L11a78

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 5 v u 4 -5 u 4 -11 v u 3 + 11 u 3 + 11 v u 2 -11 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-10 q^{9/2}+15 q^{7/2}-20 q^{5/2}+22 q^{3/2}-22 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{13}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -4 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 6 a z 3 -9 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 -2 a 3 z + 8 a z -9 z a -1 + 3 z a -3 - a 3 z -1 + 4 a z -1 -4 a -1 z -1 + a -3 z -1$ (db)
Kauffman polynomial	$- z 10 a -2 - z 10 -3 a z 9 -8 z 9 a -1 -5 z 9 a -3 -4 a 2 z 8 -18 z 8 a -2 -10 z 8 a -4 -12 z 8 -3 a 3 z 7 -5 a z 7 -3 z 7 a -1 -11 z 7 a -3 -10 z 7 a -5 - a 4 z 6 + 5 a 2 z 6 + 32 z 6 a -2 + 10 z 6 a -4 -5 z 6 a -6 + 23 z 6 + 8 a 3 z 5 + 25 a z 5 + 36 z 5 a -1 + 35 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 3 a 4 z 4 + 5 a 2 z 4 -6 z 4 a -2 + 4 z 4 a -4 + 5 z 4 a -6 -3 z 4 -8 a 3 z 3 -28 a z 3 -38 z 3 a -1 -23 z 3 a -3 -5 z 3 a -5 -3 a 4 z 2 -10 a 2 z 2 -12 z 2 a -2 -5 z 2 a -4 -14 z 2 + 4 a 3 z + 15 a z + 17 z a -1 + 6 z a -3 + a 4 + 4 a 2 + 4 a -2 + a -4 + 7- a 3 z -1 -4 a z -1 -4 a -1 z -1 - a -3 z -1$ (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 L11a78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$