A Phenomenological Approach to Interactive Knot Diagrams.

Area of Science:

• Topology
• Computational Mathematics
• Computer Graphics

Background:

• Knot diagrams are fundamental visual tools in topology.
• Digital tools aid knot theory research and education, but interactive manipulation is lacking.
• Existing methods often rely on underlying 3D structures.

Purpose of the Study:

• To develop a method for real-time, interactive manipulation of knot diagrams.
• To enable direct operation on the geometry of 2D knot diagrams.
• To facilitate computation of knot invariants using interactive diagram manipulation.

Main Methods:

• Operating directly on the geometry of vector graphics knot diagrams.
• Developing an algorithm that bypasses the need for underlying 3D structures.
• Integrating real-time invariant computation with interactive diagram manipulation.

Main Results:

• A novel method for direct, real-time manipulation of knot diagrams is presented.
• The method allows interaction with the 2D geometry of knot diagrams.
• Knot invariants can be computed concurrently with diagram manipulation.

Conclusions:

• The developed method provides a unique tool for knot theory.
• This approach enhances the usability of digital knot diagrams for research and education.
• An implementation of the described method is available.