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A 3,10 torus knot is a knot that loops around a 2-torus (or doughnut) three times with one continuously strand of string that winds through the center hole ten times. Is this 3,10 torus considered topologically equivalent to a 2-torus?

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    $\begingroup$ In the phrase "3,10 torus knot", the "3,10" modifies the noun phrase "torus knot". There is no such thing as a "3,10 torus", so your question does not really make any sense. If I have misunderstood something, please edit the question to clarify it. $\endgroup$
    – MJD
    Commented May 31, 2012 at 2:08
  • $\begingroup$ @MarkDominus Perhaps OP is referring to the solid creating by turning the "strings" in a $(p,q)$-knot into three-dimensional tubes with nonzero volume, as in the Wikipedia illustration? $\endgroup$
    – anon
    Commented May 31, 2012 at 2:28
  • $\begingroup$ That would be a sensible question! I wonder if that is what is being asked, and I hope OP will clear things up. $\endgroup$
    – MJD
    Commented May 31, 2012 at 2:29

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Under the interpretation given by anon in the comments, yes, what you have is a knotted torus, but a torus all the same (that is, homeomorphic to a torus). Think about slicing through it, then unravelling it from the torus it loops around, giving you a cylinder, then gluing it back together along the slice: that gives you a torus, so the original was just a torus, albeit embedded in 3-space in an amusing way.

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