of L11n98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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