L11a1

(Knotscape image)

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Link L11a1.

A graph, link L11a1.

A part of a link and a part of a graph.

Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,6,11,5 X₈₄₉₃ X_20,12,21,11 X_22,17,5,18 X_18,21,19,22 X_12,20,13,19 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 6, -9, 11, -2, 3, -10, 7, -8, 9, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -12 v u 3 + 12 u 3 + 12 v u 2 -12 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+15 q^{7/2}-21 q^{5/2}+24 q^{3/2}-25 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 3 a z 3 -4 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 - a z + 2 z a -3 - z a -5 + a 3 z -1 -2 a z -1 + 2 a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -7 a z 9 -13 z 9 a -1 -6 z 9 a -3 -9 a 2 z 8 -17 z 8 a -2 -9 z 8 a -4 -17 z 8 -5 a 3 z 7 + 5 a z 7 + 12 z 7 a -1 -6 z 7 a -3 -8 z 7 a -5 - a 4 z 6 + 20 a 2 z 6 + 36 z 6 a -2 + 9 z 6 a -4 -4 z 6 a -6 + 44 z 6 + 10 a 3 z 5 + 14 a z 5 + 16 z 5 a -1 + 25 z 5 a -3 + 12 z 5 a -5 - z 5 a -7 + a 4 z 4 -11 a 2 z 4 -19 z 4 a -2 - z 4 a -4 + 5 z 4 a -6 -25 z 4 -4 a 3 z 3 -9 a z 3 -14 z 3 a -1 -18 z 3 a -3 -8 z 3 a -5 + z 3 a -7 + 2 z 2 a -2 - z 2 a -4 -2 z 2 a -6 + z 2 - a 3 z -4 a z -2 z a -1 + 3 z a -3 + 2 z a -5 + 1 + a 3 z -1 + 2 a z -1 + 2 a -1 z -1 + a -3 z -1$ (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 L11a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$