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Linking number

In topology, the linking number is an integer-valued invariant that measures the degree of entanglement between two disjoint, oriented closed curves in three-dimensional Euclidean space, indicating how many times one curve winds around the other while accounting for orientation.^[1] It serves as a fundamental quantity in knot theory, remaining unchanged under continuous deformations of the curves that do not allow them to pass through each other.^[2] The concept was first introduced by Carl Friedrich Gauss in a 1833 diary entry, where he defined it as a measure of mutual encirclement for two closed curves, later formalized through the Gauss linking integral.^[3] This integral computes the linking number Lk(\gamma_1, \gamma_2) as

Lk(\gamma_1, \gamma_2) = \frac{1}{4\pi} \iint \frac{(\mathbf{r}_2 - \mathbf{r}_1) \cdot (\mathbf{r}_1' \times \mathbf{r}_2')}{|\mathbf{r}_2 - \mathbf{r}_1|^3} \, dt_1 \, dt_2,

where \mathbf{r}_1(t_1) and \mathbf{r}_2(t_2) parametrize the curves \gamma_1 and \gamma_2, and primes denote derivatives with respect to the parameters.^[3] The linking number is symmetric, Lk(\gamma_1, \gamma_2) = Lk(\gamma_2, \gamma_1), and changes sign if the orientation of one curve is reversed, but remains invariant under reversal of both.^[1] For unlinked curves separable by a plane, the value is zero.^[1] Computationally, the linking number can be determined by projecting the curves onto a plane and summing signed crossings: each crossing contributes +\frac{1}{2} or -\frac{1}{2} based on the over/under strands and orientations, yielding an integer total.^[2] Alternatively, numerical approximations of the Gauss integral are used for complex curves, with efficient algorithms like the Barnes-Hut method achieving O(N \log N) time complexity for discretized polylines of N segments.^[2] These methods ensure the invariant's practical utility in verifying topological properties without full simulations.^[2] Beyond pure mathematics, the linking number has significant applications in molecular biology, particularly in describing the topology of closed circular DNA molecules, where it quantifies supercoiling as Lk = Tw + Wr, with Tw (twist) representing helical turns and Wr (writhe) the geometric coiling.^[4] In DNA, the linking number is fixed unless strands are broken and resealed by enzymes like topoisomerases, influencing replication, transcription, and packaging.^[5] This topological framework extends to fields like polymer physics and topological quantum field theory, underscoring the linking number's interdisciplinary impact.^[3]

Core Concepts

Definition

The linking number is a topological invariant that quantifies the degree of entanglement between two disjoint oriented closed curves (components of a link), embedded in three-dimensional Euclidean space \mathbb{R}^3.^[6] For two such curves \gamma_1 and \gamma_2, denoted as their disjoint union \gamma_1 \sqcup \gamma_2, the linking number \mathrm{Lk}(\gamma_1, \gamma_2) is an integer that remains unchanged under ambient isotopy, a continuous deformation of the pair in \mathbb{R}^3 that does not allow the curves to pass through each other.^[6] Here, orientation assigns a consistent direction to each curve, enabling the distinction between positive and negative intertwinings. Intuitively, the linking number measures how many times one curve winds around the other, accounting for the handedness of crossings: a positive crossing occurs when the overpassing strand (from the perspective of \gamma_1 over \gamma_2) aligns right-handed with respect to the underpassing strand, contributing +1, while a left-handed crossing contributes -1.^[6] In a regular projection of the link onto a plane, where crossings are transverse and no three strands meet at a point, the linking number is given by half the sum of the signs of all crossings between \gamma_1 and \gamma_2:

\mathrm{Lk}(\gamma_1, \gamma_2) = \frac{1}{2} \sum \epsilon(c),

where the sum is over all such inter-component crossings c, and \epsilon(c) = \pm 1 denotes the sign based on the right-hand rule.^[6] The concept originated in a 1833 diary entry by Carl Friedrich Gauss, who proposed an integral formula to compute the linking of two closed curves in the context of electromagnetic interactions, though it remained unpublished during his lifetime.^[7] It was later formalized within the emerging field of knot theory by Peter Guthrie Tait in the late 19th century, as part of his systematic classification of knots and links.^[7]

