Immersed Curve Invariants

• Immersed curve invariants are quantities assigned to curve immersions that remain unchanged under prescribed equivalence relations, capturing essential geometric and topological features.
• They are developed in diverse settings including the Heisenberg group, regular-homotopy of plane curves, finite-order diagrammatics, and invariants on immersed surfaces.
• These invariants facilitate practical classification and reconstruction in low-dimensional topology, linking techniques from differential geometry to Floer homology and related categorical models.

Immersed curve invariants are quantities, functions, or geometric objects attached to immersions of $1$-manifolds into a specified ambient space and preserved under a chosen equivalence relation. In the literature this phrase encompasses several distinct theories: local differential invariants for curves in the three-dimensional Heisenberg group, regular-homotopy invariants of generic immersions in the plane, finite-order invariants for triple-points-free curves, invariants of special curves on immersed surfaces under isometries or conformal motions, and Floer-theoretic immersed multicurves with local systems that encode invariants of $3$-manifolds, tangles, and cobordisms (Chiu et al., 2013, Lanzat et al., 2011, Vassiliev, 2014, Cohen et al., 10 Jan 2025).

1. Differential invariants and local equivalence in the Heisenberg group

For curves in the three-dimensional Heisenberg group $H_1$, the relevant equivalence problem is local equivalence under Heisenberg-rigid motions. The basic geometric structure is $\mathbb R^3$ with coordinates $(x,y,z)$, group law

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$

standard left-invariant frame

$$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$

contact form

$\theta_0=dz+x\,dy-y\,dx,$

horizontal bundle $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$, CR-structure $J_0(e_1)=e_2$, $3$0, Levi metric $3$1, and adapted metric

$3$2

With respect to $3$3, the frame $3$4 is orthonormal (Chiu et al., 2013).

A parametrized curve $3$5 is called horizontally regular if the projection of its velocity to $3$6 never vanishes. After reparametrization by horizontal arclength $3$7, one has $3$8. Along such a curve, the canonical moving frame is

$3$9

If $H_1$0 is the dual coframe, then pulling back the Maurer–Cartan form along the lifted curve $H_1$1 gives

$H_1$2

The two scalar invariants are therefore the $H_1$3-curvature and the $H_1$4-variation,

$H_1$5

In coordinates $H_1$6, before reparametrization, these become

$H_1$7

The fundamental theorem states that the pair $H_1$8 completely determines a horizontally regular curve up to an element of $H_1$9, and conversely any smooth functions $\mathbb R^3$0 arise from a unique such curve up to $\mathbb R^3$1. When $\mathbb R^3$2, the invariant reduces to the signed curvature of the projection to the $\mathbb R^3$3-plane, recovering the classical plane-curve reconstruction theorem up to $\mathbb R^3$4-translations and rotations (Chiu et al., 2013).

These invariants are local and differential rather than regular-homotopy-theoretic. Their role is analogous to curvature in Euclidean curve theory, but adapted to the contact and CR geometry of $\mathbb R^3$5.

2. Regular-homotopy invariants of generic plane immersions

For plane curves, the standard setting is a smooth immersion $\mathbb R^3$6 with $\mathbb R^3$7 for all $\mathbb R^3$8. A generic immersion has only finitely many transversal double points. The basic curvature density is

$\mathbb R^3$9

and Hopf’s Umlaufsatz asserts

$(x,y,z)$0

where $(x,y,z)$1 is the rotation number. Thus total curvature is invariant under arbitrary regular homotopies in the immersed category (Lanzat et al., 2011).

A nontrivial refinement is the Lanzat–Polyak family

$(x,y,z)$2

where $(x,y,z)$3 is the winding-number index of a point relative to the curve, $(x,y,z)$4 is the set of double points, and $(x,y,z)$5 is the crossing angle. The integral term alone changes under local modifications, but the discrete correction over double points compensates so that $(x,y,z)$6 is invariant under regular homotopy in the class of generic immersions. At $(x,y,z)$7, all correction terms vanish and $(x,y,z)$8, so the family is a quantization of total curvature. Differentiating at $(x,y,z)$9 yields

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$0

and hence the integral formula

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$1

for Arnold’s $$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$2-invariant (Lanzat et al., 2011).

