The symmetric group is a factor of the braid group (see e.g. Surjective Group Homomorphism From Braid Group Into Symmetric Group, Symmetric group, Braid Groups, and related groups). Consequently one can visualize an element in the symmetry group like a braid with no information regarding which strand is on top at a crossing (e.g. as done here: Symmetric Group $S_n$ and Artin Braid Group $B_n$: Differences in definition of composition? or here https://arxiv.org/abs/2303.02533 on p.19). It seems to me rather intuitive, convenient and fast (and more fun) to use "wire symbols" as opposed to the cycle notation (e.g. with wire symbols one need not name the elements being permuted).
Question: Is there a relatively detailed account of the symmetric group that uses fundamentally these wire symbols, say at the undergraduate (or below) level?
I am aware that the question has ambiguities; any answer to a reasonable interpretation of the question is welcome.
Here is a humble demonstration as to what I mean by a "wire symbol":
