2004/11/30 by Hee Jung Kim · 24 citations
Chemistry · Mathematics · #Alexander polynomial #Chemistry #Combinatorics #Diffeomorphism #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Knot (papermaking) #Knot theory #Materials science #Mathematical Dynamics and Fractals #Mathematics #Physics #Pure mathematics #Quantum mechanics #Sigma #Spinning #Surface (topology) #Twist #Type (biology) #math.GT #math.SG #msc:14J80 #msc:57R57 #msc:57R95
paper · pdf · doi:10.2140/gt.2006.10.27
published in Geometry & Topology 10(1), 27-56 (Mathematical Sciences Publishers) · This is the version published by Geometry & Topology on 25 February 2006
openalex publication_date 2006/02/25 · arxiv created 2009/03/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In this paper, given a knot K , for any integer m we construct a new surface K .m/ from a smoothly embedded surface in a smooth 4-manifold X by performing a surgery on . This surgery is based on a modification of the 'rim surgery' which was introduced by Fintushel and Stern, by doing additional twist spinning. We investigate the diffeomorphism type and the homeomorphism type of .X; / after the surgery. One of the main results is that for certain pairs .X; /, the smooth type of K .m/ can be easily distinguished by the Alexander polynomial of the knot K and the homeomorphism type depends on the number of twist and the knot. In particular, we get new examples of knotted surfaces in P 2 , not isotopic to complex curves, but which are topologically unknotted.