L11n79

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,9,17,10 X_4,21,1,22 X_11,14,12,15 X_3,10,4,11 X_5,13,6,12 X_13,5,14,22 X_15,2,16,3 X_20,18,21,17 X_18,8,19,7 X_8,20,9,19
Gauss code	{1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -6 v u 3 + 4 u 3 + 4 v u 2 -6 u 2 + 4 u -1$ (db)
Jones polynomial	$q^{9/2}-3 q^{7/2}+6 q^{5/2}-8 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{2}{q^{9/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 - a 3 z 5 + 5 a z 5 -2 z 5 a -1 -4 a 3 z 3 + 9 a z 3 -7 z 3 a -1 + z 3 a -3 + a 5 z -5 a 3 z + 7 a z -7 z a -1 + 2 z a -3 + a 5 z -1 -2 a 3 z -1 + 3 a z -1 -3 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -3 z 8 a -2 -6 z 8 -3 a 3 z 7 -6 a z 7 -6 z 7 a -1 -3 z 7 a -3 - a 4 z 6 + 6 a 2 z 6 + 5 z 6 a -2 - z 6 a -4 + 13 z 6 + 7 a 3 z 5 + 24 a z 5 + 26 z 5 a -1 + 9 z 5 a -3 -2 a 4 z 4 -10 a 2 z 4 + 6 z 4 a -2 + 3 z 4 a -4 -5 z 4 -3 a 5 z 3 -15 a 3 z 3 -32 a z 3 -27 z 3 a -1 -7 z 3 a -3 + 2 a 4 z 2 + 7 a 2 z 2 -8 z 2 a -2 -3 z 2 a -4 + 4 a 5 z + 10 a 3 z + 16 a z + 13 z a -1 + 3 z a -3 -2 a 2 + 2 a -2 + a -4 - a 5 z -1 -2 a 3 z -1 -3 a z -1 -3 a -1 z -1 - a -3 z -1$ (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 L11n79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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