Arnold’s invariants $$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$3 and $$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$4 are defined for generic immersed loops $$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$5 by axioms involving the standard curves $$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$6, direct and inverse self-tangencies, triple-point crossings, additivity under connected sum, and independence of orientation. They satisfy

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$7

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$8

and

$$(x_1,y_1,z_1)\ast(x_2,y_2,z_2)=(x_1+x_2,\;y_1+y_2,\;z_1+z_2+x_1y_2-y_1x_2),$$9

where $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$0 is the number of double points. A central computational tool is Viro’s formula,

$$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$1

with $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$2 the complement components, $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$3 their winding numbers, and $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$4 the double-point indices. The rotation number is likewise recovered from winding data by

$$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$5

These formulas convert regular-homotopy invariants into explicit combinatorial sums on the planar immersion (Mai, 2022, Mai, 2022).

For immersions in $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$6, Cieliebak–Frauenfelder–van Koert’s invariants are

$$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$7

and

$$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$8

where $$e_1=\partial_x+y\partial_z,\qquad e_2=\partial_y-x\partial_z,\qquad T=\partial_z,$$9. Their behavior under $\theta_0=dz+x\,dy-y\,dx,$0-bifurcation is explicit. For a generic immersion $\theta_0=dz+x\,dy-y\,dx,$1 and any $\theta_0=dz+x\,dy-y\,dx,$2, a $\theta_0=dz+x\,dy-y\,dx,$3-bifurcation $\theta_0=dz+x\,dy-y\,dx,$4 satisfies

$\theta_0=dz+x\,dy-y\,dx,$5

with equality exactly for minimal double-point bifurcations, and in general

$\theta_0=dz+x\,dy-y\,dx,$6

Moreover,

$\theta_0=dz+x\,dy-y\,dx,$7

and $\theta_0=dz+x\,dy-y\,dx,$8 obeys the analogous lower bound and exact formula (Mai, 2022).

A frequent source of confusion is that $\theta_0=dz+x\,dy-y\,dx,$9 itself is not invariant under all local singular events: it jumps by $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$0 under direct self-tangencies. The regular-homotopy-invariant content emerges either by restricting the class of moves or by passing to corrected expressions such as $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$1.

3. Finite-order, diagrammatic, and graph-theoretic invariants

Vassiliev’s theory of finite-order invariants for plane curves reformulates immersed-curve invariants through the topology of the discriminant. A doodle is a smooth map $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$2 such that no three distinct points $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$3 satisfy any of

$\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$4

An I-doodle is an immersed doodle, so $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$5 everywhere and only triple points are forbidden. If $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$6 and $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$7 is the discriminant of maps violating these conditions, Vassiliev resolves $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$8 simplicially and filters the resolution $\xi_0=\ker\theta_0=\mathrm{span}\{e_1,e_2\}$9 by singularity complexity. An invariant $J_0(e_1)=e_2$0 has order $J_0(e_1)=e_2$1 if its Poincaré–Lefschetz dual can be represented by a cycle supported in the $J_0(e_1)=e_2$2th filtration stage $J_0(e_1)=e_2$3 (Vassiliev, 2014).

The combinatorics replacing chord diagrams are triangular diagrams and connected $J_0(e_1)=e_2$4-hypergraphs. For an ordered partition $J_0(e_1)=e_2$5, an $J_0(e_1)=e_2$6-clique is a collection of $J_0(e_1)=e_2$7 points on $J_0(e_1)=e_2$8 partitioned into groups of sizes $J_0(e_1)=e_2$9, with complexity $3$00. Over each $3$01-clique $3$02, the simplicial resolution yields an order complex $3$03, and the visible blocks $3$04 assemble into a spectral sequence. Only connected $3$05-hypergraphs contribute to the top-dimensional Borel–Moore homology; disconnected splittings produce “4T-type” relations (Vassiliev, 2014).

For doodles, there are no invariants of order $3$06, and there is exactly one nontrivial invariant of order $3$07, represented by the “two-alternating-triangles” triangular diagram on six points. The computational procedure is explicit: list $3$08-configurations of complexity $3$09, compute $3$10, assemble the $3$11-page, compute horizontal differentials, and read off surviving Borel–Moore classes in $3$12. In low degrees this gives no invariants in degrees $3$13, exactly one in $3$14 for doodles, and for immersions one in degree $3$15 for Arnold’s strangeness, one in $3$16, and five in $3$17. The same framework extends to immersions avoiding $3$18-fold points for any $3$19 (Vassiliev, 2014).

