L11a43

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$-3 q^{9/2}+\frac{2}{q^{9/2}}+5 q^{7/2}-\frac{5}{q^{7/2}}-9 q^{5/2}+\frac{8}{q^{5/2}}+11 q^{3/2}-\frac{10}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-13 \sqrt{q}+\frac{12}{\sqrt{q}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a 5 z + a 5 z -1 + z a -5 -2 a 3 z 3 -2 z 3 a -3 -3 a 3 z -2 a 3 z -1 -2 z a -3 - a -3 z -1 + a z 5 + z 5 a -1 + a z 3 + z 3 a -1 + a z + a z -1 + 2 z a -1 + a -1 z -1$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -5 a z 9 -3 z 9 a -1 -2 a 4 z 8 -2 a 2 z 8 -4 z 8 a -2 -4 z 8 - a 5 z 7 + 4 a 3 z 7 + 12 a z 7 + 3 z 7 a -1 -4 z 7 a -3 + 8 a 4 z 6 + 16 a 2 z 6 + 3 z 6 a -2 -4 z 6 a -4 + 15 z 6 + 5 a 5 z 5 + 8 a 3 z 5 -3 z 5 a -5 -9 a 4 z 4 -18 a 2 z 4 + z 4 a -2 + 3 z 4 a -4 - z 4 a -6 -12 z 4 -8 a 5 z 3 -20 a 3 z 3 -13 a z 3 + 2 z 3 a -1 + 7 z 3 a -3 + 4 z 3 a -5 + 4 a 4 z 2 + 8 a 2 z 2 + z 2 a -6 + 5 z 2 + 5 a 5 z + 13 a 3 z + 8 a z -3 z a -1 -5 z a -3 -2 z a -5 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 L11a43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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