Jones polynomial
The Jones polynomial is a knot invariant that assigns to every oriented link in three-dimensional Euclidean space a Laurent polynomial in the indeterminate t^{1/2}, uniquely determined by the link up to ambient isotopy and serving as a distinguishing feature for non-trivial knots and links.[1] It was discovered by mathematician Vaughan F. R. Jones in 1984 during his work on von Neumann algebras and subfactor theory, initially arising as a representation of the braid group tied to index computations in type II₁ factors.[2] Jones presented the polynomial on May 22, 1984, at Columbia University, linking it to knot theory through closed braids and Markov's theorem, which built upon earlier invariants like the Alexander polynomial from 1923.[2] The polynomial satisfies a skein relation that allows recursive computation: for three oriented links L_+, L_-, and L_0 differing only in a local crossing, t^{-1} V_{L_+}(t) - t V_{L_-}(t) = \left( t^{1/2} - t^{-1/2} \right) V_{L_0}(t), with normalization V_{\bigcirc}(t) = 1 for the unknot.[1] It can also be computed using the Kauffman bracket, where the unoriented bracket \langle L \rangle is defined by \langle \bigcirc \rangle = 1, \langle L \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle, and the crossing relation \langle X \rangle = A \langle \alpha \rangle + A^{-1} \langle \beta \rangle, and then V_L(t) = (-A^3)^{-w(\tilde{L})} \langle L \rangle \big|_{A=t^{-1/4}} with w denoting writhe, providing an alternative state-sum formulation independent of orientation.[3] These relations make the Jones polynomial computable for any link diagram, though its evaluation grows exponentially with complexity.[1] Unlike classical invariants such as the Alexander polynomial, the Jones polynomial can distinguish a knot from its mirror image by substituting t with t^{-1}, and it detects handedness, resolving long-standing conjectures like Tait's.[2] Its discovery revolutionized low-dimensional topology, spawning quantum topology and inspiring further invariants like the HOMFLY and Kauffman polynomials, while earning Jones the 1990 Fields Medal.[2] In physics, it connects to quantum field theory via Wilson loop expectations in Chern-Simons theory, interpreting knots as particle worldlines in three-dimensional spacetime.[4]Introduction
History and Discovery
The Jones polynomial emerged from Vaughan Jones' research on subfactors within the theory of von Neumann algebras. In early 1983, Jones introduced the concept of the index for subfactors of type II₁ factors, providing a numerical measure of the "size" of a subfactor relative to its ambient algebra. This index, now known as the Jones index, motivated his subsequent work by revealing connections to statistical mechanics and representation theory, particularly through projections satisfying the Temperley-Lieb relations. Jones' discovery of the polynomial invariant occurred in May 1984, specifically between meetings with topologist Joan Birman at Columbia University on May 14 and May 22. During this period, while exploring representations of the braid group arising from subfactor theory, Jones constructed a Laurent polynomial in one variable associated to knots and links, initially via a rescaled trace on the Hecke algebra derived from these representations. This construction lacked a direct diagrammatic interpretation at the time, relying instead on the index of subfactors and Markov traces to ensure invariance under ambient isotopy and link equivalences. Jones first shared details of this invariant through correspondence, including a letter to Birman dated May 31, 1984, where he outlined its computation for basic examples.[5] In these early computations, Jones evaluated the polynomial for simple knots such as the trefoil, demonstrating that it distinguished the right-handed and left-handed trefoils—unlike the Alexander polynomial—and yielded values like -t^4 + t^3 + t for the right-handed trefoil.[6] He formally announced the invariant in a January 1985 article in the Bulletin of the American Mathematical Society, titled "A polynomial invariant for knots via von Neumann algebras," emphasizing its origins in operator algebras without reference to knot diagrams. A more detailed exposition appeared in his 1987 paper in the Annals of Mathematics, extending the construction to a two-variable version via Hecke algebra representations.[7] The discovery rapidly gained recognition for bridging operator algebras and low-dimensional topology, leading to Jones' award of the Fields Medal in 1990 at the International Congress of Mathematicians in Kyoto. The medal citation praised his invention of the Jones polynomial as a new knot invariant and its profound applications to von Neumann algebra theory.[8]Overview and Significance
