L11a547

From Knot Atlas

Jump to: navigation, search

L11a546

L11a548

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

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a547's page at Knotilus.

Visit L11a547's page at the original Knot Atlas.

[edit L11a547 Quick Notes]

[edit L11a547 Further Notes and Views]

Link L11a547.

A graph, L11a547.

A part of a knot and a part of a graph.

[edit] Link Presentations

[edit Notes on L11a547's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$2 v 2 u 2 -3 v u 2 - v 2 w u 2 + 2 v w u 2 - w u 2 -2 v 2 x u 2 + 3 v x u 2 -2 v w x u 2 + w x u 2 - x u 2 + u 2 -3 v 2 u + 6 v u + 2 v 2 w u -5 v w u + 3 w u + 3 v 2 x u -5 v x u -2 v 2 w x u + 6 v w x u -3 w x u + 2 x u -2 u + v 2 -2 v - v 2 w + 3 v w -2 w - v 2 x + 2 v x + v 2 w x -3 v w x + 2 w x - x$ (db)
Jones polynomial	$-\frac{1}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{16}{q^{11/2}}-\frac{24}{q^{13/2}}+\frac{25}{q^{15/2}}-\frac{27}{q^{17/2}}+\frac{21}{q^{19/2}}-\frac{17}{q^{21/2}}+\frac{9}{q^{23/2}}-\frac{4}{q^{25/2}}+\frac{1}{q^{27/2}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$- z a 13 + a 13 z -3 + 3 z 3 a 11 + z a 11 -5 a 11 z -1 -3 a 11 z -3 -2 z 5 a 9 + z 3 a 9 + 11 z a 9 + 10 a 9 z -1 + 3 a 9 z -3 -4 z 5 a 7 -11 z 3 a 7 -11 z a 7 -5 a 7 z -1 - a 7 z -3 - z 5 a 5 - z 3 a 5$ (db)
Kauffman polynomial	$- z 6 a 16 + 2 z 4 a 16 - z 2 a 16 -4 z 7 a 15 + 9 z 5 a 15 -8 z 3 a 15 + 3 z a 15 -7 z 8 a 14 + 13 z 6 a 14 -6 z 4 a 14 - z 2 a 14 -6 z 9 a 13 + 23 z 5 a 13 -25 z 3 a 13 + 12 z a 13 -5 a 13 z -1 + a 13 z -3 -2 z 10 a 12 -18 z 8 a 12 + 45 z 6 a 12 -26 z 4 a 12 -3 z 2 a 12 -3 a 12 z -2 + 10 a 12 -14 z 9 a 11 + 10 z 7 a 11 + 26 z 5 a 11 -31 z 3 a 11 + 21 z a 11 -12 a 11 z -1 + 3 a 11 z -3 -2 z 10 a 10 -23 z 8 a 10 + 50 z 6 a 10 -22 z 4 a 10 -15 z 2 a 10 -6 a 10 z -2 + 19 a 10 -8 z 9 a 9 -4 z 7 a 9 + 30 z 5 a 9 -32 z 3 a 9 + 23 z a 9 -12 a 9 z -1 + 3 a 9 z -3 -12 z 8 a 8 + 15 z 6 a 8 - z 4 a 8 -12 z 2 a 8 -3 a 8 z -2 + 10 a 8 -10 z 7 a 7 + 17 z 5 a 7 -17 z 3 a 7 + 11 z a 7 -5 a 7 z -1 + a 7 z -3 -4 z 6 a 6 + 3 z 4 a 6 - z 5 a 5 + z 3 a 5$ (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 =

-5 is the signature of L11a547. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L11a547/KhovanovTable

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -6$	$i = -4$
$r = -11$	${\mathbb Z}$
$r = -10$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -9$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -8$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -7$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -6$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{15}$
$r = -5$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = -3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$