Polynomials and Homotopy of Virtual Knot Diagrams
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If a virtual knot diagram can be transformed to another virtual one by a finite sequence of crossing changes, Reidemeister moves and virtual moves then the two virtual knot diagrams are said to be \textit{homotopic}. There are infinitely many homotopy classes of virtual knot diagrams. We give necessary conditions by using polynomial invariants of virtual knots for two virtual knots to be homotopic. For a sequence $S$ of crossing changes, Reidemeister moves and virtual moves between two homotopic virtual knot diagrams,we give a lower bound for the number of crossing changes in $S$ by using the affine index polynomial introduced in \cite{Kau4}. In \cite{Je4}, the first author gave the $q$-polynomial of a virtual knot diagram to find Reidemeister moves of virtually isotopic virtual knot diagrams. We find how to apply Reidemeister moves by using the $q$-polynomial to show homotopy of two virtual knot diagrams.
Abstract
If a virtual knot diagram can be transformed to another virtual one by a finite sequence of crossing changes, Reidemeister moves and virtual moves then the two virtual knot diagrams are said to be \textit{homotopic}. There are infinitely many homotopy classes of virtual knot diagrams. We give necessary conditions by using polynomial invariants of virtual knots for two virtual knots to be homotopic. For a sequence $S$ of crossing changes, Reidemeister moves and virtual moves between two homotopic virtual knot diagrams,we give a lower bound for the number of crossing changes in $S$ by using the affine index polynomial introduced in \cite{Kau4}. In \cite{Je4}, the first author gave the $q$-polynomial of a virtual knot diagram to find Reidemeister moves of virtually isotopic virtual knot diagrams. We find how to apply Reidemeister moves by using the $q$-polynomial to show homotopy of two virtual knot diagrams.
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