Can Two-Moves Separate Components of a Link in Knot Theory?

In summary, the conversation is about showing the equivalence of every link to a trivial link with the same number of components. The speaker has a method for simple links with linking number 1, but is looking for a way to generalize it for links with any number of linking numbers. They mention a specific chapter and book for further reference, and jokingly mention that they cannot help with knot theory. However, they later think they have come up with a solution involving a specific type of move.
  • #1
de1irious
20
0
Hi, so I need to show that every link is two-equivalent to a trivial link with the same number of components. Right now I can show that if I have a simple link with linking number = 1, then it is possible to immediately separate the link into its two components. But how can I generalize this idea t links of n linking numbers? Any ideas?

For those interested, this is 3.7 in Invariants of Knots, the Knot Book by Adams.
 
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  • #2
If it's not theory we can't help you. :tongue2:

I'm sorry, I just couldn't resist the pun. I really should've because I don't know anything about knot theory and I can't actually help you. I hope someone does soon.
⚛​
 
  • #3
Haha :rolleyes:

Actually I think I figured it out. You can show two-moves contain a certain other kind of move, which more clearly is able to separate components of a link.
 

1. What is knot theory?

Knot theory is a branch of mathematics that studies mathematical knots. These are closed, non-self-intersecting curves embedded in three-dimensional space. Knot theory involves the study of the properties and classifications of knots, as well as their applications in various fields such as physics and biology.

2. How do you determine if two knots are equivalent?

In knot theory, two knots are considered equivalent if one can be deformed into the other without cutting or passing the string through itself. This is known as the ambient isotopy property. Another way to determine equivalence is by using knot invariants, which are numerical values associated with each knot that do not change under ambient isotopy.

3. What are some real-world applications of knot theory?

Knot theory has applications in various fields such as physics, biology, chemistry, and computer science. In physics, knot theory is used to study the properties of polymers and the behavior of DNA molecules. In biology, it can help understand the structure and function of proteins. In chemistry, knot theory is used to study the properties of molecules. In computer science, it has applications in data compression and cryptography.

4. Can knots have different numbers of crossings?

Yes, knots can have different numbers of crossings. The minimum number of crossings for a non-trivial knot (a knot that is not equivalent to a simple loop) is three. However, there is no upper limit to the number of crossings a knot can have. The number of crossings is one of the factors used to classify knots into different types.

5. How does knot theory relate to topology?

Knot theory is a subfield of topology, which is the study of the properties of geometric objects that do not change under continuous deformations. Knots are topological objects, and studying their properties and classifications falls under the realm of topology. This connection between knot theory and topology allows for the application of topological techniques in the study of knots.

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