L11a538

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L11a537

L11a539

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_14,7,15,8 X_8,13,5,14 X_18,12,19,11 X_22,19,17,20 X_16,21,9,22 X_20,15,21,16 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {9, -5, 6, -8, 7, -6}, {11, -2, 5, -9, 4, -3, 8, -7}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$v u 3 - v w u 3 + w u 3 - v x u 3 + x u 3 - u 3 -3 v u 2 + 4 v w u 2 -4 w u 2 + 4 v x u 2 -2 v w x u 2 + 2 w x u 2 -4 x u 2 + 3 u 2 + 2 v u -4 v w u + 4 w u -4 v x u + 3 v w x u -3 w x u + 4 x u -2 u + v w - w + v x - v w x + w x - x$ (db)
Jones polynomial	$-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{13}{q^{5/2}}-\frac{20}{q^{7/2}}+\frac{19}{q^{9/2}}-\frac{22}{q^{11/2}}+\frac{17}{q^{13/2}}-\frac{14}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$- a 11 z -1 + 4 z a 9 + 4 a 9 z -1 + a 9 z -3 -6 z 3 a 7 -11 z a 7 -9 a 7 z -1 -3 a 7 z -3 + 3 z 5 a 5 + 8 z 3 a 5 + 12 z a 5 + 10 a 5 z -1 + 3 a 5 z -3 + z 5 a 3 - z 3 a 3 -4 z a 3 -4 a 3 z -1 - a 3 z -3 - z 3 a - z a$ (db)
Kauffman polynomial	$- z 6 a 12 + 3 z 4 a 12 -3 z 2 a 12 + a 12 -3 z 7 a 11 + 8 z 5 a 11 -9 z 3 a 11 + 6 z a 11 -2 a 11 z -1 -4 z 8 a 10 + 4 z 6 a 10 + 8 z 4 a 10 -14 z 2 a 10 + 6 a 10 -3 z 9 a 9 -7 z 7 a 9 + 34 z 5 a 9 -43 z 3 a 9 + 30 z a 9 -11 a 9 z -1 + a 9 z -3 - z 10 a 8 -12 z 8 a 8 + 21 z 6 a 8 + 8 z 4 a 8 -27 z 2 a 8 -3 a 8 z -2 + 18 a 8 -7 z 9 a 7 -9 z 7 a 7 + 59 z 5 a 7 -79 z 3 a 7 + 52 z a 7 -18 a 7 z -1 + 3 a 7 z -3 - z 10 a 6 -15 z 8 a 6 + 26 z 6 a 6 + z 4 a 6 -27 z 2 a 6 -6 a 6 z -2 + 21 a 6 -4 z 9 a 5 -11 z 7 a 5 + 44 z 5 a 5 -59 z 3 a 5 + 40 z a 5 -14 a 5 z -1 + 3 a 5 z -3 -7 z 8 a 4 + 7 z 6 a 4 + 2 z 4 a 4 -12 z 2 a 4 -3 a 4 z -2 + 9 a 4 -6 z 7 a 3 + 10 z 5 a 3 -12 z 3 a 3 + 11 z a 3 -5 a 3 z -1 + a 3 z -3 -3 z 6 a 2 + 4 z 4 a 2 - z 2 a 2 - z 5 a + 2 z 3 a - z a$ (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 L11a538. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a538/KhovanovTable

Integral Khovanov Homology

(db, data source)

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