Invariance Proof

The linking number \mathrm{Lk}(\gamma_1, \gamma_2) of two oriented, disjoint closed curves \gamma_1 and \gamma_2 in \mathbb{R}^3 is a topological invariant, remaining constant under ambient isotopies that do not allow the curves to pass through each other.^[8] This invariance follows from the fact that any such isotopy induces a sequence of Reidemeister moves on regular projections of the link, and the linking number—defined as half the sum of signed crossings between the components in a regular diagram—is preserved under these moves.^[8] To establish this, consider a regular projection of the link onto a plane, where the linking number is computed from the signed inter-component crossings: positive for right-handed overcrossings and negative for left-handed ones.^[9] Reidemeister moves type I, II, and III generate the equivalence classes of link diagrams under ambient isotopy.^[8] A type I move introduces or removes a twist in a single strand, creating or eliminating a self-crossing within one component. Since this does not involve inter-component crossings, the sum of signed crossings between \gamma_1 and \gamma_2 remains unchanged, preserving the linking number.^[8] A type II move brings two parallel strands from different components adjacent, creating or removing two inter-component crossings of opposite signs (one positive and one negative). These signs cancel in the sum, adding zero to the total and thus leaving the linking number invariant.^[8] A type III move slides one strand over or under a crossing formed by the other two strands, reconfiguring three crossings without altering their signs relative to the components. The algebraic sum of the signed inter-component crossings before and after the move is identical, ensuring no change in the linking number.^[8] An alternative geometric perspective on invariance arises from viewing the linking number as the algebraic intersection number of \gamma_2 with any Seifert surface bounded by \gamma_1. Seifert surfaces exist for any oriented knot and are unique up to isotopy, and the signed intersection number is preserved under continuous deformations of the curves that keep them disjoint.^[10] A key theorem equates the linking number to the degree of the Gauss map G: [T^2](/page/T+2) \to S^2, defined by G(s,t) = \frac{\gamma_1(s) - \gamma_2(t)}{|\gamma_1(s) - \gamma_2(t)|}, where T^2 = S^1 \times S^1 parameterizes the curves. This degree is well-defined and invariant under homotopy, confirming the topological constancy of \mathrm{Lk}(\gamma_1, \gamma_2).^[11]

Computation Techniques

Projection Method

The projection method for computing the linking number relies on obtaining a regular diagram of the oriented link by projecting the two closed curves, γ₁ and γ₂, onto a plane such that no three strands meet at a single point and all crossings are transverse, with over and under information explicitly indicated.^[12] In this diagram, only the crossings where one strand from γ₁ intersects a strand from γ₂ (inter-component crossings) are considered, ignoring self-crossings within each component.^[6] To assign signs to these inter-component crossings, the orientations of both curves must be specified. A crossing is deemed right-handed and assigned ε_i = +1 if the overcrossing strand (say from γ₁) runs from bottom-left to top-right while the undercrossing strand (from γ₂) runs from top-left to bottom-right, following the standard convention in oriented diagrams; conversely, a left-handed crossing, where the overstrand runs from top-left to bottom-right and the understrand from bottom-left to top-right, is assigned ε_i = -1.^[12] This signing is determined locally at each crossing using the right-hand rule: thumb along the overstrand in its direction, with fingers curling toward the understrand's direction for a positive sign.^[13] The linking number is then given by the formula

\text{Lk}(\gamma_1, \gamma_2) = \frac{1}{2} \sum \varepsilon_i,

where the sum is taken over all inter-component crossings in the diagram, with each ε_i = ±1.^[6]^[13] The procedure is as follows: (1) Select a generic projection direction to yield a regular diagram; (2) Orient both curves consistently; (3) Locate all inter-component crossings and determine over/under relations; (4) Assign signs ε_i to each based on the handedness relative to orientations; (5) Compute the sum of signs and divide by 2 to obtain the integer linking number.^[12] The result is independent of the specific projection direction chosen, as changes under regular isotopies preserve the total signed count due to the invariance of the linking number.^[13] For illustration, consider a simple two-bridge link diagram, such as that of the Hopf link formed by two oriented circles interlocked once, projected onto the plane to show exactly two inter-component crossings. At each crossing, the over/under configuration and orientations are examined to assign +1 or -1, following the steps above, with the signs then summed and halved to yield the linking number.^[6]

