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

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SUMMARY

The discussion centers on the concept of two-moves in knot theory and their ability to separate components of a link. It is established that every link is two-equivalent to a trivial link with the same number of components. The participants reference the problem from section 3.7 in "Invariants of Knots" from the book "The Knot Book" by Dale Rolfsen. A key insight shared is that two-moves can be generalized to demonstrate the separation of links with multiple linking numbers.

PREREQUISITES
  • Understanding of knot theory fundamentals
  • Familiarity with linking numbers in knot theory
  • Knowledge of two-moves and their implications in link separation
  • Access to "The Knot Book" by Dale Rolfsen for reference
NEXT STEPS
  • Research the concept of two-moves in knot theory
  • Study linking numbers and their role in knot equivalence
  • Explore the implications of section 3.7 in "Invariants of Knots"
  • Investigate other types of moves in knot theory that facilitate component separation
USEFUL FOR

Mathematicians, knot theorists, and students studying topology who are interested in the properties and equivalences of links in knot theory.

de1irious
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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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If it's not theory we can't help you. :-p

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.
⚛​
 
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.
 

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