What Does Null-Homotopic Framing Mean in Knot Topology?

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In summary, null-homologous framing is a mathematical and topological technique used to represent surfaces in three-dimensional space. It is unique in that it preserves the topology of the surface being embedded, making it useful in various fields such as computer graphics and theoretical physics. However, it has limitations in that it can only be applied to certain types of surfaces and requires careful calculation and analysis.
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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.
 
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Hello,

Thank you for your question. A null-homotopic framing refers to a specific type of framing that is used in the study of knots and their topology. To understand this term, we first need to define what a framing is in this context.

A framing is a choice of isomorphism between the tubular neighborhood N(K) and ## S^1 \times D^2 ##. In other words, it is a way of "wrapping" the tubular neighborhood around the knot in a specific manner. This isomorphism is not unique and can be changed through isotopy, which is a continuous transformation.

Now, a null-homotopic framing is a framing that can be continuously deformed into the trivial framing, which is just the standard way of wrapping the tubular neighborhood around the knot. In other words, a null-homotopic framing is one that can be continuously transformed into the trivial framing without changing the underlying knot.

This concept is important in the study of knot topology because it allows us to understand the relationship between different framings and how they can be transformed into each other. It also has implications for the Legendrian conjecture, which states that any two Legendrian knots with the same topological type must be isotopic. A null-homotopic framing is one way to show that two knots have the same topological type.

I hope this helps to clarify the meaning of null-homotopic framing. If you have any further questions, please don't hesitate to ask.
 

Related to What Does Null-Homotopic Framing Mean in Knot Topology?

What is null-homologous framing?

Null-homologous framing is a term used in mathematics and topology to describe a set of techniques for representing surfaces in three-dimensional space. It involves taking a two-dimensional surface and embedding it in three-dimensional space in such a way that it is "flat" or has no curvature.

How is null-homologous framing different from other types of framings?

Null-homologous framing is unique in that it preserves the topology of the surface being embedded. This means that any holes or handles on the surface will also be represented in the three-dimensional space. Other types of framings may distort the topology of the surface, making it difficult to analyze or interpret.

What is the importance of null-homologous framing in mathematics?

Null-homologous framing is important in mathematics because it allows for a more accurate representation of surfaces in three-dimensional space. This can be useful in various fields, such as computer graphics, computer-aided design, and even in theoretical physics.

What are some applications of null-homologous framing?

Null-homologous framing has applications in a variety of fields, including geometry, topology, and algebraic topology. It is also used in computer graphics and computer-aided design to accurately represent surfaces and objects in three-dimensional space.

Are there any limitations to null-homologous framing?

Null-homologous framing has its limitations, as it can only be applied to surfaces that are orientable and have a single boundary component. It also requires careful calculation and analysis, making it a more complex technique compared to other types of framings.

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