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Rational tangle distances on knots and links

Published online by Cambridge University Press:  01 May 2000

ISABEL K. DARCY  and
DE WITT SUMNERS
ISABEL K. DARCY
Affiliation:
Department of Mathematics, University of Texas, Dallas, TX 75083, U.S.A. e-mail: [email protected]
DE WITT SUMNERS
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, U.S.A. e-mail: [email protected]
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Abstract

In order to model biological reactions, distances between knots/links based on tangle replacement are defined. Given a tangle R, an R-move is defined as the replacement of the zero tangle in a link L with the tangle R. The R-distance between the link L and the link M is defined to be the minimum number of R-moves required to change L into M where the minimum is taken over all diagrams of the link L. A formula is given to determine when one 4-plat knot/link can be obtained from another 4-plat via one R-move when R is a rational tangle and not equal to the 1/n tangle. The formula can also be used to find all rational tangles R for which the R-distance between two given 4-plats is 1 except in the case R = 1/n tangle. These results are also generalized to the case when any rational tangle P is replaced with any rational tangle R, P not necessarily the zero tangle.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 128 , Issue 3 , May 2000 , pp. 497 - 510
Copyright
The Cambridge Philosophical Society 2000

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