L11a540

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L11a539

L11a541

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

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[edit] Link Presentations

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

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v u 3 + 2 v w u 3 + v x u 3 - v w x u 3 + 3 v u 2 -4 v w u 2 + 3 w u 2 -4 v x u 2 + 3 v w x u 2 -2 w x u 2 + 2 x u 2 -2 u 2 -2 v u + 2 v w u -4 w u + 3 v x u -2 v w x u + 3 w x u -4 x u + 3 u + w - w x + 2 x -1$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{19}{q^{7/2}}+\frac{17}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + 2 a 9 z -1 + a 9 z -3 -3 z 3 a 7 -9 z a 7 -9 a 7 z -1 -3 a 7 z -3 + 3 z 5 a 5 + 11 z 3 a 5 + 17 z a 5 + 12 a 5 z -1 + 3 a 5 z -3 - z 7 a 3 -4 z 5 a 3 -8 z 3 a 3 -10 z a 3 -5 a 3 z -1 - a 3 z -3 + z 5 a + 2 z 3 a + z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 3 z 3 a 11 -3 z a 11 + a 11 z -1 -2 z 6 a 10 + 3 z 4 a 10 - a 10 -3 z 7 a 9 + 3 z 5 a 9 - z 3 a 9 + 3 z a 9 -3 a 9 z -1 + a 9 z -3 -3 z 8 a 8 -2 z 6 a 8 + 12 z 4 a 8 -16 z 2 a 8 -3 a 8 z -2 + 11 a 8 -3 z 9 a 7 - z 7 a 7 + 9 z 5 a 7 -20 z 3 a 7 + 20 z a 7 -12 a 7 z -1 + 3 a 7 z -3 - z 10 a 6 -9 z 8 a 6 + 19 z 6 a 6 -3 z 4 a 6 -28 z 2 a 6 -6 a 6 z -2 + 24 a 6 -7 z 9 a 5 + 4 z 7 a 5 + 23 z 5 a 5 -39 z 3 a 5 + 29 z a 5 -14 a 5 z -1 + 3 a 5 z -3 - z 10 a 4 -12 z 8 a 4 + 32 z 6 a 4 -16 z 4 a 4 -12 z 2 a 4 -3 a 4 z -2 + 13 a 4 -4 z 9 a 3 -2 z 7 a 3 + 28 z 5 a 3 -31 z 3 a 3 + 18 z a 3 -6 a 3 z -1 + a 3 z -3 -6 z 8 a 2 + 12 z 6 a 2 -2 z 4 a 2 - z 2 a 2 -4 z 7 a + 10 z 5 a -8 z 3 a + 3 z a - z 6 + 2 z 4 - z 2$ (db)

[edit] Khovanov Homology

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 L11a540. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a540/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}$
$r = -6$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$