L11a37

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,15,19,16 X_16,7,17,8 X_8,17,9,18 X_22,11,5,12 X_20,13,21,14 X_14,19,15,20 X_12,21,13,22 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 6, -9, 7, -8, 3, -4, 5, -3, 8, -7, 9, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 u 3 -4 v u 2 + 6 u 2 + 6 v u -4 u -3 v$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{5}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{5}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + 3 z a 11 + 2 a 11 z -1 -2 z 3 a 9 -2 z a 9 - a 9 z -1 -2 z 3 a 7 + a 7 z -1 -2 z 3 a 5 -2 z a 5 - a 5 z -1 - z 3 a 3 - z a 3$ (db)
Kauffman polynomial	$- z 8 a 14 + 6 z 6 a 14 -12 z 4 a 14 + 10 z 2 a 14 -3 a 14 -2 z 9 a 13 + 11 z 7 a 13 -18 z 5 a 13 + 10 z 3 a 13 -3 z a 13 + a 13 z -1 - z 10 a 12 + z 8 a 12 + 16 z 6 a 12 -39 z 4 a 12 + 28 z 2 a 12 -7 a 12 -5 z 9 a 11 + 25 z 7 a 11 -38 z 5 a 11 + 23 z 3 a 11 -8 z a 11 + 2 a 11 z -1 - z 10 a 10 - z 8 a 10 + 19 z 6 a 10 -31 z 4 a 10 + 18 z 2 a 10 -4 a 10 -3 z 9 a 9 + 11 z 7 a 9 -14 z 5 a 9 + 13 z 3 a 9 -5 z a 9 + a 9 z -1 -3 z 8 a 8 + 6 z 6 a 8 -3 z 7 a 7 + 3 z 5 a 7 + 4 z 3 a 7 -4 z a 7 + a 7 z -1 -3 z 6 a 6 + 2 z 4 a 6 + z 2 a 6 - a 6 -3 z 5 a 5 + 3 z 3 a 5 -3 z a 5 + a 5 z -1 -2 z 4 a 4 + z 2 a 4 - z 3 a 3 + z a 3$ (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 L11a37. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -7$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$