The Jones polynomial, discovered by Vaughan Jones in 1984, is a fundamental invariant in knot theory that assigns to each oriented link in three-dimensional space a Laurent polynomial in the variable t^{1/2} with integer coefficients, denoted V(L; t) \in \mathbb{Z}[t^{\pm 1/2}]. This polynomial is invariant under ambient isotopy, meaning that two links that can be continuously deformed into each other without passing through themselves (i.e., ambient isotopic) have the same polynomial, providing a powerful tool for distinguishing many distinct knot types.[9] For the unknot, the simplest link consisting of a single unknotted loop, the Jones polynomial evaluates to V(\text{unknot}; t) = 1.[9] To ensure invariance, the polynomial incorporates a normalization based on the writhe of the link diagram, which accounts for the linking number and orientation, adjusting the exponent to make the result independent of the particular diagram chosen to represent the link.[9] The variable t (sometimes reparameterized as q in related contexts) is not ambiguous in its role, as the polynomial's structure aligns consistently across different formulations, such as those arising from quantum group representations.[9] The significance of the Jones polynomial lies in its status as the first truly quantum knot invariant, surpassing the classical Alexander polynomial in discriminatory power and opening new avenues in low-dimensional topology.[9] It ignited the development of quantum topology, including the discovery of related invariants like the HOMFLY polynomial, and inspired categorifications such as Khovanov homology, which lift the polynomial to a homology theory. Furthermore, its connections to physics, particularly through Edward Witten's interpretation via Chern-Simons quantum field theory, have linked knot invariants to quantum gravity and topological quantum field theories, influencing areas from string theory to quantum computing.[10]Definitions
Kauffman Bracket Approach
The Kauffman bracket provides a combinatorial framework for defining the Jones polynomial using unoriented link diagrams, treating links as collections of immersed circles in the plane without specified orientations. This approach, developed by Louis Kauffman, relies on a state summation over all possible resolutions of crossings in the diagram, yielding a Laurent polynomial in the variable A that is invariant under regular isotopy for unoriented links. Unlike orientation-dependent methods, it operates directly on the planar embedding, making it particularly suited for computational verification through exhaustive enumeration of states. To compute the Kauffman bracket \langle L \rangle of an unoriented link diagram L, first define the possible smoothings at each crossing. A crossing in the diagram consists of two arcs intersecting transversely. The A-smoothing resolves the crossing by connecting the incoming arcs to the outgoing arcs in a way that preserves the "horizontal" alignment: specifically, the upper arc connects to the rightward continuation, and the lower arc to the leftward, effectively splitting the strands without introducing a twist (often visualized as rotating the overstrand counterclockwise relative to the understrand). The B-smoothing, in contrast, connects the arcs "vertically": the upper arc to the downward continuation, and the lower to the upward, introducing a split that aligns with the vertical direction (clockwise rotation). These rules ensure that each smoothing replaces the crossing with two non-intersecting arcs, and applying one choice at every crossing in the diagram produces a state s, which is a collection of disjoint, unknotted circles embedded in the plane. The number of such states is $2^c, where c is the number of crossings. For a given state s, let \alpha(s) denote the number of A-smoothings performed, \beta(s) the number of B-smoothings (so \alpha(s) + \beta(s) = c), and \gamma(s) the number of resulting circles in the state. The contribution of state s to the bracket is then A^{\alpha(s) - \beta(s)} (-A^2 - A^{-2})^{\gamma(s) - 1}, where the factor (-A^2 - A^{-2}) accounts for each additional circle beyond the first, mimicking the evaluation of planar matchings in the Temperley-Lieb algebra. The full Kauffman bracket is the sum over all states: \langle L \rangle = \sum_s A^{\alpha(s) - \beta(s)} (-A^2 - A^{-2})^{\gamma(s) - 1}. This polynomial satisfies the axioms \langle \bigcirc \rangle = 