L4a1

Contents

 (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"

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

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 a5z−1−a3z−a3z−1−az (db) Kauffman polynomial a5z3−3a5z + a5z−1 + a4z2−a4 + a3z3−2a3z + a3z−1 + a2z2 + az (db)

Khovanov Homology

 The coefficients of the monomials trqj 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−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
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ 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.

Read me first: Modifying Knot Pages

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