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Hi all, I was reviewing some old material on the representation of orientable surfaces
in terms of disks and bands , in page 2 of:
http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf
Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into
an upper part and a lower part. Consider pairs in the upper part ; each represents a 2-torus with a disk removed. Consider too, individual handles in the lower part; each attached handle is basically a disk D^2 with a smaller disk d^2 removed. Then:
1) The "upper part" contributes one boundary component as a whole ; every pair of handles contributes 2 to the genus, i.e., every time we attach a pair of handles as in the upper part, the genus increases by 2 , since we can add a pair of essential , i.e., non-separating loops l_1, l_2 , each going exactly once around each of the respective handles (and, of course, any other loop about these handles would be homologous to either l_1 or to l_2).
2) Attaching a single handle as in the lower part increases both the number of boundary components by 1 ( the inner loop within the handle, representing a small disk d^2 removed from
D^2 ), and also increases the overall genus by 1, since we now have a new essential loop for each handle, this loop is the one that winds around the inner hole.
Is this correct?
in terms of disks and bands , in page 2 of:
http://www.maths.ed.ac.uk/~jcollins/Knot_Theory.pdf
Please tell me if I am correct here. Assume there is a horizontal line dividing the surface into
an upper part and a lower part. Consider pairs in the upper part ; each represents a 2-torus with a disk removed. Consider too, individual handles in the lower part; each attached handle is basically a disk D^2 with a smaller disk d^2 removed. Then:
1) The "upper part" contributes one boundary component as a whole ; every pair of handles contributes 2 to the genus, i.e., every time we attach a pair of handles as in the upper part, the genus increases by 2 , since we can add a pair of essential , i.e., non-separating loops l_1, l_2 , each going exactly once around each of the respective handles (and, of course, any other loop about these handles would be homologous to either l_1 or to l_2).
2) Attaching a single handle as in the lower part increases both the number of boundary components by 1 ( the inner loop within the handle, representing a small disk d^2 removed from
D^2 ), and also increases the overall genus by 1, since we now have a new essential loop for each handle, this loop is the one that winds around the inner hole.
Is this correct?