Document Zbl 0724.57006 - zbMATH Open
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A new numerical invariant of knots induced from their regular diagrams. (English) Zbl 0724.57006
The author introduces a numerical invariant of knots, O(K), from their diagrams. For a knot K, K is defined to have an n-trivial diagram if there exist a diagram \(\tilde K\) of K and a family of mutually disjoint nonempty sets \(A_ 1,...,A_ n\) of crossings in \(\tilde K\) such that for every nonempty subfamily \(A_{j_ 1},...,A_{j_ i}\) the diagram obtained from \(\tilde K\) by changing all the crossings of \(A_{j_ 1}\cup...\cup A_{j_ i}\) is a trivial knot diagram. If a knot K has an n-trivial diagram and no m-trivial diagrams for \(m>n\), O(K) is defined to be n. If a knot K has an n-trivial diagram for any natural number n, O(K) is defined to be \(\infty\), e.g. \(O(U)=\infty\) for a trivial knot U. For every natural number n, the author gives an example of an n-trivial tangle, and furthermore gives a necessary condition for knots to have an n-trivial tangle in terms of Conway polynomials. The consecutive work of the author and Y. Ogushi can be found in “On the triviality index of knots” [Tokyo J. Math. 13, No.1, 179-190 (1990; Zbl 0714.57003)].
MSC:
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
References:
| [1] | Burde, G.; Zieschang, H., Knots, de Gruyter studies in Mathematics, 5 (1985), de Gruyter: de Gruyter Berlin · Zbl 0568.57001 |
| [2] | Conway, J. H., An enumeration of knots and links and some of their related properties, (Leech, J., Computation Problems in Abstract Algebra (1967), Pergamon Press: Pergamon Press New York), 329-358, Proceedings Conference Oxford · Zbl 0202.54703 |
| [3] | Kinoshita, S.; Terasaka, H., On unions of knots, Osaka Math. J., 9, 131-153 (1957) · Zbl 0080.17001 |
| [4] | Murasugi, K., Jones polynomials and classical conjectures in knot theory, Topology, 187-194 (1987) · Zbl 0628.57004 |
| [5] | Rolfsen, D., Knots and Links (1976), Publish or Perish: Publish or Perish Berkeley, CA · Zbl 0339.55004 |
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