Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0608172

3-coloring and other elementary invariants of knots

Jozef H. Przytycki
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We relate Fox 3-colorings to Jones and Kauffman polynomials of links and we use this connection to sketch the method of approximating the unknotting number of a knot. We discuss some elementary open problems in knot theory and show that Fox colorings can be useful in trying to solve them. Finally we demonstrate how analysis of Fox colorings can lead us to understand homology and the fundamental group of branched coverings along links.

Explore related subjects

math.GTmath.HO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jozef H. Przytycki. 2006-08-07. 3-coloring and other elementary invariants of knots. https://arxiv.org/abs/math/0608172

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries

In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation $ρ\colonπ_1(M)\to G$ into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.

math.GT↗
arXiv2026

Khovanov homology and refined bounds for Gordian distances

From Khovanov homology, we extract a new lower bound for the Gordian distance of knots, which combines and strengthens the previously existing bounds coming from Rasmussen invariants and from torsion invariants. We also improve the bounds for the proper rational Gordian distance.

math.GT↗
arXiv2026

From arcs to curves: quadratic growth of 1-systems

We show that a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ has at most $2016|χ|^2+338|χ|$ curves. Up to multiplicative constants, this resolves a thirty-year old problem (see Problem 2.12(b) from the K3 Problem List). Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of tulips, flowers, and stem systems in order to account for how certain polygons built from pairs of curves in the collection distribute area over the surface.

math.GT↗