### Extending Closed Plane Curves to Immersions of the Disk with n Handles

Keith D. Bailey
1975 Transactions of the American Mathematical Society
Let /: S -► E be a normal curve In the plane. The extensions of / to immersions of the disk with n handles (Tn) can be determined as follows. A word for / is constructed using the definitions of Blank and Marx and a combinatorial structure, called a ^-assemblage, is defined for such words. There is an immersion extending / to Tn iff the tangent winding number of / is 1 -2/t and / has a ^-assemblage. For each n, a canonical curve /" with a topologically unique extension to Tn is described (/q =
more » ... is described (/q = Jordan curve). Any extendible curve with the minimum number (In + 2 for n > 0) of self-intersections is equivalent to /". 1 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2 K. D. BAILEY immersions (presenting self-tangencies, multiple crossings and multiply covered arcs, etc.), the bounding properties of normal immersions are unaffected by sufficiently small deformations. When the domain dM of / is the circle S and N is the plane, E, the tangent winding number t(J) is the degree of the unit tangent vector. By the Whitney-Graust ein theorem, r(/) classifies / up to regular homotopy. If dM has d components and N is S2 with base point °°, then define t(/) to be the sum of the tangent winding numbers computed in E = S2 -{ °°}. The character of extensions F oí f has been found to depend on t(J), the genus n of M, the number d of components of dM, and, if F is properly interior, on a nonnegative integer u associated with the multiplicities of the branch points. Finally, letting b be the number of preimages of the spherical base point, Francis  found the constraint: