Periodic Reporting for period 1 - KNOTDYNAPP (Knots in dynamical systems with applications to electromagnetism and quantum systems)
Période du rapport: 2022-03-16 au 2024-03-15
A knot is a closed loop in 3-dimensional space that does not intersect itself. Knot Theory is a branch of mathematics that studies such objects up to continuous deformations. In this sense two knots are considered equivalent if one can be deformed into the other without any cutting. Knots and their mathematical description play an increasingly important role in biology, chemistry and physics.
Funded by the Marie Sklodowska-Curie Actions programme, the KNOTDYNAPP project studies the time evolution of knots in different dynamical systems. Of particular interest are solutions of differential equations that originate in electromagnetism or quantum mechanics. Such solutions correspond to mathematical functions that can contain knots in one way or another.
More concretely, an electromagnetic field is a mathematical function that satisfies Maxwell's equations, a set of differential equations. A closed field line forms a knot. Such an electromagnetic knot is not made up of any physical material and usually disappears almost instantaneously. However, for a small family of knots, the so-called torus knots, solutions to Maxwell's equations have been found that have a closed field line in the shape of a torus knot not only at one moment in time, but for all time. The knotted structure is "stable". As the field changes with time, the knot may move around in space, but it will always maintain its knot type, its topological shape. One objective of the project is to prove that such solutions exist not only for the family of torus knots, but for all knots, and to study the dynamics of the knots in these fields.
Similarly, a quantum system can be described by a "wavefunction", a time-dependent function that satisfies Schrödinger's equation. The zeros of this function can form knots, so-called quantum vortex knots. As the wavefunction changes with time, the knot may move around. Parts of the knot may also pass through each other, thereby changing the knot type. The time evolution of a knot is thus described by a surface in 4-dimensional space. In fact, it is known that for any surface in 4-dimensional space there exists a quantum wavefunction whose zeros are knots that undergo the time evolution described by the given surface. However, finding such solutions to Schrödinger's equation for a given surface is very challenging. The project aims to develop a construction that produces for any given surface a corresponding quantum wavefunction.
Given a physical system that contains a knot in one way or another we would hope that topological properties of the knot are somehow reflected in physical properties of the corresponding equations. Another objective of the project is to establish such a relation in the case where the physical system is the magnetic field that is induced by an electric current through a knotted wire. A knot is called fibered if the space around the knot can be filled with topologically equivalent surfaces that are attached to the knot. Not every knot is fibered. We will prove that a knot is fibered if and only if it can be placed in 3-dimensional space such that the magnetic field that is induced by an electric current through the knot vanishes nowhere. This would produce a new characterisation of the family of fibered knots in terms of physics.
Knots (and more generally topology) play an important role in modern science due to their (topological) stability. Many possible applications of the study of knots in physical systems have been suggested, such as topological quantum computing as well as new communication and information storage possibilities. Understanding which knot types can appear as stable solutions in certain physical system and describing the topological changes that can occur therefore is expected to lead to more discoveries in the research area, resulting in progress on applications of these theoretical results.
Funded by the Marie Sklodowska-Curie Actions programme, the KNOTDYNAPP project studies the time evolution of knots in different dynamical systems. Of particular interest are solutions of differential equations that originate in electromagnetism or quantum mechanics. Such solutions correspond to mathematical functions that can contain knots in one way or another.
More concretely, an electromagnetic field is a mathematical function that satisfies Maxwell's equations, a set of differential equations. A closed field line forms a knot. Such an electromagnetic knot is not made up of any physical material and usually disappears almost instantaneously. However, for a small family of knots, the so-called torus knots, solutions to Maxwell's equations have been found that have a closed field line in the shape of a torus knot not only at one moment in time, but for all time. The knotted structure is "stable". As the field changes with time, the knot may move around in space, but it will always maintain its knot type, its topological shape. One objective of the project is to prove that such solutions exist not only for the family of torus knots, but for all knots, and to study the dynamics of the knots in these fields.
Similarly, a quantum system can be described by a "wavefunction", a time-dependent function that satisfies Schrödinger's equation. The zeros of this function can form knots, so-called quantum vortex knots. As the wavefunction changes with time, the knot may move around. Parts of the knot may also pass through each other, thereby changing the knot type. The time evolution of a knot is thus described by a surface in 4-dimensional space. In fact, it is known that for any surface in 4-dimensional space there exists a quantum wavefunction whose zeros are knots that undergo the time evolution described by the given surface. However, finding such solutions to Schrödinger's equation for a given surface is very challenging. The project aims to develop a construction that produces for any given surface a corresponding quantum wavefunction.
Given a physical system that contains a knot in one way or another we would hope that topological properties of the knot are somehow reflected in physical properties of the corresponding equations. Another objective of the project is to establish such a relation in the case where the physical system is the magnetic field that is induced by an electric current through a knotted wire. A knot is called fibered if the space around the knot can be filled with topologically equivalent surfaces that are attached to the knot. Not every knot is fibered. We will prove that a knot is fibered if and only if it can be placed in 3-dimensional space such that the magnetic field that is induced by an electric current through the knot vanishes nowhere. This would produce a new characterisation of the family of fibered knots in terms of physics.
