L11a44

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-4 t(2)^3+8 t(2)^2-4 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+20 q^{3/2}-24 \sqrt{q}+\frac{23}{\sqrt{q}}-\frac{19}{q^{3/2}}+\frac{14}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 8 a z 3 -9 z 3 a -1 + 2 z 3 a -3 + a 5 z -5 a 3 z + 9 a z -8 z a -1 + 3 z a -3 + a 5 z -1 -3 a 3 z -1 + 4 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -4 a 3 z 9 -13 a z 9 -9 z 9 a -1 -3 a 4 z 8 -9 a 2 z 8 -17 z 8 a -2 -23 z 8 - a 5 z 7 + 7 a 3 z 7 + 20 a z 7 -5 z 7 a -1 -17 z 7 a -3 + 10 a 4 z 6 + 41 a 2 z 6 + 26 z 6 a -2 -10 z 6 a -4 + 67 z 6 + 4 a 5 z 5 + 8 a 3 z 5 + 24 a z 5 + 50 z 5 a -1 + 26 z 5 a -3 -4 z 5 a -5 -12 a 4 z 4 -42 a 2 z 4 -8 z 4 a -2 + 7 z 4 a -4 - z 4 a -6 -46 z 4 -6 a 5 z 3 -24 a 3 z 3 -49 a z 3 -49 z 3 a -1 -18 z 3 a -3 + 6 a 4 z 2 + 14 a 2 z 2 - z 2 a -2 -3 z 2 a -4 + 10 z 2 + 4 a 5 z + 16 a 3 z + 26 a z + 21 z a -1 + 7 z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -3 a 3 z -1 -4 a z -1 -2 a -1 z -1$ (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 L11a44. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$