Gauss Integral Method

The Gauss integral method provides a differential geometric formulation of the linking number for two disjoint oriented closed curves \gamma_1 and \gamma_2 in \mathbb{R}^3, expressed as a double line integral over their parametrizations. In 1833, Carl Friedrich Gauss introduced this integral in the context of electrodynamic interactions, defining the linking number \mathrm{Lk}(\gamma_1, \gamma_2) as

\mathrm{Lk}(\gamma_1, \gamma_2) = \frac{1}{4\pi} \int_0^{2\pi} \int_0^{2\pi} \frac{ (\mathbf{r}_2(t_2) - \mathbf{r}_1(t_1)) \cdot (\mathbf{r}_1'(t_1) \times \mathbf{r}_2'(t_2)) }{ |\mathbf{r}_2(t_2) - \mathbf{r}_1(t_1)|^3 } \, dt_1 \, dt_2,

where \mathbf{r}_1(t_1) and \mathbf{r}_2(t_2) are periodic parametrizations of \gamma_1 and \gamma_2 by arc length or unit speed, respectively.^[14] This integral measures the oriented linking between the curves by quantifying the average solid angle subtended by one curve at points along the other or, equivalently, the winding of one curve around the other in space. For closed disjoint curves, it yields an integer value representing the topological degree to which \gamma_1 threads through \gamma_2.^[14] The method's advantages include its coordinate independence, making it invariant under rigid motions of the ambient space, and its exactness for smooth curves, which facilitates analytical computations and numerical approximations in settings like molecular modeling.^[14] It is particularly useful in computational geometry for assessing entanglement in biomolecular structures, such as DNA supercoiling, where discrete approximations can be integrated efficiently.^[15] A sketch of the derivation traces back to the degree of the Gauss map, which sends pairs of points (t_1, t_2) on the torus \gamma_1 \times \gamma_2 to the normalized direction (\mathbf{r}_1(t_1) - \mathbf{r}_2(t_2))/|\mathbf{r}_1(t_1) - \mathbf{r}_2(t_2)| on the unit sphere; the linking number equals this map's degree, computed via the integral of the pullback of the sphere's area form. Gauss originally motivated it through an analogy to the Biot-Savart law in electromagnetism, where the integral represents the magnetic flux linkage between current-carrying loops. The formulation assumes the curves are disjoint, as the integrand exhibits singularities when \mathbf{r}_1(t_1) = \mathbf{r}_2(t_2), rendering the integral improper or undefined for intersecting curves.^[14]

