Algebraic topology: knots links and braids
7 hours ago
- #knot-theory
- #topology
- #mathematics
- A knot is a simple closed curve in Euclidean 3-space (E(3)), and two knots are equivalent if there's an orientation-preserving homeomorphism mapping one to the other.
- Schoenflies proved in 1908 that homeomorphisms from simple closed curves in the plane (E(2)) to the unit circle (S(1)) can be extended to the entire plane, but this doesn't hold in higher dimensions.
- Wild embeddings in E(3) exist, such as Alexander's horned sphere and Antoine's necklace, where complements are not simply connected.
- Knots are typically studied as tame embeddings, like simple closed polygonal curves, to avoid wild behavior.
- Knot diagrams are projections into E(2) with over/under-crossing indications, sufficient to reconstruct the knot up to equivalence.
- Reidemeister moves define equivalence between knot diagrams: two diagrams represent the same knot if one can be transformed into the other via these moves.
- The unknot is the unique knot with a diagram without crossings.
- Knot composition involves tying two oriented knots sequentially, with the unknot acting as a zero element.
- A knot is prime if it cannot be expressed as a sum of two non-trivial knots, and every knot has a unique prime factorization.
- Seifert surfaces are used to define the genus of a knot, which is additive under composition and zero only for the unknot.
- Alternating knots have diagrams where overcrossings and undercrossings alternate, and most prime knots with ≤8 crossings are alternating.
- Knot invariants, like the fundamental group of the complement or the Jones polynomial, help distinguish inequivalent knots.
- Links are disjoint unions of knots, and trivial links have free fundamental groups for their complements.
- Braids are collections of arcs in E(3), and the braid group B(n) is generated by elementary braids with specific relations.
- Alexander's theorem states that every link is equivalent to one obtained from a braid by identifying its endpoints.