Knots (and more generally topology) play an important role in modern science due to their (topological) stability. Many possible applications of the study of knots in physical systems have been suggested, such as topological quantum computing as well as new communication and information storage possibilities. Understanding which knot types can appear as stable solutions in certain physical system and describing the topological changes that can occur therefore is expected to lead to more discoveries in the research area, resulting in progress on applications of these theoretical results.
We showed that every knot arises as a stable electromagnetic knot. We also developed a method that allows us to predict the motion of any such electromagnetic knot. We proved that even though the knots are stable for all time, they cannot be contained in any finite region indefinitely (article in preparation). We also extended our results on knots to toroidal surfaces ("donuts") (article with D. Peralta Salas submitted to IMRN).
We were able to prove the lower-dimensional analogue of the objective on quantum vortex knots. The paraxial wave equation is a Schrödinger equation with two spatial variables and time. We developed a construction, which allowed us to obtain solutions of this equation with zeros in the shape of any given knot in the knot table of up to 8 crossings (1 article with M. Hirasawa accepted for publication in Res. Math. Sci., 1 article in preparation). We expect that the method can be applied to any knot, but unfortunately it is unlikely to work in the 3+1-dimensional setting, where we have to construct surfaces instead of knots. The found solutions correspond to laser beams with "optical vortex knots". We also developed a construction for knotted defect lines in liquid crystal materials (article with Randall Kamien and Paul Severino in preparation). We also obtained results on analytic properties of knot parametrisations (article submitted to Rev. Math. Iberoam.)
We proved that non-vanishing induced magnetic fields indicate that the knot type of the wire is fibered. For the family of so-called P-fibered braid closures we also proved the converse (article in preparation). It is conjectured that this family is identical with the family of fibered knots, which would prove our conjecture. The set of P-fibered braids was originally defined in the context of singularity theory and our attempts to prove the conjecture has led to many insights and results on knots of singularities (2 articles published, one in Rev. Math. Iberoam. and one in Eur. J. Math., 1 article with R. N. Araújo dos Santos and E. L. Sanchez Quiceno submitted to Bull. Braz. Math. Soc., 1 article with Eder L. Sanchez Quiceno submitted to Eur. J. Math., 1 article submitted to Math. Res. Lett., 1 article accepted for publication in Algebr. Geom. Topol.).
In addition to the written articles, the research results have been presented at various conferences and research seminar talks.
We were able to prove the lower-dimensional analogue of the objective on quantum vortex knots. The paraxial wave equation is a Schrödinger equation with two spatial variables and time. We developed a construction, which allowed us to obtain solutions of this equation with zeros in the shape of any given knot in the knot table of up to 8 crossings (1 article with M. Hirasawa accepted for publication in Res. Math. Sci., 1 article in preparation). We expect that the method can be applied to any knot, but unfortunately it is unlikely to work in the 3+1-dimensional setting, where we have to construct surfaces instead of knots. The found solutions correspond to laser beams with "optical vortex knots". We also developed a construction for knotted defect lines in liquid crystal materials (article with Randall Kamien and Paul Severino in preparation). We also obtained results on analytic properties of knot parametrisations (article submitted to Rev. Math. Iberoam.)
We proved that non-vanishing induced magnetic fields indicate that the knot type of the wire is fibered. For the family of so-called P-fibered braid closures we also proved the converse (article in preparation). It is conjectured that this family is identical with the family of fibered knots, which would prove our conjecture. The set of P-fibered braids was originally defined in the context of singularity theory and our attempts to prove the conjecture has led to many insights and results on knots of singularities (2 articles published, one in Rev. Math. Iberoam. and one in Eur. J. Math., 1 article with R. N. Araújo dos Santos and E. L. Sanchez Quiceno submitted to Bull. Braz. Math. Soc., 1 article with Eder L. Sanchez Quiceno submitted to Eur. J. Math., 1 article submitted to Math. Res. Lett., 1 article accepted for publication in Algebr. Geom. Topol.).
In addition to the written articles, the research results have been presented at various conferences and research seminar talks.
The project solved some of the most important questions on knotted electromagnetic fields. The work establishes new and beautiful relationships between contact topology, complex plane curves and electromagnetic fields, which should lead to interesting new research directions in the future.
The results on optical vortex knots mark significant progress in the field of topological optics. Previously, only few knots have been constructed. Now any knot, no matter how complicated, should be accessible. Even though the technique does not generalise to the higher-dimensional setting (a construction of surfaces describing the time evolution of quantum vortex knots), this insight in itself will be important for future constructions of quantum knots.
We have established the desired relation between topological properties of knots (fiberedness or P-fiberedness) and magnetic fields (being nowhere vanishing). In fact, the work goes beyond the initial proposal by establishing a further connection between these research areas and singularity theory, which has already led to several new results and should produce further interesting research interactions in the future.
The results on optical vortex knots mark significant progress in the field of topological optics. Previously, only few knots have been constructed. Now any knot, no matter how complicated, should be accessible. Even though the technique does not generalise to the higher-dimensional setting (a construction of surfaces describing the time evolution of quantum vortex knots), this insight in itself will be important for future constructions of quantum knots.
We have established the desired relation between topological properties of knots (fiberedness or P-fiberedness) and magnetic fields (being nowhere vanishing). In fact, the work goes beyond the initial proposal by establishing a further connection between these research areas and singularity theory, which has already led to several new results and should produce further interesting research interactions in the future.