Winding number and Cutting number of Harmonic cycle

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.