Parallelization of Welded Links

TL;DR

This paper introduces a parallelization method for welded link diagrams, demonstrating its well-defined nature, effects on diagram properties, and implications for quandle colorings and link decomposition.

Contribution

It presents a novel parallelization construction for welded links, analyzing its properties and effects on link invariants and structure.

Findings

Parallelization preserves welded link equivalence.

Parallel diagrams can be almost classical or admit checkerboard coloring.

Conditions for non-split parallel diagrams are identified.

Abstract

The notion of a welded link was introduced by Fenn, Rim\'anyi, and Rourke as an analogue of welded braids. A welded link is defined as an equivalence class of link diagrams that may contain virtual crossings, where the equivalence is generated by the classical and virtual Reidemeister moves together with the welded moves. In this paper, we introduce a parallelization construction for welded link diagrams and show that it is well defined: if two diagrams represent equivalent welded links, then the corresponding parallel diagrams obtained by our construction are also equivalent. When the two parallel strands are given parallel orientations, the resulting diagram admits a checkerboard coloring, whereas if they are assigned opposite orientations, the diagram is almost classical. Our construction further yields a decomposition in which one component is a copy of the original diagram and the other is a diagram representing a trivial welded link. We also investigate quandle colorings and the fundamental quandle of the parallel diagram, deriving a presentation from that of the original diagram. Finally, we examine conditions under which the parallel diagram is non-split.