Flat Knots and Invariants

Abstract

This thesis concerns flat knots and their properties. We study various invariants of flat knots, such as the crossing number, the u-polynomial, the flat arrow polynomial, the flat Jones-Krushkal polynomial, the based matrices, and the φ-invariant. We also examine the behavior of these invariants under connected sum and cabling. We give a matrix-based algorithm to calculate the flat Jones-Krushkal polynomial. We take a special interest in certain subclasses of flat knots, such as almost classical flat knots, checkerboard colorable flat knots, and slice flat knots. We explore how the invariants can be used to obstruct a flat knot from being almost classical, checkerboard colorable, or slice. We show that any minimal crossing diagram of a composite flat knot is a con- nected sum, and we introduce a skein formula for the constant term of the flat arrow polynomial. A companion project to this thesis is the interactive website, FlatKnotInfo. It provides a curated dataset of examples and invariants of flat knots. It also features a tool for searching flat knots and another tool that crossreferences flat knots with virtual knots. FlatKnotInfo was used to develop many of the results in this thesis, and we hope others find it useful for their research on flat knots. The Python code for calculating based matrices and flat Jones-Krushkal polynomials is included in an appendix.