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Example 4.3: examples of gluing two cycles along an edge -- example 4.3 in arXiv 2312.16069

The following example has two pentagons glued on an edge, and this example has 4 exotic solutions, and it is the unique graph on 8 vertices with more than 2 exotic solutions.

i1 : needsPackage "Oscillators";
i2 : printingPrecision = 3;
i3 : PentPent = graph{{0,1},{1,2},{2,3},{3,4},{4,0},{0,5},{5,6},{6,7},{7,1}}

o3 = Graph{0 => {1, 4, 5}}
           1 => {0, 2, 7}
           2 => {1, 3}
           3 => {2, 4}
           4 => {0, 3}
           5 => {0, 6}
           6 => {5, 7}
           7 => {1, 6}

o3 : Graph
i4 : stablesolsPentPent = {
         {.648, .796, .908, .977, .215, -.908, -.605, .762, .605, .42, .215, -.977, -.42, .796},
         {1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0},
         {.648, .796, .908, .977, .215, -.908, -.605, -.762, -.605, -.42, -.215, .977, .42, -.796},
         {.648, -.605, -.908, .215, .977, .908, .796, .762, .796, -.42, -.977, .215, .42, .605},
         {.648, -.605, -.908, .215, .977, .908, .796, -.762, -.796, .42, .977, -.215, -.42, -.605}}

o4 = {{.648, .796, .908, .977, .215, -.908, -.605, .762, .605, .42, .215,
     ------------------------------------------------------------------------
     -.977, -.42, .796}, {1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0}, {.648,
     ------------------------------------------------------------------------
     .796, .908, .977, .215, -.908, -.605, -.762, -.605, -.42, -.215, .977,
     ------------------------------------------------------------------------
     .42, -.796}, {.648, -.605, -.908, .215, .977, .908, .796, .762, .796,
     ------------------------------------------------------------------------
     -.42, -.977, .215, .42, .605}, {.648, -.605, -.908, .215, .977, .908,
     ------------------------------------------------------------------------
     .796, -.762, -.796, .42, .977, -.215, -.42, -.605}}

o4 : List
i5 : stablesolsHexPent

o5 = stablesolsHexPent

o5 : Symbol

The computation of these (linearly) stable solutions takes perhaps a minute:

elapsedTime stablesolsPentPent = showExoticSolutions PentPent

The following example has a hexagon and pentago glued identified on a common edge, and this example has 9 vertices and 6 exotic solutions.

i6 : HexPent = graph{{0,1},{1,2},{2,3},{3,4},{4,0},{0,5},{5,6},{6,7},{7,8},{8,1}}

o6 = Graph{0 => {1, 4, 5}}
           1 => {0, 2, 8}
           2 => {1, 3}
           3 => {2, 4}
           4 => {0, 3}
           5 => {0, 6}
           6 => {5, 7}
           7 => {6, 8}
           8 => {1, 7}

o6 : Graph

The computation of exotic solutions for this example takes several minutes:

elapsedTime stablesolsHexPent = showExoticSolutions HexPent

See also


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:906:0.