Intro to Knot Theory, for folks with a background in physical knots

The terminology used for mathematical knots is beguilingly similar to that used for physical knots, but differs in important ways. Things can get very confusing unless you're aware of some important differences.

terminology

• knot
  — within the context of knot theory, it means a closed non-physical knot
  • closed knot
    — a knot that has no ends, instead it's an unbroken loop (while unusual for physical knots, this is by far the most studied in mathematics)

    open knot

    — a knot that has two ends

    non-physical knot

    — a knot made out of line that has no thickness, mass, friction, stiffness, etc. It is an idealized curve, as seen in geometry.

    physical knot

    — a mathematical knot that has thickness, and possibly other physical properties such as: intermolecular forces, friction, stiffness, inertia, ...

    ideal knot

    — a physical knot that minimizes ropelength (minimal ropelength is an invariant)
• link
  — one or more knots that are interwoven
• tangle
  — A knot subsection that can be recombined in a few elementary ways. The main use of tangles is for Conway notation.
• arc
• slice knot, ribbon knot
• braid

applications

• knot theory — used to answer questions in topology
• physical knot theory — used to answer questions about protein folding, DNA folding, and other physical knots

overview / 101 texts

• Knot Theory, by John Nardo at Oglethrope University
• Knot Theory, published by the MEC (Math Explorers' Club) of Cornell
• knots, links, arcs
• The Theory of knots in Physics, Mathematics, and Biology, by Riccardo Longoni
• "Teaching and Learning of Knot Theory in School Mathematics" (book)

• Knot Theory and its Applications, by Kunio Murasugi
• Knot Theory, by V. O. Manturov