1 for the empty diagram or a single circle, \langle L \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle for disjoint union with an unknot, and the linear relation at crossings \langle X \rangle = A \langle X_A \rangle + A^{-1} \langle X_B \rangle (noting the conventional substitution B = A^{-1}). The resulting \langle L \rangle is invariant under Reidemeister moves of types II and III, and type I up to framing, establishing it as a regular isotopy invariant for unoriented framed links. To obtain the oriented Jones polynomial V_L(t) from the unoriented bracket, incorporate the link's orientation via the writhe w(L), defined as the algebraic sum of crossing signs (positive for right-handed overcrossings, negative for left-handed) across a diagram of the oriented link. The normalization is V_L(t) = (-A^3)^{-w(L)} \langle L \rangle \big|_{A = t^{-1/4}}, which adjusts for the framing dependence and yields an ambient isotopy invariant—a Laurent polynomial in t^{1/4} and t^{-1/4} that is unchanged under all Reidemeister moves for oriented links. This step ensures the polynomial detects orientation, distinguishing, for example, a knot from its mirror image in many cases. The writhe computation requires choosing an orientation, but the final V_L(t) is independent of the diagram as long as it represents the same oriented link.Braid Representation Approach
The Jones polynomial can be defined algebraically through unitary representations of the Artin braid group B_n, which capture the topological structure of links via their braid closures. This approach leverages the connection to quantum groups, specifically the representation theory of U_q(\mathfrak{sl}_2) at q = t^{-1}, where t is the variable of the polynomial. The fundamental 2-dimensional representation V of U_q(\mathfrak{sl}_2) provides the building blocks, with the braid group acting on the tensor powers V^{\otimes n}. The generators \sigma_i of B_n are represented by \rho(\sigma_i) = \mathrm{id}^{\otimes (i-1)} \otimes R \otimes \mathrm{id}^{\otimes (n-i-1)}, where R \in \mathrm{End}(V \otimes V) is the universal R-matrix of U_q(\mathfrak{sl}_2) satisfying the Yang-Baxter equation \hat{R}_{12} \hat{R}_{13} \hat{R}_{23} = \hat{R}_{23} \hat{R}_{13} \hat{R}_{12}. This ensures the representation respects the braid relations \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} and \sigma_i \sigma_j = \sigma_j \sigma_i for |i-j| \geq 2. This braid representation is closely tied to the Jones-Temperley-Lieb (JTL) algebra, a quotient of the group algebra of B_n generated by idempotents e_i with relations e_i^2 = \delta e_i, e_i e_{i \pm 1} e_i = e_i, and e_i e_j = e_j e_i for |i-j| \geq 2, where \delta = -t^{-1/2} - t^{1/2}. In the JTL representation, the braid generators act as \sigma_i \mapsto q^{1/2} P_+ + q^{-1/2} P_-, where P_+ and P_- are projectors onto the symmetric and antisymmetric subspaces of V \otimes V, respectively; equivalently, in matrix form on basis vectors, \rho(\sigma_i) corresponds to a combination of a projector (cup-cap idempotent) and a q-weighted overcrossing operator on the i-th and (i+1)-th strands, while acting as the identity elsewhere. This formulation embeds the braid action within the diagrammatic structure of the Temperley-Lieb algebra, facilitating computations via linear algebra on tensor spaces. The equivalence to the Kauffman bracket method arises from the isomorphism between the JTL trace and state-sum evaluations, though the braid approach emphasizes the algebraic homomorphism from B_n. To obtain the Jones polynomial V(L; t) of a link L from a braid \beta \in B_n whose closure is L, one computes the trace of the representation applied to \beta, normalized to ensure topological invariance under braid isotopies and Markov moves. Specifically, V(\beta) = (-q^{3/4})^{-w(\beta)} \operatorname{Tr}(\rho(\beta)), where w(\beta) is the writhe (algebraic crossing number) of \beta and the trace is taken over V^{\otimes n}. This formula requires further normalization by quantum factorials for the number of strands, accounting for the quantum dimension _q! = \prod_{k=1}^n _q with _q = \frac{q^{k/2} - q^{-k/2}}{q^{1/2} - q^{-1/2}}, to handle closures of the empty braid (yielding the unknot with V = [1](/page/1)) and multiplicativity over link components. The writhe factor compensates for the framing dependence in the representation, rendering V(L; t) an ambient isotopy invariant of oriented links.Properties
Skein Relations and Normalization
