• Title, Summary, Keyword: Reidemeister classes

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REIDEMEISTER SETS OF ITERATES

  • Lee, Seoung Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.16 no.1
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    • pp.15-23
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    • 2003
  • In order to compute the Nielsen number N(f) of a self-map $f:X{\rightarrow}X$, some Reidemeister classes in the fundamental group ${\pi}_1(X)$ need to be distinguished. D. Ferrario has some algebraic results which allow distinguishing Reidemeister classes. In this paper we generalize these results to the Reidemeister sets of iterates.

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IRREDUCIBLE REIDEMEISTER ORBIT SETS

  • Lee, Seoung Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.721-734
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    • 2014
  • The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, much as the Reidemeister set does in Nielsen fixed point theory. Extending our work on Reidemeister orbit sets, we obtain algebraic results such as addition formulae for irreducible Reidemeister orbit sets. Similar formulae for Nielsen type irreducible essential orbit numbers are also proved for fibre preserving maps.

Polynomials and Homotopy of Virtual Knot Diagrams

  • Jeong, Myeong-Ju;Park, Chan-Young;Park, Maeng Sang
    • Kyungpook Mathematical Journal
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    • v.57 no.1
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    • pp.145-161
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    • 2017
  • 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 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 [13]. In [10], 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.

A NOTE ON NIELSEN TYPE NUMBERS

  • Lee, Seoung-Ho
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.263-271
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    • 2010
  • The Reidemeister orbit set plays a crucial role in the Nielsen type theory of periodic orbits, such as the Reidemeister set does in Nielsen fixed point theory. In this paper, using Heath and You's methods on Nielsen type numbers, we show that these numbers can be evaluated by the set of essential orbit classes under suitable hypotheses, and obtain some formulas in some special cases.

MINIMAL SETS OF PERIODS FOR MAPS ON THE KLEIN BOTTLE

  • Kim, Ju-Young;Kim, Sung-Sook;Zhao, Xuezhi
    • Journal of the Korean Mathematical Society
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    • v.45 no.3
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    • pp.883-902
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    • 2008
  • The main results concern with the self maps on the Klein bottle. We obtain the Reidemeister numbers and the Nielsen numbers for all self maps on the Klein bottle. In terms of the Nielsen numbers of their iterates, we totally determine the minimal sets of periods for all homotopy classes of self maps on the Klein bottle.

The Forbidden Number of a Knot

  • CRANS, ALISSA S.;MELLOR, BLAKE;GANZELL, SANDY
    • Kyungpook Mathematical Journal
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    • v.55 no.2
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    • pp.485-506
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    • 2015
  • Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the forbidden number. We relate the forbidden number to several known invariants, and calculate bounds for some classes of virtual knots.