## Concerning a bound problem in knot theory

HTML articles powered by AMS MathViewer

- by L. B. Treybig PDF
- Trans. Amer. Math. Soc.
**158**(1971), 423-436 Request permission

## Abstract:

In a recent paper Treybig shows that if two knot functions $f,g$ determine equivalent knots, then $f,g$ are the ends of a simple sequence $x$ of knot functions. In an effort to bound the length of $x$ in terms of $f$ and $g$ (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 $K,L$ in regular position can “essentially” be embedded as part of the $1$-skeleton of a triangulation $T$ of a tetrahedron, where (1) all 3 cells which are unions of elements of $T$ can be shelled and (2) the number of elements in $T$ is determined by $K,L$. A number of “counting” lemmas are proved.## References

- R. H. Bing,
*An alternative proof that $3$-manifolds can be triangulated*, Ann. of Math. (2)**69**(1959), 37–65. MR**100841**, DOI 10.2307/1970092 - Richard H. Crowell and Ralph H. Fox,
*Introduction to knot theory*, Ginn and Company, Boston, Mass., 1963. Based upon lectures given at Haverford College under the Philips Lecture Program. MR**0146828**
C. F. Gauss, - Martin Greendlinger,
*Dehn’s algorithm for the word problem*, Comm. Pure Appl. Math.**13**(1960), 67–83. MR**124381**, DOI 10.1002/cpa.3160130108 - Martin Greendlinger,
*On Dehn’s algorithms for the conjugacy and word problems, with applications*, Comm. Pure Appl. Math.**13**(1960), 641–677. MR**125020**, DOI 10.1002/cpa.3160130406 - Martin Grindlinger,
*On the word problem and the conjugacy problem*, Izv. Akad. Nauk SSSR Ser. Mat.**29**(1965), 245–268 (Russian). MR**0174620** - Julius Nagy,
*Über ein topologisches Problem von Gauß*, Math. Z.**26**(1927), no. 1, 579–592 (German). MR**1544876**, DOI 10.1007/BF01475475 - L. P. Neuwirth,
*Knot groups*, Annals of Mathematics Studies, No. 56, Princeton University Press, Princeton, N.J., 1965. MR**0176462**
D. E. Penney, - 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**90052**, DOI 10.1090/S0002-9939-1957-0090052-8 - D. E. Sanderson,
*Isotopy in 3-manifolds. II. Fitting homeomorphisms by isotopy*, Duke Math. J.**26**(1959), 387–396. MR**107231**
P. G. Tait, - L. B. Treybig,
*Prime mappings*, Trans. Amer. Math. Soc.**130**(1968), 248–253. MR**217790**, DOI 10.1090/S0002-9947-1968-0217790-X - 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**217789**, DOI 10.1090/S0002-9947-1968-0217789-3 - L. B. Treybig,
*An approach to the polygonal knot problem using projections and isotopies*, Trans. Amer. Math. Soc.**158**(1971), 409–421. MR**279800**, DOI 10.1090/S0002-9947-1971-0279800-3

*Werke*(8), Teubner, Leipzig, (1900), pp. 272, 282-286.

*An algorithm for establishing isomorphism between tame prime knots in*${E^3}$, Dissertation, Tulane University, New Orleans, La., 1965.

*On knots*, Scientific Paper I, Cambridge Univ. Press, London, 1898.

## Additional Information

- © Copyright 1971 American Mathematical Society
- 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