L11a63

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -9 v u 3 + 9 u 3 + 9 v u 2 -9 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$q^{17/2}-4 q^{15/2}+9 q^{13/2}-13 q^{11/2}+17 q^{9/2}-20 q^{7/2}+18 q^{5/2}-16 q^{3/2}+11 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -6 z 3 a -1 + 8 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + 2 a z -6 z a -1 + 8 z a -3 -5 z a -5 + z a -7 + a z -1 -2 a -1 z -1 + 3 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 a -4 -3 z 9 a -1 -8 z 9 a -3 -5 z 9 a -5 -10 z 8 a -2 -17 z 8 a -4 -10 z 8 a -6 -3 z 8 - a z 7 + 4 z 7 a -1 + 8 z 7 a -3 -9 z 7 a -5 -12 z 7 a -7 + 40 z 6 a -2 + 46 z 6 a -4 + 8 z 6 a -6 -9 z 6 a -8 + 11 z 6 + 4 a z 5 + 13 z 5 a -1 + 28 z 5 a -3 + 39 z 5 a -5 + 16 z 5 a -7 -4 z 5 a -9 -41 z 4 a -2 -31 z 4 a -4 + 7 z 4 a -6 + 9 z 4 a -8 - z 4 a -10 -13 z 4 -6 a z 3 -26 z 3 a -1 -41 z 3 a -3 -31 z 3 a -5 -9 z 3 a -7 + z 3 a -9 + 13 z 2 a -2 + 7 z 2 a -4 -5 z 2 a -6 -4 z 2 a -8 + 5 z 2 + 4 a z + 13 z a -1 + 18 z a -3 + 12 z a -5 + 3 z a -7 -2 a -2 + 2 a -6 + a -8 - a z -1 -2 a -1 z -1 -3 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 =

3 is the signature of L11a63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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