L11a531

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L11a530

L11a532

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

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v x u 3 - w x u 3 + 2 x u 3 - u 3 -3 v u 2 + 4 v w u 2 -3 w u 2 + 3 v x u 2 -2 v w x u 2 + 3 w x u 2 -4 x u 2 + 2 u 2 + 3 v u -4 v w u + 3 w u -3 v x u + 2 v w x u -3 w x u + 4 x u -2 u - v + 2 v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{14}{q^{11/2}}-\frac{21}{q^{13/2}}+\frac{19}{q^{15/2}}-\frac{22}{q^{17/2}}+\frac{16}{q^{19/2}}-\frac{11}{q^{21/2}}+\frac{6}{q^{23/2}}-\frac{2}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- a 15 z -3 + 5 a 13 z -1 + 5 a 13 z -3 -10 z a 11 -19 a 11 z -1 -9 a 11 z -3 + 10 z 3 a 9 + 26 z a 9 + 23 a 9 z -1 + 7 a 9 z -3 -4 z 5 a 7 -13 z 3 a 7 -16 z a 7 -9 a 7 z -1 -2 a 7 z -3 - z 5 a 5 - z 3 a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 4 z 4 a 16 -6 z 2 a 16 - a 16 z -2 + 4 a 16 -2 z 7 a 15 + 5 z 5 a 15 -4 z 3 a 15 + 2 z a 15 -2 a 15 z -1 + a 15 z -3 -2 z 8 a 14 -3 z 6 a 14 + 22 z 4 a 14 -34 z 2 a 14 -7 a 14 z -2 + 24 a 14 -2 z 9 a 13 -5 z 7 a 13 + 19 z 5 a 13 -22 z 3 a 13 + 16 z a 13 -12 a 13 z -1 + 5 a 13 z -3 - z 10 a 12 -7 z 8 a 12 + 3 z 6 a 12 + 38 z 4 a 12 -75 z 2 a 12 -18 a 12 z -2 + 58 a 12 -7 z 9 a 11 -3 z 7 a 11 + 34 z 5 a 11 -45 z 3 a 11 + 37 z a 11 -24 a 11 z -1 + 9 a 11 z -3 - z 10 a 10 -15 z 8 a 10 + 19 z 6 a 10 + 29 z 4 a 10 -73 z 2 a 10 -19 a 10 z -2 + 60 a 10 -5 z 9 a 9 -10 z 7 a 9 + 40 z 5 a 9 -47 z 3 a 9 + 39 z a 9 -23 a 9 z -1 + 7 a 9 z -3 -10 z 8 a 8 + 10 z 6 a 8 + 12 z 4 a 8 -26 z 2 a 8 -7 a 8 z -2 + 23 a 8 -10 z 7 a 7 + 19 z 5 a 7 -19 z 3 a 7 + 16 z a 7 -9 a 7 z -1 + 2 a 7 z -3 -4 z 6 a 6 + 3 z 4 a 6 - z 5 a 5 + z 3 a 5$ (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 L11a531. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a531/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -8$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{6}$
$r = -7$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -6$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = -4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{14}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$