61 research outputs found
Homotopy Classification of Generalized Phrases in Turaev's Theory of Words
In 2005 V. Turaev introduced the theory of topology of words and phrases.
Turaev defined an equivalence relation on generalized words and phrases which
is called homotopy. This is suggested by the Reidemeister moves in the knot
theory. Then Turaev gave the homotopy classification of generalized words with
less than or equal to five letters. In this paper we give the classification of
generalized phrases up to homotopy with less than or equal to three letters. To
do this we construct a new homotopy invariant for nanophrases over any
.Comment: 12 page
Homotopy classification of nanophrases with less than or equal to four letters
In this paper we give the stable classification of ordered, pointed, oriented
multi-component curves on surfaces with minimal crossing number less than or
equal to 2 such that any equivalent curve has no simply closed curves in its
components. To do this, we use the theory of words and phrases which was
introduced by V. Turaev. Indeed we give the homotopy classification of
nanophrases with less than or equal to 4 letters. It is an extension of the
classification of nanophrases of length 2 with less than or equal to 4 letters
which was given by the author in a previous paper. This is a corrected version
of Hokkaido University Preprint Series in Mathematics #921. I corrected the
subsection 5.3 and added proofs of propositions.Comment: 15 pages, 2 figures. This is a corrected version of Hokkaido
University Preprint Series in Mathematics #92
Khovanov homology and words
This paper is concerned with nanowords, a generalization of links, introduced
by Turaev. It is shown that the system of bigraded homology groups is an
invariant of nanowords by introducing a new notion. This paper gives two
examples which show the independence of this invariant from some of Turaev's
homotopy invariants.Comment: 39 pages; 2 figur
Framed Surfaces in the Euclidean Space
A framed surface is a smooth surface in the Euclidean space with a moving frame.The framed surfaces may have singularities. We treat smooth surfaces with singular points,that is, singular surfaces more directly. By using the moving frame, the basic invariants and curvatures of the framed surface are introduced. Then we show that the existence and uniqueness for the basic invariants of the framed surfaces. We give properties of framed surfaces and typical examples. Moreover, we construct framed surfaces as one-parameter families of Legendre curves along framed curves. We give a criteria for singularities of framed surfaces by using the curvature of Legendre curves and framed curves
Existence conditions of framed curves for smooth curves
A framed curve is a smooth curve in the Euclidean space with a moving frame. We call the smooth curve in the Euclidean space the framed base curve. In this paper, we give an existence condition of framed curves. Actually, we construct a framed curve such that the image of the framed base curve coincides with the image of a given smooth curve under a condition. As a consequence, polygons in the Euclidean plane can be realised as not only a smooth curve but also a framed base curve
On convexity of simple closed frontals
We study convexity of simple closed frontals in the Euclidean plane by using the curvature of Legendre curves. We show that for a Legendre curve, the simple closed frontal is convex if and only if the sign of both functions of the curvature of the Legendre curve does not change. We also give some examples of convex simple closed frontals
Evolutes and involutes of frontals in the Euclidean plane
We have already defined the evolutes and the involutes of fronts without inflection points. For regular curves or fronts, we can not define the evolutes at inflection points. On the other hand, the involutes can be defined at inflection points. In this case, the involute is not a front but a frontal at inflection points. We define evolutes of frontals under conditions. The definition is a generalisation of both evolutes of regular curves and of fronts. By using relationship between evolutes and involutes of frontals, we give an existence condition of the evolute with inflection points. We also give properties of evolutes and involutes of frontals
Simplified numerical form of universal finite type invariant of Gauss words
In the present paper, we study the finite type invariants of Gauss words. In
the Polyak algebra techniques, we reduce the determination of the group
structure to transformation of a matrix into its Smith normal form and we give
the simplified form of a universal finite type invariant by means of the
isomorphism of this transformation. The advantage of this process is that we
can implement it as a computer program. We obtain the universal finite type
invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we
give the complete classification of Gauss words of rank 4 and the partial
classification of Gauss words of rank 5 where the distinction of only one pair
remains.Comment: 12 pages, 3 table
Involutes of fronts in the Euclidean plane
For a regular plane curve, an involute of it is the trajectory described by the end of a stretched string unwinding from a point of the curve. Even for a regular curve, the involute always has a singularity. By using a moving frame along the front and the curvature of the Legendre immersion in the unit tangent bundle, we define an involute of the front in the Euclidean plane and give properties of it. We also consider a relationship between evolutes and involutes of fronts without inflection points. As a result, the evolutes and the involutes of fronts without inflection points are corresponding to the differential and the integral of the curvature of the Legendre immersion.Mathematics Subject Classification : 58K05; 53A04; 57R4
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