L11a53

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_20,9,21,10 X_8,19,9,20 X_4,21,1,22 X_14,6,15,5 X_10,4,11,3 X_16,12,17,11 X_12,16,13,15 X_22,14,5,13 X_2,18,3,17
Gauss code	{1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 8, -9, 10, -6, 9, -8, 11, -2, 4, -3, 5, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 5 + 2 u 5 + 5 v u 4 -5 u 4 -7 v u 3 + 7 u 3 + 7 v u 2 -7 u 2 -5 v u + 5 u + 2 v -2$ (db)
Jones polynomial	$-q^{11/2}+4 q^{9/2}-7 q^{7/2}+11 q^{5/2}-16 q^{3/2}+17 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 + z 7 a -1 - a 3 z 5 + 4 a z 5 + 3 z 5 a -1 - z 5 a -3 -3 a 3 z 3 + 7 a z 3 -2 z 3 a -3 -3 a 3 z + 8 a z -6 z a -1 + z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -5 a z 9 -10 z 9 a -1 -5 z 9 a -3 -8 a 2 z 8 -3 z 8 a -2 -4 z 8 a -4 -7 z 8 -9 a 3 z 7 -2 a z 7 + 25 z 7 a -1 + 17 z 7 a -3 - z 7 a -5 -6 a 4 z 6 + 8 a 2 z 6 + 23 z 6 a -2 + 15 z 6 a -4 + 22 z 6 -3 a 5 z 5 + 15 a 3 z 5 + 26 a z 5 -11 z 5 a -1 -16 z 5 a -3 + 3 z 5 a -5 - a 6 z 4 + 6 a 4 z 4 + 6 a 2 z 4 -18 z 4 a -2 -15 z 4 a -4 -4 z 4 + 2 a 5 z 3 -15 a 3 z 3 -29 a z 3 -7 z 3 a -1 + 3 z 3 a -3 -2 z 3 a -5 + a 6 z 2 -4 a 4 z 2 -12 a 2 z 2 -3 z 2 a -2 + z 2 a -4 -11 z 2 + 6 a 3 z + 15 a z + 11 z a -1 + 2 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L11a53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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