The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots
by Colin C. Adams · American Mathematical Society
The consensus first book for newcomers: rigorous enough to reach current research questions but with minimal prerequisites. Colin Adams's standard entry text on knot theory, needing only basic college algebra. Covers knot tabulation, Reidemeister moves, invariants, the Jones and other polynomials, hyperbolic and braid perspectives, and applications to chemistry, biology and physics, with many exercises and open problems.
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More resources on Knot Theory
KnotInfo: Table of Knot Invariants
Searchable database of prime knots up to 13 crossings with diagrams, notations (Conway, DT, PD) and invariants such as genus, unknotting number, Alexander, Jones and HOMFLY polynomials, Khovanov homology and hyperbolic volume. Tables are downloadable.
Knots and Links
Dale Rolfsen's classic topological treatment of knots, from knot groups and covering spaces to Seifert surfaces, the Alexander invariant, surgery descriptions of 3-manifolds and branched covers. Contains the Rolfsen knot table and exercises; emphasises geometric constructions and pictures.
An Introduction to Knot Theory (Graduate Texts in Mathematics 175)
Graduate text by W. B. Raymond Lickorish based on his Cambridge course. Develops Seifert surfaces, the Alexander polynomial, the Jones, HOMFLY and Kauffman polynomials, skein theory and quantum invariants of 3-manifolds. Assumes the fundamental group and basic homology.
Knot Theory (Math 487 lecture series, Andrews University)
Recorded upper-level undergraduate course taught by topologist Anthony Bosman (Spring 2019). Lectures cover colorings, the Alexander polynomial, Seifert surfaces and genus, the Jones polynomial and further invariants, worked at the board with examples. The strongest free full-length video course on knot theory; complements the written sources for learners who want lectures.
Knots Knotes (lecture notes on elementary knot theory)
Best free written course: proofs are complete, explanations patient, and it bridges Adams-level intuition to Lickorish-level rigor. 149-page undergraduate lecture notes by Justin Roberts (UC San Diego), first written for an Edinburgh course. Covers formal definitions, Reidemeister moves, linking number, p-colourings, the Kauffman bracket and Jones polynomial, alternating knots, manifolds and classification of surfaces, with exercises.