L11a545

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L11a544

L11a546

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_14,5,15,6 X_12,4,13,3 X_2,9,3,10 X_18,7,19,8 X_8,17,9,18 X_10,13,5,14 X_22,20,17,19 X_16,21,11,22 X_20,11,21,12 X_4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -6, 4, -7}, {10, -3, 7, -2, 11, -9}, {6, -5, 8, -10, 9, -8}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 2 + 2 v u 2 + 2 v 2 w u 2 -3 v w u 2 + w u 2 + v 2 x u 2 -2 v x u 2 - v 2 w x u 2 + 2 v w x u 2 - w x u 2 + 3 v 2 u -5 v u -2 v 2 w u + 5 v w u -2 w u -2 v 2 x u + 5 v x u + 2 v 2 w x u -5 v w x u + 3 w x u -2 x u + 2 u - v 2 + 2 v -2 v w + w + v 2 x -3 v x + 2 v w x -2 w x + 2 x -1$ (db)
Jones polynomial	$q^{3/2}-5 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{20}{q^{5/2}}-\frac{25}{q^{7/2}}+\frac{21}{q^{9/2}}-\frac{21}{q^{11/2}}+\frac{13}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z a 9 + a 9 z -1 + a 9 z -3 -3 z 3 a 7 -6 z a 7 -6 a 7 z -1 -3 a 7 z -3 + 3 z 5 a 5 + 8 z 3 a 5 + 10 z a 5 + 9 a 5 z -1 + 3 a 5 z -3 - z 7 a 3 -3 z 5 a 3 -4 z 3 a 3 -4 z a 3 -4 a 3 z -1 - a 3 z -3 + z 5 a + z 3 a - z a$ (db)
Kauffman polynomial	$- z 5 a 11 + 2 z 3 a 11 - z a 11 -3 z 6 a 10 + 4 z 4 a 10 - z 2 a 10 -6 z 7 a 9 + 10 z 5 a 9 -12 z 3 a 9 + 11 z a 9 -5 a 9 z -1 + a 9 z -3 -7 z 8 a 8 + 6 z 6 a 8 + 4 z 4 a 8 -14 z 2 a 8 -3 a 8 z -2 + 10 a 8 -5 z 9 a 7 -7 z 7 a 7 + 35 z 5 a 7 -50 z 3 a 7 + 33 z a 7 -12 a 7 z -1 + 3 a 7 z -3 -2 z 10 a 6 -13 z 8 a 6 + 25 z 6 a 6 -26 z 2 a 6 -6 a 6 z -2 + 19 a 6 -12 z 9 a 5 + 10 z 7 a 5 + 27 z 5 a 5 -46 z 3 a 5 + 33 z a 5 -12 a 5 z -1 + 3 a 5 z -3 -2 z 10 a 4 -15 z 8 a 4 + 37 z 6 a 4 -11 z 4 a 4 -14 z 2 a 4 -3 a 4 z -2 + 10 a 4 -7 z 9 a 3 + 6 z 7 a 3 + 13 z 5 a 3 -14 z 3 a 3 + 11 z a 3 -5 a 3 z -1 + a 3 z -3 -9 z 8 a 2 + 20 z 6 a 2 -10 z 4 a 2 - z 2 a 2 -5 z 7 a + 10 z 5 a -4 z 3 a - z a - z 6 + z 4$ (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 L11a545. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a545/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -4$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{11}$
$r = -3$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$