L11n22

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,21,10,20 X₃₈₄₉ X_21,18,22,19 X_11,14,12,15 X_5,13,6,12 X_13,5,14,22 X_19,11,20,10 X_15,2,16,3
Gauss code	{1, 11, -5, -3}, {-8, -1, 2, 5, -4, 10, -7, 8, -9, 7, -11, -2, 3, 6, -10, 4, -6, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 5 + 4 v u 4 -4 v u 3 -4 u 2 + 4 u -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-4 \sqrt{q}+\frac{5}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{1}{q^{13/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- z a 7 + z 5 a 5 + 4 z 3 a 5 + 3 z a 5 + 2 a 5 z -1 - z 7 a 3 -5 z 5 a 3 -8 z 3 a 3 -8 z a 3 -4 a 3 z -1 + 2 z 5 a + 8 z 3 a + 7 z a + 3 a z -1 - z 3 a -1 -3 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -3 a 4 z 8 -5 a 2 z 8 -2 z 8 -3 a 5 z 7 -2 a 3 z 7 - z 7 a -1 - a 6 z 6 + 11 a 4 z 6 + 21 a 2 z 6 + 9 z 6 + 12 a 5 z 5 + 23 a 3 z 5 + 16 a z 5 + 5 z 5 a -1 + a 6 z 4 -11 a 4 z 4 -23 a 2 z 4 -11 z 4 -3 a 7 z 3 -20 a 5 z 3 -37 a 3 z 3 -28 a z 3 -8 z 3 a -1 - a 8 z 2 - a 6 z 2 + 4 a 4 z 2 + 9 a 2 z 2 + 5 z 2 + 3 a 7 z + 15 a 5 z + 23 a 3 z + 16 a z + 5 z a -1 - a 6 -2 a 4 -3 a 2 -1-2 a 5 z -1 -4 a 3 z -1 -3 a z -1 - a -1 z -1$ (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 L11n22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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