On the Semi-threading of Knot Diagrams with Minimal Overpasses
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Given a knot diagram D, we construct a semi-threading circle of it which can be an axis of D as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles of minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we try to give another proof of the fact that, for every nontrivial knot K, the braid index b(K) of K is not less than the minimum number l(K) of overpasses of diagrams. Also, they are the same for a torus knot.
Abstract
Given a knot diagram D, we construct a semi-threading circle of it which can be an axis of D as a closed braid depending on knot diagrams. In particular, we consider semi-threading circles of minimal diagrams of a knot with respect to overpasses which give us some information related to the braid index. By this notion, we try to give another proof of the fact that, for every nontrivial knot K, the braid index b(K) of K is not less than the minimum number l(K) of overpasses of diagrams. Also, they are the same for a torus knot.
참고문헌 (7)
- C. Adams, "The Knot Book" (1994), W. H. Freeman & Co
- J. W. Alexander, "A lemma on systems of knotted curves", Proc. Nat. Acad. Sci. U. S. A, 9, pp.93~95 (1923)
- J. -W. Chung, "On the bridge number of knot diagrams with minimal crossings", MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 137, pp.632~3 (2004), CAMBRIDGE UNIV PRESS
- A. A. Markov, "Uber die freie Aquivalenz geschlossener Zopfe", Recueil Mathematique Moscou, 1, pp.73~78 (1935)
- H. R. Morton, "Threading knot diagrams", Math. Proc. Camb. Phil. Soc, 99, pp.247~260 (1986)
- K. Murasugi, "On the braid index of alternating links", Trans. Amer. Math. Soc, 326, pp.237~260 (1991)
- H. Schubert, "Uber eine numerische Knoteninvariante", Math. Z, 61, pp.245~288 (1954)