Future Projects
For my research prospectus see my Research Page. Here is short, more informal, treatment of some ideas and things of interest, it is by no means exhaustive. If anyone is interested in working on these please contact me!
For my research prospectus see my Research Page. Here is short, more informal, treatment of some ideas and things of interest, it is by no means exhaustive. If anyone is interested in working on these please contact me!
Research Idea 1
Research Idea 1
Brief Background: Two knots K and K’ are equivalent if there is a deformation between them that respects their embeddings. Reidemeister's Theorem says two knot diagrams represent equivalent knots if and only if the diagrams are related by a sequence of Reidemeister moves. The Reidemeister Graph of a knot is the collection of all isotopy classes of diagrams of a knot, two diagrams are connected by an edge if and only if their corresponding diagrams are transformed into each other by a single Reidemeister move. In a similar fashion, Schultens defined the width complex. This complex considered the various positions of a knot with respect to a height function.
Brief Background: Two knots K and K’ are equivalent if there is a deformation between them that respects their embeddings. Reidemeister's Theorem says two knot diagrams represent equivalent knots if and only if the diagrams are related by a sequence of Reidemeister moves. The Reidemeister Graph of a knot is the collection of all isotopy classes of diagrams of a knot, two diagrams are connected by an edge if and only if their corresponding diagrams are transformed into each other by a single Reidemeister move. In a similar fashion, Schultens defined the width complex. This complex considered the various positions of a knot with respect to a height function.
Idea: Inspired by these constructions, we could consider the Plat Graph. The Plat Graph of a knot K, is the graph whose vertices are double coset classes of plat presentations for K. We define an edge between vertices in the Plat Graph if they are related by a single stabilization or destabilization. By Markov’s theorem for Plats, we know that the Plat Graph is connected. Also in contrast to the Reidemeister Graph, this graph is hyperbolic! By results of Johnson and Tomova there are n-bridge plats that are arbitrarily far apart (they require a lot of stabilizing!).
Idea: Inspired by these constructions, we could consider the Plat Graph. The Plat Graph of a knot K, is the graph whose vertices are double coset classes of plat presentations for K. We define an edge between vertices in the Plat Graph if they are related by a single stabilization or destabilization. By Markov’s theorem for Plats, we know that the Plat Graph is connected. Also in contrast to the Reidemeister Graph, this graph is hyperbolic! By results of Johnson and Tomova there are n-bridge plats that are arbitrarily far apart (they require a lot of stabilizing!).
Research Idea 2
Research Idea 2
Brief Background: Using the algebra coming from a plat presentation of a link, Birman in “On the equivalence of Heegaard splittings of closed, orientable 3-manifolds” has given a way to demonstrate that two plat presentations are in distinct Hilden double coset classes. This was heavily utilized to demonstrate Heegaard splittings of 3-manifolds that are not isotopic.
Brief Background: Using the algebra coming from a plat presentation of a link, Birman in “On the equivalence of Heegaard splittings of closed, orientable 3-manifolds” has given a way to demonstrate that two plat presentations are in distinct Hilden double coset classes. This was heavily utilized to demonstrate Heegaard splittings of 3-manifolds that are not isotopic.
Idea: The problem with her method is that it is not a very strong invariant. The problem was that, Birman used a homological method for finding invariants. In the end this method amounted to a finding a representation of the link into the symmetric group and making arguments about the corresponding representations that come out of this. However, the representation of a braid word into the symmetric group loses a lot of information. A natural generalization of this approach is to use the Burau representation of the braid groups. This would potentially give a better invariant.
Idea: The problem with her method is that it is not a very strong invariant. The problem was that, Birman used a homological method for finding invariants. In the end this method amounted to a finding a representation of the link into the symmetric group and making arguments about the corresponding representations that come out of this. However, the representation of a braid word into the symmetric group loses a lot of information. A natural generalization of this approach is to use the Burau representation of the braid groups. This would potentially give a better invariant.
Also, something to note is that in both Birmans example (See figure #2 above) and Montesinos’ the non-isotopic plats differ by a flype move! Do all distinct isotopy classes have a flype move between them?
Also, something to note is that in both Birmans example (See figure #2 above) and Montesinos’ the non-isotopic plats differ by a flype move! Do all distinct isotopy classes have a flype move between them?
Research Idea 3
Research Idea 3
Idea: Given a braid, we can view it as a map from the punctured disk to itself. From this, we can see what the braid does to a special collection of fixed curves. (See figure #3 above.) The image of these curves can then be represented as words in the generators for the fundamental group of the punctured disk. If the curves are chosen properly, the resulting words will be invariant under plat isotopy. Moreover, adding these words results in a polynomial! This could be used to distinguish between different plat isotopy (and bridge isotopy) classes. It would be interesting to see how this polynomial relates to other knot polynomials.
Idea: Given a braid, we can view it as a map from the punctured disk to itself. From this, we can see what the braid does to a special collection of fixed curves. (See figure #3 above.) The image of these curves can then be represented as words in the generators for the fundamental group of the punctured disk. If the curves are chosen properly, the resulting words will be invariant under plat isotopy. Moreover, adding these words results in a polynomial! This could be used to distinguish between different plat isotopy (and bridge isotopy) classes. It would be interesting to see how this polynomial relates to other knot polynomials.