Intrinsic Properties of Spatial Graphs - Denison University

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$Math 232 syllabus - Denison University$

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Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and KnotsRather than asking how complicated the links and knots in aspatial graph must be, we could instead ask how many theremust be.Conway and Gordon found an embedding of K 6 with exactlyone link, and an embedding of K 7 with exactly one (trefoil) knot.But larger graphs will be forced to contain many links and knots.Theorem (Blain*, Bowlin*, Foisy, Hendricks*, LaCombe*, 2007)For n > 8, every embedding of K n contains at leastknotted Hamiltonian cycles.(n − 1)!7!Blake MellorIntrinsic Properties of Spatial Graphs
Introduction Linking and Knotting Other Intrinsic Properties ConclusionCounting Links and Knots in Complete Partite GraphsFleming and M (2009) gave upper and lower bounds for theminimum number of links in complete partite graphs on 7 or 8vertices.Abrams* and M (2010) extended this to counting links incomplete partite graphs on 9 vertices and knots in completepartite graphs on 8 vertices.In most cases, the bounds are not sharp, and there isconsiderable room for improvement. The table below gives afew results.graph K 7 K 3,3,1,1 K 8 K 4,4,1 K 9mnl 21 25 217 ≤ mnl ≤ 305 74 ≤ 3987mnk 1 1 15 ≤ mnk ≤ 29 ?? ??Blake MellorIntrinsic Properties of Spatial Graphs
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