Braiding link cobordisms and non-ribbon surfaces

Hughes, Mark C

doi:10.2140/agt.2015.15.3707

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Braiding link cobordisms and non-ribbon surfaces

Mark C Hughes

Algebraic & Geometric Topology 15 (2015) 3707–3729

DOI: 10.2140/agt.2015.15.3707

Abstract

We define the notion of a braided link cobordism in $S^{3} \times [0, 1]$ , which generalizes Viro’s closed surface braids in $ℝ^{4}$ . We prove that any properly embedded oriented surface $W \subset S^{3} \times [0, 1]$ is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when $\partial W$ already consists of closed braids. These surfaces are closely related to another notion of surface braiding in $D^{2} \times D^{2}$ , called braided surfaces with caps, which are a generalization of Rudolph’s braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in $4$ –space, as well as constructing singular fibrations on smooth $4$ –manifolds from a given handle decomposition.

Keywords

braids, links, knot cobordisms

Mathematical Subject Classification 2010

Primary: 57M12

Secondary: 57M25, 57R52

References

Publication

Received: 2 March 2015

Revised: 31 March 2015

Accepted: 12 April 2015

Published: 12 January 2016

Authors

Mark C Hughes
Department of Mathematics Brigham Young University 312 TMCB Provo, UT 84602 USA
http://math.byu.edu/~hughes/

Volume 15, issue 6 (2015)