trace of horizontal to round chord diagrams

**Examples/classes:**

**Types**

**Related concepts:**

category: knot theory

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:

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

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)

Created on November 28, 2019 at 02:47:05. See the history of this page for a list of all contributions to it.