The paper simplifies knot and link diagrams with triple-crossings.
problem Generating and classifying minimal triple-crossing knot and link diagrams.
method Systematic method to generate minimal triple-crossing projections, introducing new diagrammatic moves.
result Classification of knots and links with triple-crossing number up to five, derivation of minimal generating set of moves.
Triple-crossing number bound for knots and links, especially torus knots.
problem Finding bounds for triple-crossing numbers of knots and links.
method Using the genus of a knot or link, we derive bounds for the triple-crossing number.
result Triple-crossing number of torus knots and many other knots is at least twice their genus.
A triple crossing is a crossing in a projection of a knot or link that has three strands of the knot passing straight through it. A triple crossing projection is a projection such that all of the crossings are triple crossings. We prove that every knot and link has a triple crossing projection and then investigate c_3(…
Enumerates knots up to five crossings and describes moves between them.
problem Counting and classifying knots up to a specific number of crossings.
method Generated tables of minimal diagrams and derived moves between knots.
result Conjecture about a lower bound for the triple-crossing number based on Alexander polynomial.
Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Re…
Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and th…
The paper improves bounds on knot crossings and tabulates minimal diagrams.
problem Improving bounds on knot crossings and tabulating minimal diagrams.
method Analyzing triple-crossing and delta-crossing numbers, proving tangle existence, generating tables.
result Improved bounds on knot crossings and tabulated minimal diagrams for prime knots up to delta-crossing number 4.
Integrable dynamics explained via geometric maps and cluster algebras.
problem Integrable dynamics in projective geometry.
method Twisted triple crossing diagram maps and cluster integrable systems.
result Cross-ratio dynamics described by geometric R-matrices. Develops TCD maps to relate discrete differential geometry and cluster algebras.
problem Capturing constraints and dynamics in discrete differential geometry.
method Triple crossing diagram maps (TCD maps) and geometric operations.
result Establishes a hierarchy of cluster structures on TCD maps.
Unified framework for various geometric constructions.
problem Organizing diverse geometric constructions.
method Introducing TCD maps and defining local moves.
result Two distinct cluster structures on TCD maps.
Harmonic unit normal sections studied for Grassmannians induced by cross products.
problem Energy of maps assigning unit vectors to subspaces of Grassmannians.
method Analyzing cross products to induce harmonic sections into sphere bundles.
result All unit normal sections of Grassmannians associated with cross products are harmonic.