Geometries are defined by their objects of study. Euclidean geometry is naturally preoccupied wit... more Geometries are defined by their objects of study. Euclidean geometry is naturally preoccupied with distance and angle. Otherwise how could we distinguish between an acute and an obtuse triangle? In the discipline of topology a straight line is as good as a curved one. Projective geometry falls between these two extremes. We have already met projections; projective geometry studies properties unchanged by these.
We know very little about Euclid, who wrote theElements of Geometryone of the most famous and inf... more We know very little about Euclid, who wrote theElements of Geometryone of the most famous and influential books of mathematics ever. Most of what we know is due to Proclus (AD 410-485). He wrote, “This man lived at the time of the first Ptolemy, aGreek King.For Archimedes, who came immediately after the first, makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry.”
If we think of lines, planes and general affine subspaces as sets of points satisfying a linear e... more If we think of lines, planes and general affine subspaces as sets of points satisfying a linear equation then circles and spheres are examples of sets of points which satisfy a quadratic equation. The solutions to a quadratic equation in the plane are calledconic sectionsor conics for short. These were known to the ancient Greeks and were given this name because they can be thought of as the intersection of a plane with a circular cone. This is the definition we shall start with and we shall end with their definition in terms of a focus and directrix. The latter, introduces us to all sorts of pretty properties of a conic. The focus, as the name suggests, involves rays of light reflected by the conic curve with important practical consequences.
The use of registered names, trademarks etc. in this publication does not imply, even in the abse... more The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.
The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and it... more The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and its Applications. Pp. viii + 128. (Gordon and Breach: London and New York, February 1970.) 95s; $11.40.
Algebraic Topology. By C. R. F. Maunder. (The New University Mathematics Series.). Pp. ix+375. (V... more Algebraic Topology. By C. R. F. Maunder. (The New University Mathematics Series.). Pp. ix+375. (Van Nostrand Reinhold: London and New York, October 1970.) 140s. Differentiable Manifolds: An Introduction. By F. Brickell R. S. Clark. (The New University Mathematics Series.) Pp. xi+289. (Van Nostrand Reinhold: London and New York, October 1970.) 110s.
The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and it... more The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and its Applications. Pp. viii + 128. (Gordon and Breach: London and New York, February 1970.) 95s; $11.40.
The present paper gives a quick survey of virtual and classical knot theory and presents a list o... more The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.
A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation genera... more A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.
In this paper we give the results of a computer search for biracks of small size and we give vari... more In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles.
Geometries are defined by their objects of study. Euclidean geometry is naturally preoccupied wit... more Geometries are defined by their objects of study. Euclidean geometry is naturally preoccupied with distance and angle. Otherwise how could we distinguish between an acute and an obtuse triangle? In the discipline of topology a straight line is as good as a curved one. Projective geometry falls between these two extremes. We have already met projections; projective geometry studies properties unchanged by these.
We know very little about Euclid, who wrote theElements of Geometryone of the most famous and inf... more We know very little about Euclid, who wrote theElements of Geometryone of the most famous and influential books of mathematics ever. Most of what we know is due to Proclus (AD 410-485). He wrote, “This man lived at the time of the first Ptolemy, aGreek King.For Archimedes, who came immediately after the first, makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry.”
If we think of lines, planes and general affine subspaces as sets of points satisfying a linear e... more If we think of lines, planes and general affine subspaces as sets of points satisfying a linear equation then circles and spheres are examples of sets of points which satisfy a quadratic equation. The solutions to a quadratic equation in the plane are calledconic sectionsor conics for short. These were known to the ancient Greeks and were given this name because they can be thought of as the intersection of a plane with a circular cone. This is the definition we shall start with and we shall end with their definition in terms of a focus and directrix. The latter, introduces us to all sorts of pretty properties of a conic. The focus, as the name suggests, involves rays of light reflected by the conic curve with important practical consequences.
The use of registered names, trademarks etc. in this publication does not imply, even in the abse... more The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.
The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and it... more The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and its Applications. Pp. viii + 128. (Gordon and Breach: London and New York, February 1970.) 95s; $11.40.
Algebraic Topology. By C. R. F. Maunder. (The New University Mathematics Series.). Pp. ix+375. (V... more Algebraic Topology. By C. R. F. Maunder. (The New University Mathematics Series.). Pp. ix+375. (Van Nostrand Reinhold: London and New York, October 1970.) 140s. Differentiable Manifolds: An Introduction. By F. Brickell R. S. Clark. (The New University Mathematics Series.) Pp. xi+289. (Van Nostrand Reinhold: London and New York, October 1970.) 110s.
The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and it... more The Topology of Classical Groups and Related TopicsBy S. Y. Husseini. Notes on Mathematics and its Applications. Pp. viii + 128. (Gordon and Breach: London and New York, February 1970.) 95s; $11.40.
The present paper gives a quick survey of virtual and classical knot theory and presents a list o... more The present paper gives a quick survey of virtual and classical knot theory and presents a list of unsolved problems about virtual knots and links. These are all problems in low-dimensional topology with a special emphasis on virtual knots. In particular, we touch new approaches to knot invariants such as biquandles and Khovanov homology theory. Connections to other geometrical and combinatorial aspects are also discussed.
A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation genera... more A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this paper we discuss colorings of virtual diagrams using an algebra, called a doodle switch, and define an invariant of virtual doodles. Besides usual colorings of diagrams, we also introduce doubled colorings.
In this paper we give the results of a computer search for biracks of small size and we give vari... more In this paper we give the results of a computer search for biracks of small size and we give various interpretations of these findings. The list includes biquandles, racks and quandles together with new invariants of welded knots and examples of welded knots which are shown to be non-trivial by the new invariants. These can be used to answer various questions concerning virtual and welded knots. As an application we reprove the result that the Burau map from braids to matrices is non injective and give an example of a non-trivial virtual (welded) knot which cannot be distinguished from the unknot by any linear biquandles.
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Papers by Roger Fenn