# On a problem in effective knot theory

- Volume: 9, Issue: 4, page 299-306
- ISSN: 1120-6330

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topGalatolo, Stefano. "On a problem in effective knot theory." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 9.4 (1998): 299-306. <http://eudml.org/doc/252452>.

@article{Galatolo1998,

abstract = {The following problem is investigated: «Find an elementary function \( F (n) : \mathbf\{ Z \}\rightarrow \mathbf\{ Z\} \) such that if \( \Gamma \) is a knot diagram with \( n \) crossings and the corresponding knot is trivial, then there is a sequence of Reidemeister moves that proves triviality such that at each step we have less than \( F (n) \) crossings». The problem is shown to be equivalent to a problem posed by D. Welsh in [7] and solved by geometrical techniques (normal surfaces).},

author = {Galatolo, Stefano},

journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},

keywords = {Knots; Complexities; Normal surfaces; trivial knot; knot diagram; number of crossings; Reidemeister moves; polygonal knot; triangulation; normal surface; algorithm},

language = {eng},

month = {12},

number = {4},

pages = {299-306},

publisher = {Accademia Nazionale dei Lincei},

title = {On a problem in effective knot theory},

url = {http://eudml.org/doc/252452},

volume = {9},

year = {1998},

}

TY - JOUR

AU - Galatolo, Stefano

TI - On a problem in effective knot theory

JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

DA - 1998/12//

PB - Accademia Nazionale dei Lincei

VL - 9

IS - 4

SP - 299

EP - 306

AB - The following problem is investigated: «Find an elementary function \( F (n) : \mathbf{ Z }\rightarrow \mathbf{ Z} \) such that if \( \Gamma \) is a knot diagram with \( n \) crossings and the corresponding knot is trivial, then there is a sequence of Reidemeister moves that proves triviality such that at each step we have less than \( F (n) \) crossings». The problem is shown to be equivalent to a problem posed by D. Welsh in [7] and solved by geometrical techniques (normal surfaces).

LA - eng

KW - Knots; Complexities; Normal surfaces; trivial knot; knot diagram; number of crossings; Reidemeister moves; polygonal knot; triangulation; normal surface; algorithm

UR - http://eudml.org/doc/252452

ER -

## References

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- Fáry, I., On straight line representation of planar-graphs. Acta Sci. Math. (Szeged), 1948, 11. Zbl0030.17902MR26311
- Hemion, G., The Classification of Knots and 3-Dimensional Spaces. Oxford University Press, 1992. Zbl0771.57001MR1211184
- Hempel, J., 3-Manifolds. Princeton University Press and University of Tokyo Press, 1976. Zbl0345.57001MR415619
- Nabutowsky, A. - Weinberger, S., Algorithmic unsolvability of the triviality problem for multidimensional Knots. Comment. Math. Helv., 71, n.3, 1996, 426-434. Zbl0862.57017MR1418946DOI10.1007/BF02566428
- Jaco, W. - Oertel, U., An algorithm to decide if a manifold is a Haken manifold. Topology, vol. 23, No.2, 1984, 195-209. Zbl0545.57003MR744850DOI10.1016/0040-9383(84)90039-9
- Welsh, D. J. A., The complexity of Knots. Annals of Discrete Mathematics, 55, 1993, 159-172. Zbl0801.68086MR1217989DOI10.1016/S0167-5060(08)70385-6

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