Involution of the 3 and 4-holed torus and its effects on some knots and links

Question

I've been working with the intersection between the topology of knots and surfaces and I have some specific questions about the involution of torus with multiple holes and the effect on the knots written on then.

This is what I call the involution of a simple torus, by turning it inside-out through a hole (if you know a more precise term, please share it). So, if I turn inside-out a torus wich inscribes a 4,3 torus knot, it becomes a torus that presents a 3,4 torus knot. Right? (It seems logical to me but I don't have the demonstration for this and would apreciate if anyone could help me with it)

But the real deal is: Can anyone tell - and if possible prove it by showing the procedure through drawing - what results from the involution made through a hole in:

1. a triple torus which inscribes a borromean link

2. a 4-holed torus which inscribes the 8_18 knot

Thank you very much in advance, any help in this matter is welcome.

Answer 1

The torus involution that you linked to is taking place inside of $\Bbb{R}^3$ (or $S^3$). Therefore, a knot on the initial torus is isotopic to the image of the knot after the involution of the torus. Therefore, the $(4,3)$ and $(3,4)$ torus knots are actually the same knot. Here is a graphic showing this more explicitly for the $(2,3)$ and $(3,2)$ torus knot (taken from Colin Adams' The Knot Book).

In your more general cases, as long as your surface involutions are inducing isotopies of the knots contained on those surfaces, then the resulting knots will be the same as the initial knots. Perhaps their diagrams will look a bit different, but it will be the same knot or link.

If the involutions of the higher genus surfaces are not inducing isotopies on their embedded knots, then we'd need to know exactly what the involution map is before we can say what it does to an embedded knot.

Answer 2

I think I found out a answer to my own question that I'd like to share with you all. Following the tip given by Adam Lowrance, I went looking for the triple torus map, wich I found in the work of Hilbert and Cohn-Vossen: HILBERT, David; COHN-VOSSEN, Stephan. Geometry and the imagination. Providence, R.I.: AMS Chelsea Pub, 2nd ed., 1999

The first three figures above show how the borromean rings are embedded on the triple torus map, a 2-dimensional dodecahedron. The next three images below show the eversion (seems to be the correct term to use, rather than involution) of the triple torus using the same borromean rings pattern embedded map, by changing the way it closes to form the three-dimensional triple torus.

I'm not sure if it's the only way to perform the triple torus eversion or either if it's equivalent to the procedure of puncturing a hole on the surface. Maybe someone could add some knowledge about that. It also leaves out the problem of the different embeddings that emerge when we exchange the outside edge with one of the holes (what we can see if we consider the tetrahedral structure of the 3-D triple torus), but I might also open another topic about that.