Winding number and Cutting number of Harmonic cycle
Abstract
A harmonic cycle $\lambda$, also called a discrete harmonic form, is a solution of the Laplace's equation with the combinatorial Laplace operator obtained from the boundary operators of a chain complex. By the combinatorial Hodge theory, harmonic spaces are isomorphic to the homology groups with real coefficients. In particular, if a cell complex has a one dimensional reduced homology, it has a unique harmonic cycle up to scalar, which we call the \emph{standard harmonic cycle}. In this paper, we will present a formula for the standard harmonic cycle $\lambda$ of a cell complex based on a high-dimensional generalization of cycletrees. Moreover, by using duality, we will define the standard harmonic cocycle $\lambda^*$, and show intriguing combinatorial properties of $\lambda$ and $\lambda^*$ in relation to (dual) spanning trees, (dual) cycletrees, winding numbers $w(\cdot)$ and cutting numbers $c(\cdot)$ in high dimensions.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1812.04930
- arXiv:
- arXiv:1812.04930
- Bibcode:
- 2018arXiv181204930K
- Keywords:
-
- Mathematics - Combinatorics;
- Mathematics - Algebraic Topology;
- Mathematics - Numerical Analysis;
- 05C30;
- 05C50;
- 52C99;
- 55M25;
- 65F40
- E-Print:
- 27 pages, 6 figures