What mathematical descriptors are sought to distinguish non-equivalent knots?

Invariants

The core challenge inherent in knot theory is determining precisely when two seemingly different knot diagrams actually represent the identical mathematical knot, meaning one can be manipulated into the other without cutting the string. To solve this problem, mathematicians seek 'invariants.' An invariant is a mathematical descriptor or property that remains constant for all equivalent knots but yields a different value for knots that are truly non-equivalent. While early attempts included simple metrics like the crossing number—the minimum number of crossings required in any diagram of that knot—these were often insufficient. The development of more robust tools, such as polynomial invariants, provided much more powerful and reliable signatures for classifying knots based on these invariant properties.