Knot Theory: Diagrams and Reidemeister Moves Explained

The Alexander polynomial (1923) was the first major knot invariant polynomial, relating three versions of a knot obtained by changing a single crossing. It satisfies Alexander(unknot)=1. Vaughan Jones discovered the Jones polynomial in 1984 while working on statistical mechanics algebra, earning the Fields Medal in 1990. Six mathematicians independently found the HOMFLY polynomial (two variables) shortly after. No single invariant distinguishes all knots; multiple invariants are combined like checking multiple identifiers. The unknot has Alexander polynomial 1 but Jones polynomial 1 as well. These polynomials provide increasingly powerful tools for distinguishing knots, though none are perfect alone.

This section introduces fundamental concepts of knot theory and the Alexander polynomial. A knot invariant is a function assigning the same value to knots differing only by ambient isotopy. The Alexander polynomial, discovered in 1923, is constructed by taking the cyclic branched covering space of the knot complement and computing the first homology group with module structure over Laurent polynomials. The polynomial has the form Δ(t) = Σ a_i t^i with integer coefficients and potentially negative exponents. The polynomial serves as a tool to distinguish knots: different polynomials imply different knots, though different knots can share the same polynomial. The symmetry property Δ(t) = Δ(t⁻¹) reflects the topological nature of the knot complement. For links, orientations on components must be specified, and the polynomial may involve variables of the form t^k with integer k, potentially including square roots. The symmetry property does not hold for links in general.

The Alexander polynomial (1923) is a knot invariant based on rules about how the polynomial changes when crossings are modified. The Jones polynomial (1984), discovered by Vaughan Jones (Fields Medal 1990), is more powerful and can distinguish more knots. Shortly after, the HOMFLY polynomial (two-variable version) was discovered. These polynomial invariants represent the most effective tools for distinguishing knots. Together with Reidemeister moves, they provide a comprehensive approach to knot classification.

Knot invariants are quantities unchanged under ambient isotopy. Three-coloring assigns colors to arcs such that at each crossing, either all three colors meet or only one appears—if a knot can be colored but another cannot, they differ. Polynomial invariants provide more powerful tools: Alexander polynomial, Jones polynomial, and Kauffman bracket. The Jones polynomial, discovered by Vaughan Jones, was transformative because it distinguished many knots that simpler invariants couldn't. It is computed using smoothing rules at crossings, with each crossing independently A-smoothed or B-smoothed. For an n-crossing diagram, there are 2ⁿ states forming a cube structure. The polynomial is a weighted sum over all states, with weights depending on circle counts. The Jones polynomial is called 'strong' but remains incomplete as some distinct knots share identical polynomials.

A Seifert surface is an orientable surface whose boundary is a knot, and the genus of a knot (the minimum number of handles in any such surface) serves as a powerful invariant that characterizes knots: the unknot has genus zero, while non-trivial knots have positive genus; for alternating knots, the genus can be computed simply by counting the number of crossings minus the number of Seifert circles and dividing by two.

The genus of a knot, defined as the minimum genus among all compact orientable surfaces bounded by the knot, serves as a powerful invariant in knot theory; it distinguishes knots by showing that the unknot has genus 0 while prime knots have genus 1, and it is additive under knot sum operations, meaning the genus of the sum of two knots equals the sum of their individual genera, thereby proving that the unknot cannot be expressed as the sum of two non-trivial knots.

A Seifert surface is a compact, orientable surface embedded in R³ whose boundary is the knot. The genus of a Seifert surface is a topological invariant. The minimum genus over all Seifert surfaces is the knot genus. The Alexander polynomial provides a lower bound on the genus: degree plus number of components minus one is at most twice the genus. The standard Seifert surface is constructed by filling each Seifert circle with a disk and connecting disks along crossings with bands. This surface has genus equal to the first Betti number of the Seifert graph. The Alexander polynomial is related to the first homology group of the Seifert graph. The degree of the polynomial equals the rank of this homology group. This provides an algebraic interpretation of the polynomial in terms of graph topology.

Topology studies spatial properties preserved under continuous deformation. An n-dimensional manifold locally resembles Euclidean space but may have non-trivial global structure. Compact orientable surfaces with boundary are classified by genus and boundary components. A Seifert surface is a compact orientable surface whose boundary is a knot, constructed by resolving crossings and attaching disks with half-twisted bands. The genus of a knot is the minimal genus among all its Seifert surfaces, and unlike crossing number, genus behaves predictably under connected sum: g(K1 + K2) = g(K1) + g(K2).

A Seifert surface is an oriented surface whose boundary is a knot, and its genus (the minimum number of handles required) serves as a powerful knot invariant: a knot has genus zero if and only if it is the unknot, and for alternating knots, the genus can be computed using Seifert's algorithm by counting the number of circles and crossings in the resolved diagram with the formula (1 + c - s)/2, where c is the number of crossings and s is the number of circles.

