स्नातक (उन्नत)Geometry & Topologyपाठ उपलब्ध
Knot theory
Knots and links in space and their diagrams, Reidemeister moves, invariants that tell knots apart (tricolourability and Fox colourings, linking number, the Kauffman bracket and Jones polynomial), Seifert surfaces and genus, and an honest map of what is known and open.
math.geom.knot-theory
इस पर आधारित
44 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Represent knots and links by diagrams and state Reidemeister's theorem.
- Prove that tricolourability and the linking number are invariants, and use them to distinguish knots and links.
- Compute the Kauffman bracket and Jones polynomial of small knots, and show that the trefoil is not equivalent to its mirror image.
परिभाषा
- A knot is a smooth (or tame) embedding of the circle into ℝ³ or S³; two knots are equivalent if there is an ambient isotopy taking one to the other.
- A knot invariant is a quantity assigned to knots that takes the same value on equivalent knots.
- A diagram is tricolourable if its arcs can be coloured with three colours, using at least two, so that at each crossing the three arcs meeting are all the same colour or all different.
- A Seifert surface for a knot is a compact connected orientable surface whose boundary is the knot.
प्रमेय
- Reidemeister: two diagrams represent equivalent knots if and only if they are related by planar isotopy and the three Reidemeister moves.
- Tricolourability is a knot invariant; the trefoil is tricolourable and the unknot is not, so the trefoil is knotted.
- Every knot bounds a Seifert surface (Seifert's algorithm).
- It is unknown whether the Jones polynomial detects the unknot.
आम भ्रांतियाँ
- Thinking a knot invariant that agrees on two knots proves they are the same; the figure-eight knot and the unknot are both not tricolourable, yet they are different.
- Believing crossing number can be read off any diagram; a given diagram may have more crossings than necessary.
आगे ले जाता है
वैकल्पिक खोजें
- Low-dimensional topology: 3- and 4-manifolds Every closed orientable 3-manifold can be obtained by surgery on a link in the 3-sphere (Lickorish–Wallace), so knots are a gateway to 3-manifold topology.
- Mathematical foundations of quantum mechanics Witten showed in 1989 that the Jones polynomial arises from a quantum field theory (Chern–Simons theory), a striking link between topology and physics.
- ← Dots and lines: networks and puzzles
- ← Gauge theory and the geometry of physics
अधिगम पथ
आगे पढ़ने के लिए
- Colin C. Adams, The Knot Book (1994)
- Dale Rolfsen, Knots and Links (1976)