L11a67

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 2 u 5 + 5 v u 4 -9 u 4 -12 v u 3 + 16 u 3 + 16 v u 2 -12 u 2 -9 v u + 5 u + 2 v -1$ (db)
Jones polynomial	$-q^{7/2}+5 q^{5/2}-13 q^{3/2}+20 \sqrt{q}-\frac{26}{\sqrt{q}}+\frac{30}{q^{3/2}}-\frac{29}{q^{5/2}}+\frac{24}{q^{7/2}}-\frac{18}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 -3 z 5 a 3 -6 z 3 a 3 -7 z a 3 -4 a 3 z -1 + z 7 a + 3 z 5 a + 6 z 3 a + 5 z a + 3 a z -1 - z 5 a -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-3 a 4 z 10 -3 a 2 z 10 -7 a 5 z 9 -19 a 3 z 9 -12 a z 9 -7 a 6 z 8 -18 a 4 z 8 -29 a 2 z 8 -18 z 8 -4 a 7 z 7 + 2 a 5 z 7 + 20 a 3 z 7 + a z 7 -13 z 7 a -1 - a 8 z 6 + 12 a 6 z 6 + 50 a 4 z 6 + 70 a 2 z 6 -5 z 6 a -2 + 28 z 6 + 9 a 7 z 5 + 22 a 5 z 5 + 27 a 3 z 5 + 31 a z 5 + 16 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -6 a 6 z 4 -37 a 4 z 4 -45 a 2 z 4 + 2 z 4 a -2 -14 z 4 -8 a 7 z 3 -31 a 5 z 3 -45 a 3 z 3 -29 a z 3 -7 z 3 a -1 - a 8 z 2 + a 6 z 2 + 9 a 4 z 2 + 11 a 2 z 2 + 4 z 2 + 3 a 7 z + 16 a 5 z + 24 a 3 z + 15 a z + 4 z a -1 - a 6 -2 a 4 -3 a 2 -1-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L11a67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -2$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16}$	${\mathbb Z}^{16}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$