nLab trace of horizontal to round chord diagrams

Idea

In knot theory, a certain trace-like operation turns horizontal chord diagrams into round chord diagrams:

Given a horizontal chord diagram on $n$ strands and given any choice of cyclic permutation of $n$ elements, the trace is the round chord diagram obtained by gluing the ends of the strands according to the cyclic permutation, and retaining the chords in the evident way.

The following graphics shows basic examples of the trace operation for cyclic permutation of strands one step to the right:

Properties

Relation to 4T relations

The trace operation sends the 4T relations on horizontal chord diagrams to those for round chord diagrams:

References

The trace operation is implicit in

• Dror Bar-Natan, Vassiliev and Quantum Invariants of Braids, Geom. Topol. Monogr. 4 (2002) 143-160 (arxiv:q-alg/9607001)

It is made a little more explicit in

• Toshitake Kohno, around Figure 2 of: Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)