L8a21

From Knot Atlas

Jump to: navigation, search

L8a20

L8n1

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 L8a21's page at Knotilus.

Visit L8a21's page at the original Knot Atlas.

[edit L8a21 Quick Notes]

L8a21 is a closed four-link chain. It is $8^4_{1}$ in the Rolfsen table of links.

[edit L8a21 Further Notes and Views]

Four linked squares.

Floor decoration in the Biblioteca Medicea Laurenziana, Florence, intended to contain four rings interlinked in this maner (but there is an interlacing error at upper right).

Ornamental circular arcs in square.

[edit] Link Presentations

[edit Notes on L8a21's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_16,11,13,12 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_12,15,9,16
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -8}, {7, -6, 8, -3}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{t(2) t(1)-t(2) t(3) t(1)+t(3) t(1)-t(2) t(4) t(1)-t(3) t(4) t(1)+2 t(4) t(1)-t(1)-t(2)+2 t(2) t(3)-t(3)+t(2) t(4)-t(2) t(3) t(4)+t(3) t(4)-t(4)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}}$ (db)
Jones polynomial	$\frac{4}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 11 z -3 -3 a 9 z -3 -4 a 9 z -1 + 3 a 7 z -3 + 6 a 7 z + 8 a 7 z -1 -3 a 5 z 3 - a 5 z -3 -6 a 5 z -4 a 5 z -1 - a 3 z 3$ (db)
Kauffman polynomial	$- z 5 a 11 + 4 z 3 a 11 -6 z a 11 + 4 a 11 z -1 - a 11 z -3 - z 6 a 10 + 6 z 2 a 10 + 3 a 10 z -2 -8 a 10 - z 7 a 9 -2 z 5 a 9 + 11 z 3 a 9 -14 z a 9 + 9 a 9 z -1 -3 a 9 z -3 -5 z 6 a 8 + 5 z 4 a 8 + 12 z 2 a 8 + 6 a 8 z -2 -15 a 8 - z 7 a 7 -7 z 5 a 7 + 17 z 3 a 7 -14 z a 7 + 9 a 7 z -1 -3 a 7 z -3 -4 z 6 a 6 + 2 z 4 a 6 + 6 z 2 a 6 + 3 a 6 z -2 -8 a 6 -6 z 5 a 5 + 9 z 3 a 5 -6 z a 5 + 4 a 5 z -1 - a 5 z -3 -3 z 4 a 4 - z 3 a 3$ (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 =

-3 is the signature of L8a21. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Data:L8a21/KhovanovTable

Integral Khovanov Homology

(db, data source)

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