L11a58

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 4 v u 4 -6 u 4 -9 v u 3 + 11 u 3 + 11 v u 2 -9 u 2 -6 v u + 4 u + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{17}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 - a 7 z -1 + 3 z 3 a 5 + 6 z a 5 + 4 a 5 z -1 -3 z 5 a 3 -9 z 3 a 3 -11 z a 3 -4 a 3 z -1 + z 7 a + 4 z 5 a + 8 z 3 a + 6 z a + a z -1 - z 5 a -1 -2 z 3 a -1 -2 z a -1$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -3 a 5 z 9 -8 a 3 z 9 -5 a z 9 -4 a 6 z 8 -12 a 4 z 8 -17 a 2 z 8 -9 z 8 -3 a 7 z 7 -5 a 5 z 7 - a 3 z 7 -7 a z 7 -8 z 7 a -1 - a 8 z 6 + 5 a 6 z 6 + 24 a 4 z 6 + 33 a 2 z 6 -4 z 6 a -2 + 11 z 6 + 8 a 7 z 5 + 25 a 5 z 5 + 30 a 3 z 5 + 26 a z 5 + 12 z 5 a -1 - z 5 a -3 + 3 a 8 z 4 + 5 a 6 z 4 -5 a 4 z 4 -15 a 2 z 4 + 5 z 4 a -2 -3 z 4 -8 a 7 z 3 -27 a 5 z 3 -30 a 3 z 3 -19 a z 3 -7 z 3 a -1 + z 3 a -3 -3 a 8 z 2 -10 a 6 z 2 -12 a 4 z 2 -3 a 2 z 2 -2 z 2 a -2 + 4 a 7 z + 14 a 5 z + 13 a 3 z + 5 a z + 2 z a -1 + a 8 + 4 a 6 + 7 a 4 + 4 a 2 + 1- a 7 z -1 -4 a 5 z -1 -4 a 3 z -1 - a z -1$ (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 L11a58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$