arXiv subjects

Andrew Gibson

Publications and source records attributed to Andrew Gibson.

6 recordsLinked to original sources

arXiv2010

Finite type invariants of nanowords and nanophrases

Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree one and degree two respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.

math.GT↗

arXiv2010

Factorization of homotopies of nanophrases

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a homotopy data triple is composite, we show that equivalence of nanophrases under the corresponding homotopy can be calculated just by using the homotopies given by its prime factors.

math.GT↗

arXiv2009

Coverings, composites and cables of virtual strings

A virtual string can be defined as an equivalence class of planar diagrams under certain kinds of diagrammatic moves. Virtual strings are related to virtual knots in that a simple operation on a virtual knot diagram produces a diagram for a virtual string. In this paper we consider three operations on a virtual string or virtual strings which produce another virtual string, namely covering, composition and cabling. In particular we study virtual strings unchanged by the covering operation. We also show how the based matrix of a composite virtual string is related to the based matrices of its components, correcting a result by Turaev. Finally we investigate what happens under cabling to some invariants defined by Turaev.

math.GT↗

arXiv2009

Homotopy invariants of Gauss words

By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, we show that there are an infinite number of equivalence classes of Gauss words under homotopy.

math.GT↗

arXiv2009

Homotopy invariants of Gauss phrases

Equivalence relations can be defined on Gauss phrases using combinatorial moves. In this paper we consider two closely related equivalence relations on Gauss phrases, homotopy and open homotopy. In particular, in each case, we define a new invariant and determine the values that it can attain.

math.GT↗

arXiv2008

On tabulating virtual strings

A virtual string can be defined as a closed curve on a surface modulo certain equivalence relations. Turaev defined several invariants of virtual strings which we use to produce a table of virtual strings up to 4 crossings. We discuss progress in extending the tabulation to 5 crossings. We also provide a counter-example to a statement of Kadokami.

math.GT↗