The Jones polynomial V(L) of an oriented link L satisfies the following skein relation, which relates the values of the polynomial on three links that differ locally at a single crossing: t^{-1} V(L_+) - t V(L_-) = (t^{1/2} - t^{-1/2}) V(L_0), where L_+ is a link diagram with a positive crossing, L_- has the corresponding negative crossing, and L_0 is the diagram obtained by smoothing the crossing in the oriented manner (replacing the crossing with two parallel strands following the orientation).[11] This relation allows recursive computation of the polynomial by resolving crossings step by step. This skein relation arises from the Kauffman bracket polynomial, an unoriented regular isotopy invariant defined via a state-sum expansion over all possible smoothings of a link diagram D. The bracket \langle D \rangle obeys its own local relation: \langle L_+ \rangle = A \langle L_0 \rangle + A^{-1} \langle L_\infty \rangle, along with \langle \bigcirc \rangle = 1 for the empty diagram and \langle L \sqcup \bigcirc \rangle = -(A^2 + A^{-2}) \langle L \rangle for a disjoint unknot, where L_\infty denotes the alternative (unnormalized) smoothing and A is a formal variable.[11] To obtain an ambient isotopy invariant, the Jones polynomial is defined as V(D) = (-A^3)^{-w(D)} \langle D \rangle, where w(D) is the writhe (algebraic crossing number) of the oriented diagram D. Substituting A = t^{-1/4} yields the Laurent polynomial in t. The writhe adjustment compensates for the bracket's sensitivity to crossing type: resolving from L_+ to L_- changes the writhe by -2, and combining this with the bracket's linear relation produces the oriented skein for V.[11] The Jones polynomial is normalized by two axioms to ensure uniqueness and consistency. First, V(\bigcirc) = 1 for the unknot \bigcirc. Second, for a link L disjoint from an unknot, V(L \sqcup \bigcirc) = -(t^{1/2} + t^{-1/2}) V(L). These follow directly from the bracket definition: the unknot has bracket 1 and writhe 0, equaling 1 after substitution and normalization; the disjoint union multiplies the bracket by the same factor.[11] Any link invariant taking values in the ring of Laurent polynomials in t that satisfies the skein relation and these normalization axioms coincides with the Jones polynomial. This uniqueness holds because the relations allow reduction of any link diagram to the unknot via a sequence of crossing resolutions and disjoint unknottings, uniquely determining the value recursively.Invariance and Multiplicativity
The invariance of the Jones polynomial under ambient isotopy for oriented links in three-dimensional space is established through its behavior under the Reidemeister moves, which generate equivalence classes of link diagrams. The polynomial is derived from the Kauffman bracket, a Laurent polynomial in a variable A defined recursively via local smoothing rules at crossings: the A-smoothing follows the orientation, and the B-smoothing (B = A^{-1}) crosses it, with the relation \langle L \rangle = A \langle L_A \rangle + A^{-1} \langle L_B \rangle, disjoint union \langle L \sqcup O \rangle = (-A^2 - A^{-2}) \langle L \rangle, and empty link \langle \emptyset \rangle = 1. The bracket itself is invariant under the second and third Reidemeister moves (RII and RIII). For RII, which introduces or removes two consecutive crossings of opposite type, the recursive application of the smoothing relation yields identical bracket values on both sides, as the intermediate states cancel pairwise. For RIII, which slides a strand over or under a crossing without altering over/under information, the bracket equates through a sequence of smoothing applications that preserve the state sum, leveraging the RII invariance in subcalculations.[12] The bracket is not invariant under the first Reidemeister move (RI), which adds or removes a twist (curl). A positive twist (right-handed curl) multiplies the bracket by -A^3, while a negative twist multiplies by -A^{-3}. To achieve full invariance, the Jones polynomial incorporates a normalization using the writhe w(D), defined for an oriented diagram D as the sum of crossing signs (+1 for positive crossings, -1 for negative, determined by orientation via right-hand rule). Under RI, the writhe shifts by \pm 1, and the normalized form f(D) = (-A^3)^{-w(D)} \langle D \rangle compensates exactly: for a positive twist addition, the factor (-A^3)^{-1} \cdot (-A^3) = 1, and similarly for negative. Substituting A = t^{-1/4} yields the Jones polynomial V(L, t) = f(L), which is thus invariant under all three Reidemeister moves and hence under oriented ambient isotopy.