Intrinsic Properties of Spatial Graphs - Denison University

More documents

Recommendations

Info

$Math 232 syllabus - Denison University$

$Math 102 syllabus - Personal Pages - Denison University$

Introduction Linking and Knotting Other Intrinsic Properties ConclusionColoring matricesSystem of linear equations: variable for each arc; equation foreach crossing and vertex.⎡ ⎤⎡⎤ x−1 2 −1 0 0 0 0 0 0 1 ⎡ ⎤0 −1 2 −1 0 0 0 0 0x 200 0 −1 0 2 −1 0 0 0x 300 −1 0 2 −1 0 0 0 0x 400 0 0 0 0 −1 2 −1 0x 5=0⎢ 0 0 0 2 0 0 −1 −1 0x 60⎥⎣ 0 0 0 −1 0 0 0 2 −1⎦⎢x 7⎢0⎥⎥ ⎣⎣x ⎦ 0⎦1 0 0 0 −1 0 1 0 −1 80x 9Matrix has c + e columns and c + v rows, where c, e, v are thenumbers of crossings, edges, vertices. Colorings are non-trivialsolutions where at least two colors used.Blake MellorIntrinsic Properties of Spatial Graphs
Introduction Linking and Knotting Other Intrinsic Properties ConclusionDeterminant of a graphObservation: Any graph with e > v is intrinsically colorablewith at least e − v independent non-trivial colorings.Define the determinant of an embedded graph as the greatestcommon divisor of the determinants of all(c + v − 1) × (c + v − 1) minors.Pre-Theorem: An embedded graph has e − v + 1 independentnon-trivial colorings mod p if and only if p divides thedeterminant.We are currently working to extend this to define an Alexanderpolynomial for graphs. (Joint work with Terry Kong*, AlecLewald*, Vadim Pigrish*.)Blake MellorIntrinsic Properties of Spatial Graphs
Page 1 and 2: Introduction Linking and Knotting O
Page 39: Introduction Linking and Knotting O

Intrinsic Properties of Spatial Graphs - Denison University

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?