Concerning a bound problem in knot theory

Author:
L. B. Treybig

Journal:
Trans. Amer. Math. Soc. **158** (1971), 423-436

MSC:
Primary 55.20

DOI:
https://doi.org/10.1090/S0002-9947-1971-0278289-8

MathSciNet review:
0278289

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Abstract: In a recent paper Treybig shows that if two knot functions determine equivalent knots, then are the ends of a simple sequence of knot functions. In an effort to bound the length of in terms of and (1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots in regular position can ``essentially'' be embedded as part of the -skeleton of a triangulation of a tetrahedron, where (1) all 3 cells which are unions of elements of can be shelled and (2) the number of elements in is determined by . A number of ``counting'' lemmas are proved.

**[1]**R. H. Bing,*An alternative proof that 3-manifolds can be triangulated*, Ann. of Math. (2)**69**(1959), 37–65. MR**0100841**, https://doi.org/10.2307/1970092**[2]**Richard H. Crowell and Ralph H. Fox,*Introduction to knot theory*, Based upon lectures given at Haverford College under the Philips Lecture Program, Ginn and Co., Boston, Mass., 1963. MR**0146828****[3]**C. F. Gauss,*Werke*(8), Teubner, Leipzig, (1900), pp. 272, 282-286.**[4]**Martin Greendlinger,*Dehn’s algorithm for the word problem*, Comm. Pure Appl. Math.**13**(1960), 67–83. MR**0124381**, https://doi.org/10.1002/cpa.3160130108**[5]**Martin Greendlinger,*On Dehn’s algorithms for the conjugacy and word problems, with applications*, Comm. Pure Appl. Math.**13**(1960), 641–677. MR**0125020**, https://doi.org/10.1002/cpa.3160130406**[6]**Martin Grindlinger,*On the word problem and the conjugacy problem*, Izv. Akad. Nauk SSSR Ser. Mat.**29**(1965), 245–268 (Russian). MR**0174620****[7]**Julius Nagy,*Über ein topologisches Problem von Gauß*, Math. Z.**26**(1927), no. 1, 579–592 (German). MR**1544876**, https://doi.org/10.1007/BF01475475**[8]**L. P. Neuwirth,*Knot groups*, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR**0176462****[9]**D. E. Penney,*An algorithm for establishing isomorphism between tame prime knots in*, Dissertation, Tulane University, New Orleans, La., 1965.**[10]**D. E. Sanderson,*Isotopy in 3-manifolds. I. Isotopic deformations of 2-cells and 3-cells*, Proc. Amer. Math. Soc.**8**(1957), 912–922. MR**0090052**, https://doi.org/10.1090/S0002-9939-1957-0090052-8**[11]**D. E. Sanderson,*Isotopy in 3-manifolds. II. Fitting homeomorphisms by isotopy*, Duke Math. J.**26**(1959), 387–396. MR**0107231****[12]**P. G. Tait,*On knots*, Scientific Paper I, Cambridge Univ. Press, London, 1898.**[13]**L. B. Treybig,*Prime mappings*, Trans. Amer. Math. Soc.**130**(1968), 248–253. MR**0217790**, https://doi.org/10.1090/S0002-9947-1968-0217790-X**[14]**L. B. Treybig,*A characterization of the double point structure of the projection of a polygonal knot in regular position*, Trans. Amer. Math. Soc.**130**(1968), 223–247. MR**0217789**, https://doi.org/10.1090/S0002-9947-1968-0217789-3**[15]**L. B. Treybig,*An approach to the polygonal knot problem using projections and isotopies*, Trans. Amer. Math. Soc.**158**(1971), 409–421. MR**0279800**, https://doi.org/10.1090/S0002-9947-1971-0279800-3

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1971-0278289-8

Keywords:
Polyhedron,
polygonal knot,
piecewise linear homeomorphism,
simplicial isotopy,
free cell,
shelling order

Article copyright:
© Copyright 1971
American Mathematical Society