Properties and Examples

Key Properties

The linking number possesses several fundamental algebraic and topological properties that underscore its role as a classical link invariant. One key property is its additivity: for disjoint oriented curves \gamma_1 and \gamma_1' in \mathbb{R}^3 minus another oriented curve \gamma_2, the linking number satisfies \mathrm{Lk}(\gamma_1 \cup \gamma_1', \gamma_2) = \mathrm{Lk}(\gamma_1, \gamma_2) + \mathrm{Lk}(\gamma_1', \gamma_2).^[16] This additivity extends naturally to multi-component links, allowing the total linking number to be decomposed across components. The linking number is symmetric with respect to interchanging the components of the link, meaning \mathrm{Lk}(\gamma_1, \gamma_2) = \mathrm{Lk}(\gamma_2, \gamma_1) for oriented curves \gamma_1 and \gamma_2.^[16] However, it exhibits antisymmetry under orientation reversal of one component: reversing the orientation of \gamma_1 yields \mathrm{Lk}(-\gamma_1, \gamma_2) = -\mathrm{Lk}(\gamma_1, \gamma_2).^[16] These symmetry properties arise from the diagrammatic or integral definitions and hold invariantly under ambient isotopies. For a single unframed knot, the self-linking number is undefined, as the linking number requires two distinct components. In the context of framed knots, however, the self-linking number is well-defined as the linking number between the knot and a parallel push-off curve determined by the framing, and it equals the framing integer representing the twist of the framing vector field around the knot.^[17] For the blackboard framing induced by a knot diagram, this self-linking number coincides with the writhe of the diagram, the signed sum of its crossings.^[17] Topologically, the linking number can be interpreted as a pairing in homology: for a two-component link L = L_1 \cup L_2 in S^3, \mathrm{Lk}(L_1, L_2) is the unique integer k such that the homology class [L_2] = k [m_1] in H_1(S^3 \setminus L_1; \mathbb{Z}), where [m_1] is the class of a meridian to L_1.^[18] This formulation positions the linking number as the intersection form on the first homology of the link complement. Regarding uniqueness, the linking number is the primitive Vassiliev invariant of degree 1 for two-component oriented links, in the sense that any other such invariant is a scalar multiple of it.^[19] It also serves as a complete invariant for link homotopy of two-component links, classifying them up to homotopy by its value.^[18]

Illustrative Examples

The Hopf link represents the simplest non-trivial two-component link in three-dimensional space, formed by two interlocked circles that cannot be separated without breaking one. With consistent orientations on both components, the linking number of the Hopf link is +1; reversing the orientation of one component yields -1, illustrating how the invariant depends on the chosen directions along the curves. This example can be visualized through a standard projection diagram showing a single positive crossing, where the signed crossing contributes to the overall linking number when computed via the projection method.^[20] Another basic example is Solomon's knot, also known as the (2,2)-torus link and denoted as L_{4a1} in Rolfsen's notation, which consists of two unknotted components woven around each other twice. The linking number for this link is 2, reflecting the double interlinking of the components regardless of orientation choices that preserve the absolute value.^[21] In three-dimensional renderings, Solomon's knot appears as a symmetric figure resembling an ancient decorative motif, with projections revealing two positive crossings that emphasize its higher degree of entanglement compared to the Hopf link.^[22] The Whitehead link provides a striking counterexample to the idea that visual interlocking implies non-zero linking, as its two components appear intertwined but possess a linking number of 0. Discovered by J. H. C. Whitehead in 1934, this link demonstrates that the linking number does not detect all forms of linking, necessitating additional invariants for full classification.^[23] Diagrams of the Whitehead link typically show a more complex projection with both positive and negative crossings that cancel out in the linking calculation, while 3D models highlight the subtle clasping without net linkage.^[24] For multi-component links, the Borromean rings offer an intriguing case of three mutually interlocked circles where each pair has a pairwise linking number of 0, yet the overall configuration is non-trivial and inseparable. This property underscores the limitations of pairwise linking numbers, introducing the need for higher-order invariants like Milnor's triple linking number, which is non-zero for the Borromean rings.^[22] Visual aids such as signed crossing diagrams reveal no net pairwise crossings, but 3D renderings capture the interdependent triple entanglement that falls apart only if any one ring is removed. The additivity of linking numbers in multi-component cases means the total linking for Borromean rings sums to zero across pairs, aligning with its pairwise values.^[20] Early explorations of linking numbers in knot theory trace back to the late 19th century, when Peter Guthrie Tait compiled the first systematic tabulations of knots and links in the context of vortex atom theory, including basic examples like interlocked rings to study their topological properties.^[25] These historical efforts laid the groundwork for recognizing linking as a fundamental invariant, with Tait's diagrams providing initial visualizations of signed crossings for simple links.

