THE MAPPING CLASS GROUP OF (S3, K), WHERE K IS A TWO-BRIDGE KNOT OR LINK

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Let K be a two-bridge knot or link in S3. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S3. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.