L11a543

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L11a542

L11a544

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_12,3,13,4 X_10,13,5,14 X_8,17,9,18 X_14,7,15,8 X_18,9,19,10 X_22,19,17,20 X_16,21,11,22 X_20,15,21,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, {11, -2, 3, -5, 9, -8}, {4, -6, 7, -9, 8, -7}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v 2 u 2 -2 v u 2 -2 v 2 w u 2 + 3 v w u 2 - w u 2 + v x u 2 -2 v w x u 2 + w x u 2 - x u 2 + u 2 -2 v 2 u + 5 v u + 3 v 2 w u -4 v w u + w u + v 2 x u -4 v x u -2 v 2 w x u + 5 v w x u -2 w x u + 3 x u -2 u + v 2 -2 v - v 2 w + v w - v 2 x + 3 v x + v 2 w x -2 v w x + w x -2 x$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{12}{q^{11/2}}-\frac{19}{q^{13/2}}+\frac{19}{q^{15/2}}-\frac{22}{q^{17/2}}+\frac{17}{q^{19/2}}-\frac{14}{q^{21/2}}+\frac{8}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 + a 13 z -3 + 3 z 3 a 11 + 2 z a 11 -5 a 11 z -1 -3 a 11 z -3 -2 z 5 a 9 - z 3 a 9 + 8 z a 9 + 10 a 9 z -1 + 3 a 9 z -3 -3 z 5 a 7 -8 z 3 a 7 -8 z a 7 -5 a 7 z -1 - a 7 z -3 - z 5 a 5 -2 z 3 a 5 - z a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 2 z 4 a 16 - z 2 a 16 -4 z 7 a 15 + 10 z 5 a 15 -8 z 3 a 15 + 3 z a 15 -6 z 8 a 14 + 12 z 6 a 14 -4 z 4 a 14 - z 2 a 14 -4 z 9 a 13 -4 z 7 a 13 + 31 z 5 a 13 -33 z 3 a 13 + 15 z a 13 -5 a 13 z -1 + a 13 z -3 - z 10 a 12 -15 z 8 a 12 + 38 z 6 a 12 -23 z 4 a 12 -4 z 2 a 12 -3 a 12 z -2 + 10 a 12 -8 z 9 a 11 - z 7 a 11 + 38 z 5 a 11 -49 z 3 a 11 + 29 z a 11 -12 a 11 z -1 + 3 a 11 z -3 - z 10 a 10 -15 z 8 a 10 + 31 z 6 a 10 -12 z 4 a 10 -18 z 2 a 10 -6 a 10 z -2 + 19 a 10 -4 z 9 a 9 -7 z 7 a 9 + 29 z 5 a 9 -39 z 3 a 9 + 29 z a 9 -12 a 9 z -1 + 3 a 9 z -3 -6 z 8 a 8 + 3 z 6 a 8 + 9 z 4 a 8 -15 z 2 a 8 -3 a 8 z -2 + 10 a 8 -6 z 7 a 7 + 11 z 5 a 7 -13 z 3 a 7 + 11 z a 7 -5 a 7 z -1 + a 7 z -3 -3 z 6 a 6 + 4 z 4 a 6 - z 2 a 6 - z 5 a 5 + 2 z 3 a 5 - z a 5$ (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 =

-5 is the signature of L11a543. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a543/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -8$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -7$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -6$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{13}$
$r = -5$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -1$	${\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$