Advanced Topics

Role in Quantum Field Theory

In abelian Chern-Simons theory, the expectation value of the product of two Wilson loops corresponding to disjoint closed curves equals the exponential of $2\pi i k times the linking number of the curves, where k is the level of the theory. This arises from the topological nature of the theory, where the path integral over gauge fields yields a phase factor directly proportional to the linking number \mathrm{Lk}, capturing the intertwined topology of the loops.^[26] In the quantum regime, this formulation highlights how linking numbers encode non-trivial braiding statistics for anyons emerging in the theory.^[27] Edward Witten's 1989 work established a connection between quantum field theory and knot invariants by showing that the path integral of Chern-Simons theory produces non-perturbative invariants for knots and links.^[28] In this framework, the linking number contributes to the abelian limit of these invariants, influencing the evaluation of knot polynomials through the topological term in the action.^[29] Specifically, for linked curves, the theory's observables incorporate \mathrm{Lk} as a phase in the partition function, bridging classical topology with quantum amplitudes.^[30] In quantum electrodynamics, the linking number appears analogously in the Aharonov-Bohm phase for a charged particle encircling a closed magnetic flux line, where the phase shift is $2\pi times the linking number between the particle's path and the flux tube multiplied by the reduced flux quantum.^[31] This topological phase persists even in field-free regions, mirroring how linking quantifies solenoid-particle entanglement.^[32] Similarly, magnetic helicity in electromagnetic fields relates to the average linking number of field lines, providing a gauge-invariant measure conserved in ideal magnetohydrodynamics, with quantum extensions in QED conserving helicity modulo anomalies.^[33] Topological quantum field theories (TQFTs), such as those based on Chern-Simons actions, incorporate the linking number modulo framing anomalies, where the self-linking of a knot depends on the choice of framing to resolve the theory's gravitational coupling. This anomaly shifts the effective linking number by an integer tied to the central charge, ensuring consistency under diffeomorphisms but requiring a framing to define observables precisely.^[34] In recent developments during the 2020s, linking numbers contribute to understanding braiding statistics in non-Abelian anyons within topological insulators. Such applications extend to fault-tolerant quantum computing, where topological invariants support anyon fusion and braiding in condensed matter systems.

Generalizations to Broader Structures

The linking number, originally defined for two oriented closed curves in three-dimensional space, extends to multi-component links through higher-order invariants that capture more intricate interdependencies. For three components, John Milnor introduced the triple linking number \bar{\mu}_{123} in 1957, which measures the extent to which the longitudes of the components penetrate deeper levels of the lower central series in the link group. This invariant is antisymmetric in its indices, invariant under link homotopy for distinct components, and defined modulo the greatest common divisor of the pairwise linking numbers when those are non-zero; it vanishes if the pairwise linking numbers are all zero unless the components exhibit higher-order braiding. Geometrically, \bar{\mu}_{ijk} can be computed from the intersection patterns of Seifert surfaces F_i, F_j, F_k bounding the link components, as \bar{\mu}_{ijk} \equiv m_{ijk}(F) - t_{ijk}(F) \pmod{\delta}, where t_{ijk}(F) counts signed triple points of the surfaces, m_{ijk}(F) arises from coefficients in the Magnus expansion of boundary words along double curves, and \delta is the gcd of pairwise links.^[35] For links involving surfaces rather than curves, the linking number generalizes to intersection numbers in three-manifolds. In the three-sphere, the linking number between a knot and a Seifert surface for another component equals the algebraic intersection number of the knot with the surface, providing a topological measure of how the knot threads through the surface's bounded region. This extends to arbitrary three-manifolds, where the linking number between two curves is computed via the intersection number with a surface spanning one curve in the manifold's fundamental class, adjusted for homology; for instance, in irregular dihedral covers, explicit algorithms derive it from branched cover presentations. Such generalizations facilitate the study of link concordance and cobordism, where surface intersections detect obstructions beyond classical linking.^[36]^[37] In higher dimensions, linking invariants for codimension-2 submanifolds in \mathbb{R}^n or S^n analogize the classical case by quantifying mutual "threading" via degrees of Gauss maps or asymptotic intersections. For two disjoint (n-2)-spheres in S^n, the linking number is the degree of the map from their complement to S^1 induced by normalized position vectors, generalizing to non-Abelian gauge fields in higher-dimensional BF theories. A prominent example is the Hopf invariant in S^3, which for maps S^3 \to S^2 counts the linking of preimages of regular points and serves as a higher-dimensional linking measure for codimension-2 spheres, isomorphic to \mathbb{Z} for n=2 but torsion in higher cases. These invariants, extended via flat connections with nontrivial holonomy, apply to flows and foliations, yielding average asymptotic linking via Hopf-type integrals over submanifolds.^[38]^[39] The Sato-Levine invariant provides a refinement for higher links, particularly two-component links with even linking number, as a third-order Vassiliev invariant detecting triple intersections in immersed surfaces. Defined via the Blanchfield form or Whitney tower filtrations in four-manifolds, it extends to higher orders SL_{2n-1} as an obstruction to framing twisted Whitney towers, classifying string links and homology cylinders up to concordance. For classical links, it vanishes on higher-dimensional analogs with simply connected components but captures non-triviality in low dimensions, such as the Whitehead link.^[40]^[41] Post-2000 advances incorporate linking data into sophisticated invariants like Casson-Gordon signatures, which refine concordance obstructions for algebraically slice knots by evaluating metabolic forms over \mathbb{Z}[t^{\pm 1}]-covers, revealing non-slice examples via secondary linking obstructions in metabolizers. Similarly, refinements of Khovanov homology, such as triply-graded theories for links with involutions, enhance detection of linking by incorporating mutation invariance and equivariant structures, distinguishing periodic links through Burnside functors and localizations that encode pairwise and higher intercomponent relations.^[42]^[43] Despite these extensions, linking invariants have limitations: the triple linking number is ill-defined without vanishing pairwise links and fails to distinguish all homotopy classes, necessitating higher Milnor invariants or total quotients for multi-component links with six or more parts. Completeness requires combining with torsion-valued or homological invariants, as classical linking alone cannot resolve Brunnian links or detect all concordance classes.^[44]

