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HAMILTONIAN CYCLES AND ROPE LENGTHS OF CONWAY ALGEBRAIC KNOTS.
- Source :
- Journal of Knot Theory & Its Ramifications; Jan2006, Vol. 15 Issue 1, p121-142, 22p, 15 Diagrams
- Publication Year :
- 2006
-
Abstract
- For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c<subscript>1</subscript>, c<subscript>2</subscript> such that for any knot or link K, c<subscript>1</subscript> · (Cr(K))<superscript>3/4</superscript> ≤ L(K) ≤ c<subscript>2</subscript> · (Cr(K))<superscript>3/2</superscript>. It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {K<subscript>n</subscript>} with the property that Cr(K<subscript>n</subscript>) → ∞ (as n → ∞) such that L(K<subscript>n</subscript>) = O(Cr(K<subscript>n</subscript>)<superscript>p</superscript>). However, it is still an open question whether there exists a family of knots {K<subscript>n</subscript>} with the property that Cr(K<subscript>n</subscript>) → ∞ (as n → ∞) such that L(K<subscript>n</subscript>) = O(Cr(K<subscript>n</subscript>)<superscript>p</superscript>) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {K<subscript>n</subscript>} with the property that Cr(K<subscript>n</subscript>) → ∞ (as n → ∞) such that L(K<subscript>n</subscript>) can grow no faster than linearly with respect to Cr(K<subscript>n</subscript>). [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02182165
- Volume :
- 15
- Issue :
- 1
- Database :
- Complementary Index
- Journal :
- Journal of Knot Theory & Its Ramifications
- Publication Type :
- Academic Journal
- Accession number :
- 19632351
- Full Text :
- https://doi.org/10.1142/S0218216506004348