L11a49

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -13 v u 3 + 13 u 3 + 13 v u 2 -13 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-11 q^{9/2}+17 q^{7/2}-23 q^{5/2}+26 q^{3/2}-26 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 4 a z 3 -5 z 3 a -1 + 3 z 3 a -3 - z 3 a -5 - a 3 z + 3 a z -3 z a -1 + z a -3 + a z -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -6 a z 9 -14 z 9 a -1 -8 z 9 a -3 -7 a 2 z 8 -23 z 8 a -2 -13 z 8 a -4 -17 z 8 -4 a 3 z 7 + 2 a z 7 + 12 z 7 a -1 -5 z 7 a -3 -11 z 7 a -5 - a 4 z 6 + 14 a 2 z 6 + 51 z 6 a -2 + 17 z 6 a -4 -5 z 6 a -6 + 44 z 6 + 9 a 3 z 5 + 17 a z 5 + 20 z 5 a -1 + 28 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 2 a 4 z 4 -8 a 2 z 4 -29 z 4 a -2 -5 z 4 a -4 + 4 z 4 a -6 -30 z 4 -7 a 3 z 3 -18 a z 3 -22 z 3 a -1 -16 z 3 a -3 -5 z 3 a -5 - a 4 z 2 + a 2 z 2 + 3 z 2 a -2 + 5 z 2 + 2 a 3 z + 6 a z + 6 z a -1 + 2 z a -3 + 1- a z -1 - a -1 z -1$ (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 L11a49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$