From: lrudolph@panix.com (Lee Rudolph) Subject: Re: A curve bounds a surface Date: 19 Oct 2000 10:43:32 -0400 Newsgroups: sci.math,sci.math.research Summary: [missing] Sacha Blumen writes: >Note that the surface doesn't have to be orientable (eg Mobius band). What "the surface" do you mean to refer to? Certainly a (tame) simple closed curve in R^3 always bounds both orientable and non-orientable surfaces. "The surface" produced by Seifert's algorithm is always orientable, and I think that good usage requires "Seifert surface" always to refer to an orientable (preferably, an oriented) surface (with no closed components). >David G Radcliffe wrote: >> >> In sci.math Will Self wrote: >> : There is a theorem that any simple closed curve in R^3 is the >> : boundary of some orientable surface. Can anyone direct me to >> : a proof of this? >> >> This is called a Seifert surface. Proofs can be found in introductory >> knot theory texts, such as "The Knot Book" by Colin Adams, and also on >> the web at http://www.ai.sri.com/~goldwate/math/knots/ . Lee Rudolph ============================================================================== From: "W. Dale Hall" Subject: Re: A curve bounds a surface Date: Wed, 25 Oct 2000 03:55:27 -0700 Newsgroups: sci.math "Zdislav V. Kovarik" wrote: > > How simple should the curve be? > If there is such a theorem, I am curious about a surface whose > boundary is the trefoil knot. > > Cheers, ZVK(Slavek). > Sorry for the delay, but here's the trefoil knot (use constant-spaced font here), with the Seifert surface shown (one side is ., other side is o). The curve itself is the dashed line, and the over/under crossings are indicated by a break in the line. Note that the surface passes from front to back (or . to o) as you pass through one of these crossings, just as you would expect if you had a strip of paper of two different colors, and twisted it by 180 degrees. ------ ------ |......\ /oooooo| |.......\ooooooo| |....../ \oooooo| |.....< >ooooo| |......\ /oooooo| |.......\ooooooo| |....../ \oooooo| |.....< >ooooo| |......\ /oooooo| |.......\ooooooo| |....../ \oooooo| |...../ \ooooo| ----- ----- Dale.