L11a82

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 6 u 4 + 10 v u 3 -14 u 3 -14 v u 2 + 10 u 2 + 6 v u -2 u - v$ (db)
Jones polynomial	$-q^{7/2}+3 q^{5/2}-8 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z a 7 + 3 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 -2 z 5 a 3 -4 z 3 a 3 -8 z a 3 -4 a 3 z -1 - z 5 a + 3 z 3 a + 6 z a + 3 a z -1 + 2 z 3 a -1 - z a -1 - a -1 z -1 - z a -3$ (db)
Kauffman polynomial	$- a 4 z 10 - a 2 z 10 -4 a 5 z 9 -9 a 3 z 9 -5 a z 9 -6 a 6 z 8 -17 a 4 z 8 -19 a 2 z 8 -8 z 8 -4 a 7 z 7 -5 a 5 z 7 + a 3 z 7 -4 a z 7 -6 z 7 a -1 - a 8 z 6 + 12 a 6 z 6 + 47 a 4 z 6 + 51 a 2 z 6 -3 z 6 a -2 + 14 z 6 + 10 a 7 z 5 + 34 a 5 z 5 + 40 a 3 z 5 + 26 a z 5 + 9 z 5 a -1 - z 5 a -3 + 2 a 8 z 4 -5 a 6 z 4 -40 a 4 z 4 -53 a 2 z 4 + 4 z 4 a -2 -16 z 4 -8 a 7 z 3 -36 a 5 z 3 -53 a 3 z 3 -34 a z 3 -7 z 3 a -1 + 2 z 3 a -3 - a 8 z 2 + a 6 z 2 + 14 a 4 z 2 + 24 a 2 z 2 - z 2 a -2 + 11 z 2 + 3 a 7 z + 15 a 5 z + 24 a 3 z + 17 a z + 4 z a -1 - z a -3 - a 6 -2 a 4 -3 a 2 -1-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L11a82. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -2$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 2$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$