I'm not a topologist, which I suppose does not excuse my incorrect nomenclature. What I am talking about is solid objects, so if "ball" is the right word, then use it. Sorry for using the word "sphere" which would certainly be misleading, as would using "torus" instead of "donut" had I done that. When I talk about "cutting apart", I mean stretching a portion of the object into a thin strand and cutting that strand. It is understood that the cross section of the cut is simple closed curve, not a cut through a 3D hole (bubble) that might be inside the object. If this strand happens to be part of a topological handle, cutting the strand removes the handle, decreasing the object's genus by 1. "Join together" would be the opposite operation, pinching the object in two places and pulling out two strands which you then join at the ends, creating a handle as it were, and increasing the object's genus by 1. "punching a hole" is stretching part of the object into a sheet and punching a hole in the sheet, so it's adding a 2D hole in your nomenclature, increasing the genus by 1. "Filling a hole" would be the inverse operation, pinching off a 2D hole, decreasing the genus by 1. My observation is that "cutting apart" is a local operation. If you see only the small section of strand you are cutting, you don't know if you are cutting a handle, leaving the object connected, or cutting the object into two disconnected pieces. In the former case, a topologist would say the cut reduces the genus by 1. I was exploring the consequences of having a "cut apart" operation always reduce the genus by 1. Performing a "cut apart" operation in the obvious way on a donut turns the donut into a ball, a topologist would say this is removing a handle from the donut, which decreases its genus by 1 from 1 to 0. Now start with a ball, of genus 0. If you stretch it into a strand and cut the strand, the result is two balls. If you ignore the fact that you did not cut a handle, and simply agree that the "cut apart" operation always reduces the genus by 1, you would conclude that the two balls, considered as a single topological object, have genus -1. If you had instead cut the strand at n-1 places, you would end up with n balls, and your n-1 cuts would have decremented the genus of the original ball n-1 times, so that logically, the n balls considered as a single topological object, should have genus 1-n. Similarly, if you start with a sphere, flatten it into a sheet, and blow a bubble in it, so that it looks like a balloon, it remains genus 0. If you tie off the balloon, which is locally equivalent to pinching off a 2D-hole, you end up with a hollow ball, a ball with a "3D hole" or bubble in it. If pinching off the hole had eliminated a handle, a topologist would have agreed it reduced the genus by 1. If we ignore the topologist and agree that pinching off a hole always reduces genus by 1, we conclude that the genus of a hollow ball should logically be -1. If we blow another bubble in the hollow ball and pinch it off, just as we did before, we end up with a ball with two bubbles (ball-shaped cavities). Our second pinching off operation should again have reduced the genus by 1, so a ball with two ball-shaped cavities should logically have genus -2, and by induction, a ball with n ball-shaped cavities should have genus -n. Now above, we concluded that n balls, considered as a single object, have genus 1-n. Also, a ball with n ball-shaped cavities have genus -n. One would be tempted to generalize that if we start with a ball, and remove from its interior an object N of genus n, resulting in a ball with an internal N-shaped cavity, that this latter object would have genus n-1. For example, a donut has genus 1, so a ball with an internal donut-shaped cavity would have genus 0. We confirm that if we punch a hole from the exterior of this ball to its internal donut shaped cavity, its genus should be increased by 1, and indeed the resulting object is topologically equivalent to a donut of genus 1. If we accept that for any positive n, n balls considered as a single object have genus 1-n, the obvious extrapolation is that 0 balls, that is, the empty object, has genus 1. Likewise, by playing around with these operations, we can convince ourselves that if we have have two objects X and Y of genus x and y, respectively, that the operation of juxtaposition (or would aggregation be better?), that is, considering X and Y together as a single topological object, results in an object of genus x+y-1. For example, a ball has genus 0, an object consisting of 2 balls has genus 0+0-1 = -1, aggregating another ball gives genus -1+1-1 = -2 for an object consisting of 3 balls, and in general, genus 1-n for an aggregation of n balls, consistent with our earlier conclusion. An easy induction will show that any nonnegative number of donuts aggregated together have genus 1. Since any knot or link is no more than an aggregation of donuts (links homeomorphic to the circle), it follows that any knot or linkage has genus 1. On 9/2/2011 8:33 PM, Dan Asimov wrote:
David,
Part of the time you seem to be talking about solid objects, and part of the time, about surfaces.
I can't follow your descriptions. It is standard in topology that sphere means the surface, and ball means the filled surface.
Also, you refer to punching out, or filling, holes.
There are 1D and 2D holes. A 2D hole (like a closed curve, or a hole punched from a piece of paper by an office hole punch) can be filled by shrinking its boundary to a point.
A 3D hole is like the hollow inside a sphere (or inside any closed surface in 3-space), and this can be filled, well, in the obvious way by filling it with solid stuff.
Maybe you can help me out by clarifying what you meant.
Thanks,
Dan
Sometimes the brain has a mind of its own.