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A polynomial invariant for knotoids

Yasuyuki Miyazawa

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Abstract

A polynomial invariant for multi-knotoids in S^2 is given by an elementary and combinatorial method. It is shown that the invariant is an extension of “a” HOMFLY polynomial for multi-knotoids and that there exist infinitely many non-trivial knotoids with trivial HOMFLY polynomial. Furthermore, formulas between the polynomials for a given multi-knotoid, its mirror and reverse images are given. By using the formulas, it is revealed that each knotoid with less than four crossings is invertible.

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What this paper is about

A polynomial invariant for multi-knotoids in S^2 is given by an elementary and combinatorial method. It is shown that the invariant is an extension of “a” HOMFLY polynomial for multi-knotoids and that there exist infinitely many non-trivial knotoids with trivial HOMFLY polynomial. Furthermore, formulas between the polynomials for a given multi-knotoid, its mirror and reverse images are given. By using the formulas, it is revealed that each knotoid with less than four crossings is invertible.

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Available abstract

A polynomial invariant for multi-knotoids in S^2 is given by an elementary and combinatorial method. It is shown that the invariant is an extension of “a” HOMFLY polynomial for multi-knotoids and that there exist infinitely many non-trivial knotoids with trivial HOMFLY polynomial. Furthermore, formulas between the polynomials for a given multi-knotoid, its mirror and reverse images are given. By using the formulas, it is revealed that each knotoid with less than four crossings is invertible.

Key concepts: Mathematics, HOMFLY polynomial, Invariant (physics), Invertible matrix, Bracket polynomial, Invariant polynomial, Extension (predicate logic), Matrix polynomial

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