What Does Null-Homotopic Framing Mean in Knot Topology?

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SUMMARY

The term "null-homotopic framing" refers to a specific choice of isomorphism between the tubular neighborhood N(K) of a knot K and the product space S^1 × D^2, where K is embedded in a manifold. This concept is crucial in understanding the relationship between knot theory and the Legendrian conjecture, particularly regarding the existence of tubular neighborhoods as stated in the theorem concerning compact submanifolds. The discussion highlights the importance of framing in knot topology and its implications for homology, specifically regarding cycles that are null-homologous.

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  • Understanding of knot theory and basic topology concepts
  • Familiarity with tubular neighborhoods in manifold theory
  • Knowledge of homology and cycles in algebraic topology
  • Awareness of the Legendrian conjecture and its significance
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  • Research the implications of null-homotopic framings in knot theory
  • Study the properties of tubular neighborhoods in manifold topology
  • Explore the Legendrian conjecture and its applications in modern topology
  • Learn about homology theories and their definitions related to cycles
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Mathematicians, particularly those specializing in topology and knot theory, as well as researchers interested in the applications of the Legendrian conjecture and homological concepts in manifold theory.

WWGD
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Hi, all:

I'm trying to understand the meaning of the term "null-homotopic framing".

Say K is a knot embedded in a manifold , and N(K) is a tubular neighborhood of K

( there is a theorem that a compact submanifold allows for the existence of a tubular

neighborhood).

I read about it here, under " motivation for the Legendrian conjecture " :

http://electrichandleslide.wordpress.com/2013/05/17/the-legendrian-surgery-conjecture/

A framing here is a choice ( up to isotopy, I believe; not specified in Rolfsen's book) of

isomorphism between the tubular neighborhood N(K) and ## S^1 \times D^2 ##.

(Though K can be any manifold for which a tubular neighborhood exists, e.g., maybe

the normal bundle N(K) is trivial, or K is compact, embedded in some ambient manifold M)

I understand the obvious meaning of null-homologous for a cycle in any space, meaning

that the cycle is , e.g., a bounding cycle (cycle with non-trivial boundary) in simplicial homology,

so homologically trivial, or different definitions for different choices of homology, but I cannot see what a

null-homologous framing is. Any ideas?

Thanks.
 
Last edited:

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