L11a27

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -2 v u 4 + 6 u 4 + 9 v u 3 -11 u 3 -11 v u 2 + 9 u 2 + 6 v u -2 u - v$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{17}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{19}{q^{11/2}}+\frac{16}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 3 a 9 z -1 -5 z 3 a 7 -7 z a 7 -3 a 7 z -1 + 2 z 5 a 5 + 3 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 + z 5 a 3 - z a 3 - a 3 z -1 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 8 z 5 a 11 -7 z 3 a 11 + 3 z a 11 - a 11 z -1 -4 z 8 a 10 + 6 z 6 a 10 + 3 z 4 a 10 -6 z 2 a 10 + 2 a 10 -3 z 9 a 9 -3 z 7 a 9 + 22 z 5 a 9 -24 z 3 a 9 + 13 z a 9 -3 a 9 z -1 - z 10 a 8 -10 z 8 a 8 + 25 z 6 a 8 -17 z 4 a 8 + 5 z 2 a 8 -7 z 9 a 7 + 3 z 7 a 7 + 21 z 5 a 7 -31 z 3 a 7 + 17 z a 7 -3 a 7 z -1 - z 10 a 6 -12 z 8 a 6 + 29 z 6 a 6 -27 z 4 a 6 + 11 z 2 a 6 -2 a 6 -4 z 9 a 5 -2 z 7 a 5 + 15 z 5 a 5 -20 z 3 a 5 + 11 z a 5 -2 a 5 z -1 -6 z 8 a 4 + 8 z 6 a 4 -5 z 4 a 4 + z 2 a 4 -5 z 7 a 3 + 7 z 5 a 3 -4 z 3 a 3 + 3 z a 3 - a 3 z -1 -3 z 6 a 2 + 5 z 4 a 2 -2 z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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