{"id":2895,"date":"2016-10-15T01:00:44","date_gmt":"2016-10-15T01:00:44","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2895"},"modified":"2016-10-29T16:22:06","modified_gmt":"2016-10-29T16:22:06","slug":"laves-graph","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/10\/15\/laves-graph\/","title":{"rendered":"Laves Graph"},"content":{"rendered":"
\n
\"Laves<\/a>

Laves Graph – Greg Egan<\/p><\/div>\n<\/div>\n

This picture by Greg Egan<\/a> shows the Laves graph<\/a>, a structure discovered by the crystallographer Fritz Laves<\/a> in 1932. It is also called the ‘\\(\\mathrm{K}_4\\) crystal’, since is an embedding of the maximal abelian cover of the complete graph on 4 vertices<\/a> in 3-dimensional Euclidean space. It is also called the ‘triamond’, since it is a theoretically possible — but never yet seen — crystal structure for carbon. <\/p>\n

In the Laves graph, each vertex is connected to three others with edges at 120° angles. These edges lie in a plane, so we get a plane for each atom. However, for any two neighboring vertices, these planes are different. In fact, these planes come in four equally spaced families, parallel to the four faces of a regular tetrahedron.<\/p>\n

The Laves graph is highly symmetrical. There is a symmetry carrying any vertex and any of its edges to any other vertex and any of its edges. However, the Laves graph has an inherent handedness, or chirality<\/a>: it comes in two different mirror-image forms.<\/p>\n

The smallest cycles<\/a> in the Laves graph have 10 edges. Each vertex lies in 15 of these 10-cycles.<\/p>\n

Some chemists have argued that the triamond should be \u2018metastable\u2019 at room temperature and pressure: that is, it should last for a while but eventually turn to graphite. Diamonds are also considered metastable. However, diamonds are formed naturally under high pressure — while triamonds, it seems, are not. <\/p>\n

Nonetheless, the mathematics behind the Laves graph does find its way into nature. The minimal surface called a gyroid<\/a> is topologically the boundary of a tubular neighborhood of the Laves graph, and this surface shows up in the structure of certain butterfly wings. For details, see:<\/p>\n

• John Baez, The physics of butterfly wings<\/a>, Azimuth<\/i>, August 11, 2015.<\/p>\n

• S. T. Hyde, M. O’Keeffe, and D. M. Proserpio, A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics, Angew. Chem. Int. Ed.<\/i> 47<\/b> (2008), 7996–8000.<\/p>\n

Mathematically, the most interesting way to construct the Laves graph is as a ‘topological crystal’. This method was introduced by Sunada:<\/p>\n

• Toshikazu Sunada, Crystals that nature might miss creating<\/a>, Notices of the American Mathematical Society<\/i> 55<\/b> (2008), 208–215.<\/p>\n

• Toshikazu Sunada, Topological Crystallography<\/i>, Springer, Berlin, 2012.<\/p>\n

Just as the universal cover<\/a> of a connected graph \\(X\\) has the fundamental group<\/a> \\(\\pi_1(X)\\) as its group of deck transformations<\/a>, its maximal abelian cover<\/b>, denoted \\(\\overline{X}\\), has the abelianization<\/a> of \\(\\pi_1(X)\\) as its group of deck transformations. It thus covers every other connected cover of \\(X\\) whose group of deck transformations is abelian. Since the abelianization of \\(\\pi_1(X)\\) is the first homology group<\/a> \\(H_1(X,\\mathbb{Z})\\), there is a close connection between the maximal abelian cover and homology theory. <\/p>\n

For a large class of graphs there is a systematic way to embed the maximal abelian cover in the vector space \\(H_1(X,\\mathbb{R})\\), the first homology group with real coefficients. We call this embedded copy of \\(\\overline{X}\\) a ‘topological crystal’. When \\(X\\) is the complete graph with 4 vertices, \\(H_1(X,\\mathbb{R})\\) is 3-dimensional Euclidean space, and the topological crystal is the Laves graph!<\/p>\n

In more detail, the construction of topological crystals proceeds as follows. Any graph \\(X\\) has a space \\(C_0(X,\\mathbb{R})\\) of 0-chains<\/b>, which are formal linear combinations of vertices, and a space \\(C_1(X,\\mathbb{R})\\) of 1-chains<\/b>, which are formal linear combinations of edges. There is a boundary operator<\/p>\n

$$ \\partial \\colon C_1(X,\\mathbb{R}) \\to C_0(X,\\mathbb{R}) , $$<\/p>\n

the linear operator sending any edge to the formal difference of its two endpoints. The kernel of this operator is the space of 1-cycles<\/b>, \\(Z_1(X,\\mathbb{R})\\). There is an inner product on the space of 1-chains such that edges form an orthonormal basis. This determines an orthogonal projection <\/p>\n

$$ \\pi \\colon C_1(X,\\mathbb{R}) \\to Z_1(X,\\mathbb{R}) . $$<\/p>\n

For a graph we have \\(H_1(X,\\mathbb{R}) \\cong Z_1(X,\\mathbb{R})\\). Thus, to build the topological crystal of \\(X\\), we only need to embed its maximal abelian cover \\(\\overline{X}\\) in \\(Z_1(X,\\mathbb{R})\\). We do this by embedding \\(\\overline{X}\\) in \\(C_1(X,\\mathbb{R})\\) and then projecting it down via \\(\\pi\\). <\/p>\n

To accomplish this, we need to fix a basepoint for \\(X\\). Each path \\(\\gamma\\) in \\(X\\) starting at this basepoint determines a 1-chain \\(c_\\gamma\\). It is easy to show that these 1-chains correspond to the vertices of \\(\\overline{X}\\). Furthermore, the graph \\(\\overline{X}\\) has an edge from \\(c_\\gamma\\) to \\(c_{\\gamma’}\\) whenever the path \\(\\gamma’\\) is obtained by adding an extra edge to \\(\\gamma\\). We can think of this edge as a straight line segment from \\(c_\\gamma\\) to \\(c_{\\gamma’}\\).<\/p>\n

The hard part is checking that the projection \\(\\pi\\) maps this copy of \\(\\overline{X}\\) into \\(Z_1(X,\\mathbb{R})\\) in a one-to-one manner. This happens precisely when the graph \\(X\\) has no bridges<\/a><\/b> that is, edges whose removal would disconnect \\(X\\). The resulting copy of \\(\\overline{X}\\) embedded in \\(Z_1(X,\\mathbb{R})\\) is called a topological crystal<\/b>. <\/p>\n

For details, see:<\/p>\n

• John Baez, Topological crystals<\/a>.<\/p>\n

This paper arose from a long discussion here:<\/p>\n

• John Baez, Diamonds and triamonds<\/a>, Azimuth<\/i>, April 11, 2016.<\/p>\n

The image of the unit cell of sphalerite was created by Benjah-bmm27<\/a> and put into the public domain on Wikicommons<\/a>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This picture by Greg Egan<\/a> shows the Laves graph<\/a>, a structure discovered by the crystallographer Fritz Laves<\/a> in 1932. It is also called the ‘\\(\\mathrm{K}_4\\) crystal’, since is an embedding of the maximal abelian cover of the complete graph on 4 vertices<\/a> in 3-dimensional Euclidean space. It is also called the ‘triamond’, since it is a theoretically possible — but never yet seen — crystal structure for carbon. <\/p>\n

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