References

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• Linking numbers are three-dimensional analogues of winding numbers, but the standard way of computing linking numbers involves projecting down to two ...
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There are several equivalent ways to calculate (and define) the linking number, all of which will be explored in this paper. The linking number, 𝜆(𝛾1,𝛾2), is a ...
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A link invariant defined for a two-component oriented link as the sum of +1 crossings and -1 crossing over all crossings between the two links divided by 2.
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None
- **Linking Numbers and Seifert Surfaces**: Milnor’s triple linking number (μ̅_ijk) for a link in the 3-sphere is related to Seifert surfaces (F_i, F_j, F_k) via their intersection patterns. It is expressed as μ̅_ijk ≡ m_ijk(F) - t_ijk(F) (mod δ), where δ is the greatest common divisor of pairwise linking numbers, t_ijk(F) is the signed count of triple points, and m_ijk(F) is a sum of coefficients from Magnus expansions of boundary words.
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[1011.6026] Higher Order Intersections in Low-Dimensional Topology
Nov 28, 2010 · We also define higher- order Sato-Levine and Arf invariants and show that these invariants detect the obstructions to framing a twisted Whitney ...
[43]
A formula for the generalized Sato-Levine invariant - IOPscience
Abstract. Let be the generalized Sato-Levine invariant, that is, the unique Vassiliev invariant of order 3 for two-component links that is ...
[44]
[PDF] NEW EXAMPLES OF NON–SLICE, ALGEBRAICALLY SLICE ...
Oct 12, 2001 · This was first made formal in Gilmer's work [7] where certain signatures of these knots were related to Casson-Gordon invariants. In the case we ...Missing: post- | Show results with:post-
[45]
[1908.00082] A refinement of Khovanov homology - arXiv
Jul 31, 2019 · We refine Khovanov homology in the presence of an involution on the link. This refinement takes the form of a triply-graded theory, arising from a pair of ...Missing: post- 2000
[46]
[PDF] On the Indeterminacy of the Triple Linking Number
Triple Linking Number- An integer valued invariant that gives information into how three components of a link are linked together. Linking Matrix- A matrix ...