L11a46

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_20,17,21,18 X_18,10,19,9 X_8,20,9,19 X_4,21,1,22 X_14,12,15,11 X_10,4,11,3 X_12,5,13,6 X_22,13,5,14 X_2,16,3,15
Gauss code	{1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, 5, -3, 6, -10}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + u 5 + 6 v u 4 -6 u 4 -13 v u 3 + 13 u 3 + 13 v u 2 -13 u 2 -6 v u + 6 u + v -1$ (db)
Jones polynomial	$q^{9/2}-5 q^{7/2}+10 q^{5/2}-17 q^{3/2}+22 \sqrt{q}-\frac{26}{\sqrt{q}}+\frac{26}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{16}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a z 7 -2 a 3 z 5 + 3 a z 5 -2 z 5 a -1 + a 5 z 3 -4 a 3 z 3 + 5 a z 3 -3 z 3 a -1 + z 3 a -3 + a 5 z -4 a 3 z + 3 a z + a 5 z -1 -2 a 3 z -1 + a z -1 + a -1 z -1 - a -3 z -1$ (db)
Kauffman polynomial	$-2 a 2 z 10 -2 z 10 -7 a 3 z 9 -14 a z 9 -7 z 9 a -1 -11 a 4 z 8 -22 a 2 z 8 -9 z 8 a -2 -20 z 8 -9 a 5 z 7 -7 a 3 z 7 + 9 a z 7 + 2 z 7 a -1 -5 z 7 a -3 -4 a 6 z 6 + 15 a 4 z 6 + 51 a 2 z 6 + 19 z 6 a -2 - z 6 a -4 + 52 z 6 - a 7 z 5 + 14 a 5 z 5 + 35 a 3 z 5 + 32 a z 5 + 22 z 5 a -1 + 10 z 5 a -3 + 4 a 6 z 4 -9 a 4 z 4 -36 a 2 z 4 -11 z 4 a -2 + z 4 a -4 -35 z 4 + a 7 z 3 -11 a 5 z 3 -35 a 3 z 3 -36 a z 3 -18 z 3 a -1 -5 z 3 a -3 - a 6 z 2 + 3 a 4 z 2 + 12 a 2 z 2 + 2 z 2 a -2 + 10 z 2 + 5 a 5 z + 15 a 3 z + 12 a z + z a -1 - z a -3 - a 4 -3 a 2 - a -2 -2- a 5 z -1 -2 a 3 z -1 - a z -1 + a -1 z -1 + a -3 z -1$ (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 L11a46. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -6$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 0$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 2$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 4$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 5$	${\mathbb Z}_2$	${\mathbb Z}$