L11a537

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L11a536

L11a538

1 Link Presentations
2 Polynomial invariants
3 Khovanov Homology
4 Computer Talk
5 Modifying This Page

(Knotscape image)

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[edit] Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_18,12,19,11 X_10,3,11,4 X_4,9,1,10 X_8,18,5,17 X_16,8,17,7 X_14,16,9,15 X_22,20,15,19 X_20,13,21,14 X_12,21,13,22
Gauss code	{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -11, 10, -8}, {8, -7, 6, -3, 9, -10, 11, -9}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 + 2 v 2 w u 2 -3 v w u 2 + w u 2 + v 2 x u 2 -2 v x u 2 - v 2 w x u 2 + 2 v w x u 2 - w x u 2 + x u 2 + 2 v 2 u -2 v u -3 v 2 w u + 4 v w u -2 w u -2 v 2 x u + 4 v x u + v 2 w x u -2 v w x u + 2 w x u -3 x u + u - v 2 + 2 v + v 2 w -2 v w + w + v 2 x -3 v x + v w x - w x + 2 x -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+9 q^{7/2}-14 q^{5/2}+17 q^{3/2}-21 \sqrt{q}+\frac{17}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 10 a z 3 -8 z 3 a -1 + 2 z 3 a -3 + a 5 z -8 a 3 z + 14 a z -9 z a -1 + 2 z a -3 + 2 a 5 z -1 -8 a 3 z -1 + 11 a z -1 -6 a -1 z -1 + a -3 z -1 + a 5 z -3 -3 a 3 z -3 + 3 a z -3 - a -1 z -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 -2 a 3 z 9 -7 a z 9 -5 z 9 a -1 -2 a 4 z 8 -7 a 2 z 8 -11 z 8 a -2 -16 z 8 - a 5 z 7 - a 3 z 7 + a z 7 -12 z 7 a -1 -13 z 7 a -3 + 6 a 4 z 6 + 23 a 2 z 6 + 10 z 6 a -2 -9 z 6 a -4 + 36 z 6 + 5 a 5 z 5 + 24 a 3 z 5 + 48 a z 5 + 52 z 5 a -1 + 19 z 5 a -3 -4 z 5 a -5 -3 a 4 z 4 -6 a 2 z 4 + 10 z 4 a -2 + 8 z 4 a -4 - z 4 a -6 -2 z 4 -10 a 5 z 3 -45 a 3 z 3 -74 a z 3 -53 z 3 a -1 -13 z 3 a -3 + z 3 a -5 -7 a 4 z 2 -24 a 2 z 2 -14 z 2 a -2 -3 z 2 a -4 -28 z 2 + 10 a 5 z + 35 a 3 z + 49 a z + 30 z a -1 + 6 z a -3 + 9 a 4 + 21 a 2 + 6 a -2 + a -4 + 18-5 a 5 z -1 -14 a 3 z -1 -18 a z -1 -11 a -1 z -1 -2 a -3 z -1 -3 a 4 z -2 -6 a 2 z -2 -3 z -2 + a 5 z -3 + 3 a 3 z -3 + 3 a z -3 + a -1 z -3$ (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 =

1 is the signature of L11a537. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a537/KhovanovTable

Integral Khovanov Homology

(db, data source)

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