L4a1

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L2a1

L5a1

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 L4a1's page at Knotilus.

Visit L4a1's page at the original Knot Atlas.

[edit L4a1 Quick Notes]

L4a1 is $4^2_1$ in the Rolfsen table of links. It frequently occurs in late Roman mosaics and some medieval decorations. In this context, it is called the "Solomon's knot" (sigillum Salomonis) or "guilloche knot". It is also the "Kramo-bone" symbol (meaning "one being bad makes all appear to be bad") of the Adinkra symbol system. Link L10a101 contains multiple L4a1 configurations.

[edit L4a1 Further Notes and Views]

Simple squared depiction	A Kolam with two cycles[1]	Hearst Castle tile [2]	Mosaic seen at Kibbutz Lahav [3]
Carving above door of church in Italy	Decorative depiction	(crossings along one side)	Linked hearts used as symbol of Vendée region of France
Heraldic ornament.	Composed of intersecting circles.	Decorative fitting closely within square.	Ancient Roman mosaic.
Made of two (impossible) Penrose rectangles.	Array of "Solomon's knots" forming overall circular patterns.	Configuration of three L4a1	Configuration of four L4a1
Medieval manuscript	Medieval manuscript	Medieval manuscript	Rotated knotwork cross with eight L4a1 sub-configurations
Knotopologynn-diagram for "Solomon's knots"

[edit] Link Presentations

[edit Notes on L4a1's Link Presentations] Why such an ugly Braid Representative?

Planar diagram presentation	X₆₁₇₂ X₈₃₅₄ X₂₅₃₆ X₄₇₁₈
Gauss code	{1, -3, 2, -4}, {3, -1, 4, -2}

A Braid Representative

A Morse Link Presentation

[edit] Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{-u-v}{\sqrt{u} \sqrt{v}}$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 5 z -1 - a 3 z - a 3 z -1 - a z$ (db)
Kauffman polynomial	$a 5 z 3 -3 a 5 z + a 5 z -1 + a 4 z 2 - a 4 + a 3 z 3 -2 a 3 z + a 3 z -1 + a 2 z 2 + a z$ (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 L4a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

Integral Khovanov Homology

(db, data source)

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