L11n23

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 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_2,16,3,15
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 $u$ , $v$ , $w$ , ...)	$- v u 3 + u 3 + 5 v u 2 -5 u 2 -5 v u + 5 u + v -1$ (db)
Jones polynomial	$-q^{9/2}+3 q^{7/2}-6 q^{5/2}+7 q^{3/2}-8 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a z 5 - a 3 z 3 + 3 a z 3 -4 z 3 a -1 - a 3 z + 5 a z -8 z a -1 + 4 z a -3 + 2 a z -1 -4 a -1 z -1 + 3 a -3 z -1 - a -5 z -1$ (db)
Kauffman polynomial	$- a z 9 - z 9 a -1 -3 a 2 z 8 -3 z 8 a -2 -6 z 8 -3 a 3 z 7 -4 a z 7 -3 z 7 a -1 -2 z 7 a -3 - a 4 z 6 + 9 a 2 z 6 + 12 z 6 a -2 + 22 z 6 + 11 a 3 z 5 + 26 a z 5 + 22 z 5 a -1 + 7 z 5 a -3 + 3 a 4 z 4 -5 a 2 z 4 -23 z 4 a -2 -3 z 4 a -4 -28 z 4 -10 a 3 z 3 -34 a z 3 -40 z 3 a -1 -17 z 3 a -3 - z 3 a -5 - a 4 z 2 + 2 a 2 z 2 + 15 z 2 a -2 + 3 z 2 a -4 + 15 z 2 + 3 a 3 z + 15 a z + 24 z a -1 + 14 z a -3 + 2 z a -5 - a 2 -3 a -2 - a -4 -2-2 a z -1 -4 a -1 z -1 -3 a -3 z -1 - a -5 z -1$ (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 L11n23. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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