Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)

L4a1

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L2a1.gif

L2a1

L5a1.gif

L5a1

Contents

L4a1.gif
(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.

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.


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"

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
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L4a1 ML.gif

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)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-1 is the signature of L4a1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10χ
0    11
-2   110
-4     0
-6  1  1
-81    1
-101    1
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}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.

L2a1.gif

L2a1

L5a1.gif

L5a1