L11a76

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-4 u 3 -6 v u 2 + 9 u 2 + 9 v u -6 u -4 v$ (db)
Jones polynomial	$-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{11}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{9}{q^{17/2}}-\frac{7}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 13 z -1 + 3 z a 11 + a 11 z -1 -2 z 3 a 9 + z a 9 + 2 a 9 z -1 -4 z 3 a 7 -4 z a 7 -2 a 7 z -1 -3 z 3 a 5 -2 z a 5 - z 3 a 3$ (db)
Kauffman polynomial	$- z 8 a 14 + 6 z 6 a 14 -13 z 4 a 14 + 12 z 2 a 14 -4 a 14 -2 z 9 a 13 + 10 z 7 a 13 -15 z 5 a 13 + 6 z 3 a 13 + a 13 z -1 - z 10 a 12 - z 8 a 12 + 23 z 6 a 12 -48 z 4 a 12 + 34 z 2 a 12 -9 a 12 -6 z 9 a 11 + 23 z 7 a 11 -20 z 5 a 11 -3 z 3 a 11 + 3 z a 11 + a 11 z -1 - z 10 a 10 -7 z 8 a 10 + 38 z 6 a 10 -46 z 4 a 10 + 19 z 2 a 10 -4 a 10 -4 z 9 a 9 + 4 z 7 a 9 + 19 z 5 a 9 -26 z 3 a 9 + 12 z a 9 -2 a 9 z -1 -7 z 8 a 8 + 13 z 6 a 8 + 2 z 4 a 8 -5 z 2 a 8 + 2 a 8 -9 z 7 a 7 + 18 z 5 a 7 -11 z 3 a 7 + 7 z a 7 -2 a 7 z -1 -8 z 6 a 6 + 10 z 4 a 6 -2 z 2 a 6 -6 z 5 a 5 + 5 z 3 a 5 -2 z a 5 -3 z 4 a 4 - z 3 a 3$ (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 L11a76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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