[12] The Jones polynomial exhibits multiplicativity for split links, i.e., disjoint unions of links L = L_1 \sqcup L_2 where the components lie in separate balls. In this case, V(L, t) = V(L_1, t) V(L_2, t), as the recursive definitions or state sums factorize independently over the separated components, with no shared crossings. This property holds without framing adjustments for the standard oriented version. For non-split links like the Hopf link (the simplest two-component link with linking number \pm 1), the polynomial does not factor as a product of individual unknot values but incorporates the linking via crossing interactions; for the positively linked Hopf link, the computation via skein recursion yields V = -t^{1/2} - t^{5/2}, reflecting the intertwined topology, while the split two-unknot case gives V = -(t^{1/2} + t^{-1/2}).[13] Under orientation reversal, the Jones polynomial of an oriented link remains unchanged if all components are reversed simultaneously, as crossing signs and writhe adjust consistently. However, reversing the orientation of exactly one component changes the polynomial by replacing t with t^{-1}, effectively altering the invariant to reflect the new crossing types. This ensures the polynomial distinguishes oriented isotopy classes while maintaining overall consistency. A limitation of the Jones polynomial is that it is not a complete invariant: there exist non-isotopic links with identical polynomials, such as mutant links obtained via Conway mutation (swapping strands at a chosen crossing). For example, the Kinoshita-Terasaka and Conway knots are mutants with the same Jones polynomial but different Seifert genera (2 and 3, respectively), demonstrating that the invariant fails to distinguish certain topological differences.[14]Computations and Examples
Basic Knot Examples
The Jones polynomial distinguishes basic knots and links through specific Laurent polynomials computed via the Kauffman bracket or skein relations. For the unknot, the polynomial is V(t) = 1. The trefoil knot, labeled $3_1 in the Rolfsen table, has Jones polynomial V(t) = t + t^3 - t^4 in the t-variable convention, where positive powers correspond to the right-handed orientation.[13] This differs from the unknot, confirming the trefoil's nontriviality, but the polynomial for the mirror image trefoil is V(t^{-1}) = t^{-1} + t^{-3} - t^{-4}, showing that the invariant detects chirality for this knot.[13] The figure-eight knot, labeled $4_1, is amphichiral and has Jones polynomial V(t) = t^{-2} - t^{-1} + 1 - t + t^2.[15] This symmetric form satisfies V(t) = V(t^{-1}), consistent with the knot being equivalent to its mirror. For links, the Hopf link (with linking number +1) has Jones polynomial V(t) = -t^{1/2} - t^{5/2}.[13] To illustrate computation, consider the right-handed trefoil knot using the Kauffman bracket polynomial \langle K \rangle, defined recursively for an unoriented diagram: \langle \bigcirc \rangle = 1, \langle L \cup \bigcirc \rangle = (-A^2 - A^{-2}) \langle L \rangle, and at each crossing, \langle \times_+ \rangle = A \langle \text{smooth} \rangle + A^{-1} \langle \text{split} \rangle. The full bracket is the state sum over all $2^n resolutions (for n crossings), where each state s contributes A^{a(s) - b(s)} (-A^2 - A^{-2})^{|s| - 1}, with a(s) the number of A-smoothings, b(s) the number of B-smoothings (splits), and |s| the number of circles in state s. For the standard trefoil diagram with three positive crossings and writhe w = 3, the eight states yield \langle 3_1 \rangle = -A^5 - A^{-3} + A^{-7}.[16] The Jones polynomial is then obtained by normalizing: substitute A = t^{-1/4} to get the Kauffman polynomial X(K) = (-A^3)^{-w} \langle K \rangle, resulting in V(t) = t + t^3 - t^4.[16] This process highlights how the bracket detects the topology via smoothing contributions, distinguishing the trefoil from the unknot (whose bracket is 1 after normalization). The following table lists the Jones polynomials for prime knots up to five crossings in the t-convention (adjusted for consistency with the trefoil example above; polynomials for higher knots follow from torus knot formulas or direct computation). These illustrate distinguishability: all differ from the unknot, the trefoils from figure-eight and twist knots, but mirrors share V(t^{-1}) up to the sign of coefficients for chiral pairs like $3_1 and $5_1, while amphichiral knots like $4_1 satisfy V(t) = V(t^{-1}).[17][15][18][19]| Knot Notation | Name | Jones Polynomial V(t) |
|---|---|---|
| $0_1 | Unknot | $1 |
| $3_1 | Trefoil | t + t^3 - t^4 |
| $4_1 | Figure-eight | t^{-2} - t^{-1} + 1 - t + t^2 |
| $5_1 | Cinquefoil | t^2 + t^4 - t^5 + t^6 - t^7 |
| $5_2 | Three-twist | t - t^2 + 2t^3 - t^4 + t^5 - t^6 |