L11a60

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 4 u 3 + 8 v u 2 -11 u 2 -11 v u + 8 u + 4 v -2$ (db)
Jones polynomial	$q^{9/2}-4 q^{7/2}+7 q^{5/2}-11 q^{3/2}+14 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{16}{q^{3/2}}-\frac{13}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 z -1 -3 z a 5 - a 5 z -1 + 3 z 3 a 3 + z a 3 - a 3 z -1 - z 5 a + z 3 a + 2 z a + 2 a z -1 - z 5 a -1 - z 3 a -1 -2 z a -1 - a -1 z -1 + z 3 a -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -6 a z 9 -4 z 9 a -1 -3 a 4 z 8 -5 a 2 z 8 -6 z 8 a -2 -8 z 8 -3 a 5 z 7 -4 a 3 z 7 + 8 a z 7 + 5 z 7 a -1 -4 z 7 a -3 -2 a 6 z 6 + a 4 z 6 + 10 a 2 z 6 + 17 z 6 a -2 - z 6 a -4 + 25 z 6 - a 7 z 5 + 3 a 5 z 5 + 12 a 3 z 5 + 5 a z 5 + 8 z 5 a -1 + 11 z 5 a -3 + 3 a 6 z 4 + a 4 z 4 -9 a 2 z 4 -12 z 4 a -2 + 2 z 4 a -4 -21 z 4 + 3 a 7 z 3 + a 5 z 3 -14 a 3 z 3 -18 a z 3 -12 z 3 a -1 -6 z 3 a -3 + 2 a 4 z 2 + 6 a 2 z 2 + 2 z 2 a -2 + 6 z 2 -3 a 7 z -2 a 5 z + 8 a 3 z + 12 a z + 5 z a -1 - a 6 -2 a 4 -3 a 2 -1 + a 7 z -1 + a 5 z -1 - a 3 z -1 -2 a z -1 - a -1 z -1$ (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 L11a60. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}^{2}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 3$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$