Every knot can be represented as the closure of a braid (Alexander's theorem), and two braids represent the same knot if and only if they are related by Markov moves (Markov's theorem), establishing a complete correspondence between knots and braids up to braid equivalence.

The braid group is defined with generators σ₁ through σₙ₋₁ and relations σᵢσᵢ₊₁σᵢ = σᵢ₊₁σᵢσᵢ₊₁ (braid relations) and σᵢσⱼ = σⱼσᵢ for |i-j| > 1 (commutativity). Each generator σᵢ represents a positive crossing between strand i and strand i+1. Alexander's theorem states that any link can be obtained as the closure of some braid. Markov's theorem provides conditions when two braids close to the same link: they must be related by conjugation (αβ ~ βα) or stabilization (adding a strand with a crossing at the top or bottom). These operations change the braid but preserve the underlying link.

The braid group Bₙ consists of n vertical strands that must always move downward, with equivalence defined by continuous deformations without cutting or passing strands through each other. The group is generated by n-1 elementary braids σ₁ through σₙ₋₁, where each σᵢ represents crossing the i-th strand over the (i+1)-th strand. Alexander's theorem states that every knot can be represented as the closure of some braid, establishing a fundamental connection between braids and knots. The group is completely determined by two relations: commutativity for non-adjacent generators and the braid relation for adjacent generators. Markov's theorem characterizes when two braids have equivalent closures: β' can be obtained from β by conjugation or stabilization/destabilization. This enables constructing knot invariants by defining functions on braid groups that respect these operations.

Braids connect to knot theory through closure operations that transform braids into links by connecting corresponding endpoints planarly. Alexander's theorem establishes that every link can be obtained as the closure of some braid, providing a complete correspondence between braids and links. Markov's theorem characterizes when two braids close to give equivalent links: they must be related by conjugation (exchanging braid order on a circle) and stabilization (adding crossings that don't change the link). Together, these theorems enable computational approaches to knot theory—representing braids algebraically allows computers to determine knot equivalence. However, limitations exist: non-equivalent knots may require diagram complication before simplification, preventing definitive negative answers in all cases. These theorems, developed around the 1930s, remain fundamental tools in modern topology for studying knots and links through algebraic methods.

This lecture introduces fundamental concepts in knot theory, including the definition of knots and links as closed curves in R³ up to isotopy, regular projections without triple intersections, and Reidemeister's theorem that all regular projections of the same knot are related by a finite set of Reidemeister moves. The lecture explains Alexander's theorem that every link can be represented as the closure of a braid, and demonstrates Alexander's algorithm for constructing braid representations from knot diagrams. It also covers Ziesler's circles method for simplifying knot projections and Markov's theorem that two braids represent the same link if and only if they are related by Markov moves of conjugation or stabilization.

Knot theory, which studies continuous loops in space, evolved from Lord Kelvin's 1860s hypothesis that chemical elements might be knotted vortices in the ether, leading mathematicians to develop sophisticated invariants like the Jones polynomial to classify knots; these mathematical tools now find critical applications in quantum computing (where topological quantum computers use knot polynomials for inherently noise-resistant computations) and molecular biology (where DNA topology and enzyme action depend on knot properties).

Parallel translation measures curvature: on flat surfaces vectors return unchanged, but on spheres they rotate (90 degrees on Earth). Witten applied gauge theory to knot theory, deriving invariants by averaging particle transformations around knots in various fields. DNA knots reveal enzyme mechanisms through electron microscopy, identifying specific knot types. Kelvin's 1800s vortex atom theory proposed atoms as knotted ether vortices, later revived with successful creation of knotted vortex rings in water using 3D-printed templates. These applications demonstrate knot theory's deep connections to geometry, physics, and molecular biology.

DNA molecules are long and flexible, making them prone to knotting when confined in cellular environments—approximately 90-94% become knotted under certain conditions. Enzymes called recombinases perform recombination events by cutting and rejoining DNA strands. Knot theorists study these events by examining how DNA knot structures change before and after enzyme action. For certain proteins, mathematical theorems prove that enzyme action can be uniquely determined from input-output behavior, though proving such results requires advanced mathematics including double branched covers and homology theory.

Knot theory has important applications in biology, particularly in DNA structure. DNA molecules in bacteria like Escherichia coli form knots, and enzymes called topoisomerases can modify these knots. The behavior of these organisms changes depending on the type of knot their DNA forms. Mathematicians have used knot theory to model how these enzymes modify knots at the molecular level, with theoretical predictions confirmed by experimental evidence.