The Knot Book

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1. MarkusQSep 2025
   For a gentile introduction, Colin Adams's "The Knot Book" is very accessible (and builds up the notion of polynomial invariants step by step).
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2. mathgeniusMar 2022
   Modern calculus (analysis) was invented because people shot themselves in the foot working with topology and wondering exactly what is a "curve" ? I am a big fan of this approach to learning mathematics, just forge ahead and when (if) things fall apart then go back and fix up the foundations. To this end I recommend a couple of books. "The Knot Book" by Adams is a very interesting exploration in topology (without requiring all the years of study at university before you are allowed to learn exactly what a topology is). And in another direction, group theory was invented because the study of symmetry gets very tricky! But if you want to dive in anyway then have a look at Conway's "The symmetries of things". It is a lot of fun. Most modern group theory (or algebra) books don't actually have any pictures of symmetric things, just endless formulas and lemmas. If you want to be a pro, then you gotta learn that stuff, but there's definitely pathways into higher mathematics that don't require you to learn that.
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3. elil17Jul 2020
   I took an experimental class in high school which taught topology and knot theory to people with a pre-calculus background. We used an early draft of a textbook (not sure of the title or if it’s even been released - our teacher got access because she knew the author) and a lovely knots textbook called “The Knot Book.”

   I think we gained a really good intuitive understanding of homeomorphism, neighborhoods, and homology groups. Knowing about these topics has been super useful in my studies and career.

   On the other hand, some of the algebraic components of topology went totally over my head. So I do think there’s a limit to how much you can understand without diving into other fields.

   Also, we had the benefit of a trusty excellent mathematician as our teacher, which helped quite a bit. Good luck diving into this field, I hope you find it rewarding!

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4. tgbJun 2020
   Related: do people have good recommendations of "casual" math books for more advanced topics? I.e. ones that an educated reader with a background in math can read through in one pass (unlike most textbooks) but nonetheless have real math in them? (And has fun exercises!)

   I can start with some. Feynman's Lectures on Computing. Scott Aaronson's Quantum Computing Since Democritus (though it assumes some background knowledge of quantum computing). I think Colin Adam's "The Knot Book" (on knot theory in topology) as well.

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5. kmillMay 2020
   One reason people care about knots in low-dimensional topology is that every compact ("finite-volume") 3-dimensional manifold without boundary can be constructed by taking a 3-sphere (the set of points in R^4 unit distance from the origin), boring out tubes along a collection of disjoint knots, then gluing solid tori ("donuts") back in in a different way, a process called Dehn surgery. This is the Lickorish-Wallace theorem. Sort of the intuition is that if you take a 3-manifold and have worms eat out enough closed loops, the manifold loses its integrity and becomes indistinguishable from the complement of a collection of disjoint knots. (Lickorish's version of the proof involves a theorem that's colloquially known as the Lickorish Twist Theorem.)

   In particular, every 3-manifold is the boundary of a 4-manifold obtained in a way that's reminiscent of knot traces from the article. You take a disjoint collection of knots in the boundary of a 4-ball, then glue in the 1-handles ("caps" in the article) along these knots, but with slight change: you glue in 1-handles with any framing whatsoever, not just the 0-framing like in knot traces (and actually using just +1-framing and -1-framing is sufficient). It's actually a remarkable fact in its own right that every 3-manifold bounds a 4-manifold; this is saying the 3-dimensional cobordism group is trivial. Other-dimensional cobordism groups are not trivial in general.

   Every 3-manifold has a diagram, then, consisting of a multi-component knot (known as a link) with each component labeled by an integer (or a rational number if you are ok with "fake" surgeries). There is a whole thing called the Kirby calculus that gives a sufficient set of moves to go between any two such representations of a particular 3-manifold. An extension to this calculus went into Piccirillo's calculations with knot traces -- she cites the classic Gompf and Stipsicz for details.

   One use of this representation of a 3-manifold is to construct Reshetikhin-Turaev invariants, which are sequences of numbers associated to a 3-manifold. This is related to the Jones polynomial, and these invariants satisfy a number of wonderful properties that together mean they form a topological quantum field theory (TQFT). I don't know the physics, but I'm under the impression you can interpret it as having something to do with quantum states of anyonic particles.

   For books, you might look at Adams "The Knot Book" or Prasolov "Intuitive Topology" to get a substantial taste of knots and low-dimensional topology.

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6. wolfgkeFeb 2017
   The variant of this joke for mathematicians (source: https://jcdverha.home.xs4all.nl/scijokes/1_12.html):

   I saw the following in "The Knot Book" by Colin Adams (a very readable introduction to Knot Theory). The joke is attributed to Joel Haas.

   A woman walks into a bar accompanied by a dog and a cow. The bartender says, "Hey, no animals are allowed in here." The woman replies, "These are very special animals." "How so?" "They're knot theorists." The bartender raises his eyebrows and says, "I've met a number of knot theorists who I thought were animals, but never an animal that was a knot theorist." "Well, I'll prove it to you. Ask them them anything you like." So the bartender asks the dog, "Name a knot invariant." "Arf, arf" barks the dog. The bartender scowls and turns to the cow asking, "Name a topological invariant." The cow says, "Mu, mu." At this point the bartender turns to the woman, says, "Just what are you trying to pull" and throws them out of the bar.

   Outside, the dog turns to the woman and asks, "Do you think I should have said the Jones polynomial?"

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