Abstract This paper provides a convenient and practical method to compute the homology intersection pairing of branched double cover 4-ball. To projections links in 3-ball, surfaces 4-ball into boundary sphere, we associate sequence groups, called disoriented homology. We show that is isomorphic link or surface. define on first group surface this equal cover. These results generalize work Gordo...

The study of branched covers of the Riemann sphere has connections to many fields. We recall the classical relationship between branched covers and group theory via the Riemann existence theorem, which then leads to represention-theoretic formulas for Hurwitz numbers, counting the number of branched covers of prescribed types. We also review the Hurwitz spaces parametrizing branched covers as w...