L11a548

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L11a547

L11n1

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_22,20,17,19 X_16,22,13,21 X_20,16,21,15 X_12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 10, -9}, {11, -3, 8, -10, 9, -8}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v u -3 v w u + 2 w u -2 v x u + 2 v w x u -2 w x u + 2 x u -2 v y u + 2 v w y u - w y u + 2 v x y u - v w x y u + w x y u -2 x y u + y u - u - v + 2 v w -2 w + v x - v w x + 2 w x -2 x + 2 v y -2 v w y + 2 w y -2 v x y + v w x y -2 w x y + 3 x y -2 y + 1$ (db)
Jones polynomial	$- q 2 + 5 q -10 + 15 q -1 -15 q -2 + 21 q -3 -15 q -4 + 16 q -5 -6 q -6 + 6 q -7 - q -8 + q -9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a 10 z -2 + a 10 z -4 -7 a 8 z -2 -4 a 8 z -4 -4 a 8 + 6 z 2 a 6 + 15 a 6 z -2 + 6 a 6 z -4 + 14 a 6 -4 z 4 a 4 -10 z 2 a 4 -13 a 4 z -2 -4 a 4 z -4 -16 a 4 + z 6 a 2 + 2 z 4 a 2 + 4 z 2 a 2 + 4 a 2 z -2 + a 2 z -4 + 6 a 2 - z 4$ (db)
Kauffman polynomial	$z 6 a 10 -5 z 4 a 10 + 10 z 2 a 10 + 5 a 10 z -2 - a 10 z -4 -10 a 10 + z 7 a 9 -10 z 3 a 9 + 20 z a 9 -15 a 9 z -1 + 4 a 9 z -3 + z 8 a 8 + 4 z 6 a 8 -20 z 4 a 8 + 30 z 2 a 8 + 14 a 8 z -2 -4 a 8 z -4 -25 a 8 + z 9 a 7 + 3 z 7 a 7 -30 z 3 a 7 + 55 z a 7 -41 a 7 z -1 + 12 a 7 z -3 + z 10 a 6 + z 8 a 6 + 9 z 6 a 6 -31 z 4 a 6 + 40 z 2 a 6 + 18 a 6 z -2 -6 a 6 z -4 -31 a 6 + 6 z 9 a 5 -8 z 7 a 5 + 11 z 5 a 5 -30 z 3 a 5 + 55 z a 5 -41 a 5 z -1 + 12 a 5 z -3 + z 10 a 4 + 10 z 8 a 4 -14 z 6 a 4 -11 z 4 a 4 + 30 z 2 a 4 + 14 a 4 z -2 -4 a 4 z -4 -25 a 4 + 5 z 9 a 3 -5 z 5 a 3 -10 z 3 a 3 + 20 z a 3 -15 a 3 z -1 + 4 a 3 z -3 + 10 z 8 a 2 -15 z 6 a 2 + 10 z 2 a 2 + 5 a 2 z -2 - a 2 z -4 -10 a 2 + 10 z 7 a -15 z 5 a + 5 z 6 -5 z 4 + z 5 a -1$ (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 =

-2 is the signature of L11a548. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a548/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -3$	$i = -1$
$r = -8$	${\mathbb Z}$	${\mathbb Z}$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{5}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -4$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2$	${\mathbb Z}^{11}$
$r = -3$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{9}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$