The linking number in knot theory is a topological invariant that quantifies the number of times two curves wind around each other. It is always an integer due to the discrete nature of crossings between the curves, which can only be counted as whole crossings. The discussion emphasizes the binary nature of crossings, asserting that curves are either crossing or not, leaving no room for fractional values. Various proofs exist that demonstrate this property, often involving algebraic topology concepts. Understanding the linking number's integer nature is fundamental in knot theory and its applications.