L11a532

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L11a531

L11a533

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_18,12,19,11 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_20,16,21,15 X_22,18,13,17 X_16,22,17,21 X_12,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 8, -10, 9, -3, 11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 v w u 3 -2 w u 3 + 2 v x u 3 - v w x u 3 + w x u 3 - x u 3 + u 3 + 2 v u 2 -2 v w u 2 + 3 w u 2 -3 v x u 2 + v w x u 2 -2 w x u 2 + 2 x u 2 - u 2 -2 v u + 2 v w u -3 w u + 3 v x u - v w x u + 2 w x u -2 x u + u + v - v w + 2 w -2 v x + v w x -2 w x + 2 x -1$ (db)
Jones polynomial	$q^{11/2}-4 q^{9/2}+10 q^{7/2}-14 q^{5/2}+17 q^{3/2}-19 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 3 a z 5 -4 z 5 a -1 + z 5 a -3 -3 a 3 z 3 + 12 a z 3 -9 z 3 a -1 + 2 z 3 a -3 + a 5 z -11 a 3 z + 22 a z -15 z a -1 + 3 z a -3 + 3 a 5 z -1 -14 a 3 z -1 + 22 a z -1 -14 a -1 z -1 + 3 a -3 z -1 + 2 a 5 z -3 -7 a 3 z -3 + 9 a z -3 -5 a -1 z -3 + a -3 z -3$ (db)
Kauffman polynomial	$- a 2 z 10 - z 10 - a 3 z 9 -6 a z 9 -5 z 9 a -1 - a 4 z 8 - a 2 z 8 -11 z 8 a -2 -11 z 8 - a 5 z 7 -3 a 3 z 7 + 6 a z 7 -6 z 7 a -1 -14 z 7 a -3 + a 4 z 6 -2 a 2 z 6 + 9 z 6 a -2 -10 z 6 a -4 + 16 z 6 + 6 a 5 z 5 + 23 a 3 z 5 + 21 a z 5 + 28 z 5 a -1 + 20 z 5 a -3 -4 z 5 a -5 + 10 a 4 z 4 + 36 a 2 z 4 + 19 z 4 a -2 + 10 z 4 a -4 - z 4 a -6 + 34 z 4 -14 a 5 z 3 -42 a 3 z 3 -42 a z 3 -22 z 3 a -1 -8 z 3 a -3 -26 a 4 z 2 -73 a 2 z 2 -34 z 2 a -2 -6 z 2 a -4 -75 z 2 + 16 a 5 z + 39 a 3 z + 37 a z + 16 z a -1 + 2 z a -3 + 23 a 4 + 60 a 2 + 24 a -2 + 4 a -4 + 58-9 a 5 z -1 -23 a 3 z -1 -24 a z -1 -12 a -1 z -1 -2 a -3 z -1 -7 a 4 z -2 -19 a 2 z -2 -7 a -2 z -2 - a -4 z -2 -18 z -2 + 2 a 5 z -3 + 7 a 3 z -3 + 9 a z -3 + 5 a -1 z -3 + a -3 z -3$ (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 L11a532. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a532/KhovanovTable

Integral Khovanov Homology

(db, data source)

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