L11n77

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_7,20,8,21 X_15,5,16,22 X_21,17,22,16 X_9,14,10,15 X_13,19,14,18 X_19,13,20,12 X_17,8,18,9 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 9, -6, 2, -11, 8, -7, 6, -4, 5, -9, 7, -8, 3, -5, 4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -7 v u 3 + 3 u 3 + 3 v u 2 -7 u 2 + 4 u -1$ (db)
Jones polynomial	$-q^{5/2}+3 q^{3/2}-6 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 + z 5 a 5 + 4 z 3 a 5 + 5 z a 5 + 2 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -10 z 3 a 3 -11 z a 3 -4 a 3 z -1 + 2 z 5 a + 7 z 3 a + 7 z a + 3 a z -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$-2 a 3 z 9 -2 a z 9 -7 a 4 z 8 -10 a 2 z 8 -3 z 8 -8 a 5 z 7 -6 a 3 z 7 + a z 7 - z 7 a -1 -3 a 6 z 6 + 22 a 4 z 6 + 37 a 2 z 6 + 12 z 6 + 26 a 5 z 5 + 42 a 3 z 5 + 20 a z 5 + 4 z 5 a -1 + 2 a 6 z 4 -22 a 4 z 4 -38 a 2 z 4 -14 z 4 -5 a 7 z 3 -31 a 5 z 3 -53 a 3 z 3 -33 a z 3 -6 z 3 a -1 - a 8 z 2 + 12 a 4 z 2 + 16 a 2 z 2 + 5 z 2 + a 7 z + 14 a 5 z + 26 a 3 z + 17 a z + 4 z a -1 - a 6 -2 a 4 -3 a 2 -1-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 =

-3 is the signature of L11n77. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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