- #1

PsychonautQQ

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So let ##K## be a knot and let ##u## be a tube (or neighborhood) around the knot. We can visualize this as making the knot 'thicker' and thus giving it volume.

I want to calculate the homology group ##H_i(R^3 \ u)##. And actually, I was given the homology, I just need to prove it. If ##i=0## then the group equals the infinite cyclic group ##Z##, and it also does when ##i=1##. However, it equals zero if ##i>1##

So yes, I'm trying to understand the homology groups of ##H_i(R^3 \ u)##.

In doing so I am using this relative homology group:

##H_i(R^3, cl(R^3 \ u))## where ##cl(R^3 \ u)## is the closure of ##(R^3 \ u)##. I notice that this group is isomorphic to ##H_i(u, Bd(u))##,

##H_i(R^3, cl(R^3 \ u))## ##=## ##H_i(u, Bd(u))##

where $=$ is denoting an isomorphism here. I know that these groups are isomorphic by using the excorcism theorem and excorcising ##int(R^3 \ u)## where ##int(R^3 \ u)## is the interior of ##R^3 \ u##.

Also, I think it will be worth noting that ##R^3 = cl(u) \bigcup (R^3 \ int(u))##.

Also, ##(R^3 \ int(u)) = Bd(u)## = ##S^1 \times S^1##

Anyone know much about homology? I'd appreciate some insight!