A different diagrammatic direction appears in immersed graphs in $3$20. For any $3$21 and any $3$22-chord diagram $3$23 on $3$24, every generic immersion

$3$25

of the complete graph on $3$26 vertices contains a $3$27-cycle $3$28 whose induced chord diagram $3$29 has a sub-chord diagram equivalent to $3$30. For $3$31, the averaged second-Conway-coefficient invariant is defined by

$3$32

for a generic plane immersion with $3$33 crossings, and then

$3$34

The main congruence is

$3$35

equivalently $3$36, for every generic immersion $3$37. This is presented as a two-dimensional analogue of the Conway–Gordon phenomenon in spatial graph theory (Sakamoto et al., 2012).

4. Invariants for special curves on immersed surfaces

When a curve lies on an immersed surface in $3$38, the invariant problem depends on the class of ambient surface transformations. For a unit-speed space curve $3$39, the Frenet frame $3$40 determines the osculating, normal, and rectifying planes. A rectifying curve is characterized by

$3$41

If $3$42 is an isometry of smooth surfaces in $3$43, then $3$44 preserves dot products and cross products, carries the Frenet frame of a curve $3$45 to the Frenet frame of $3$46, and preserves the rectifying decomposition: $3$47 Thus the image of a rectifying curve is again rectifying, with the same coefficient functions $3$48 and $3$49. The normal component of the position vector is also preserved: $3$50 In the coordinate expression derived in the paper, the right-hand side depends on the first fundamental form coefficients $3$51, second derivatives of the parametrization, and the curve parameters $3$52, and equality follows from preservation of the first fundamental form under the isometry (Shaikh et al., 2018).

For normal curves on immersed surfaces, the ambient definition is

$3$53

Along an arc-length parametrized curve on a surface, the acceleration decomposes as

$3$54

with normal curvature

$3$55

If $3$56 is a surface isometry, the image of a normal curve satisfies the same normal-curve equation if and only if the normal curvature is preserved along the tangent direction: $3$57 equivalently $3$58 in terms of the shape operator. The paper also gives deviations

$3$59

so failure of invariance is governed linearly by the change in normal curvature (Shaikh et al., 2019).

Under conformal transformations $3$60, the first fundamental forms satisfy

$3$61

for a positive dilation factor $3$62. If $3$63 is a normal curve, the paper derives an invariant-sufficient condition for $3$64 to remain normal and computes the normal- and tangential-component deviations. In particular,

$3$65

where

$3$66

The tangential deviation is stated as

$3$67

with $3$68 quadratic in $3$69 and $3$70 linear in $3$71, and the geodesic curvature transforms by

$3$72

If $3$73, these formulas recover the isometric case; if $3$74, they reduce to the homothetic case (Lone, 2019).

These surface-theoretic invariants are neither regular-homotopy invariants nor purely local differential invariants of the ambient curve alone. They encode compatibility between the curve and the intrinsic or conformal geometry of the supporting surface.

5. Floer-theoretic immersed multicurves with local systems

In low-dimensional topology, immersed curve invariants become categorical objects rather than scalar functions. For a compact, oriented $3$75-manifold $3$76 with torus boundary, bordered Heegaard Floer theory associates a type-$3$77 module $3$78 over the torus algebra $3$79. Hanselman–Rasmussen–Watson identify $3$80, up to homotopy, with an immersed multicurve $3$81, where $3$82 is the once-punctured torus $3$83, decorated by a finite $3$84-local system $3$85. The underlying immersion is a finite collection of embedded loops and arcs with only transverse self-intersections, and each connected component carries a representation $3$86. Crossover arrows between parallel push-offs encode the local systems (Cohen et al., 10 Jan 2025).

This geometric model is functorial with respect to the partially wrapped Fukaya category $3$87. The morphism space between objects $3$88 and $3$89 is

$3$90

and for sufficiently large wrapping this stabilizes to the transverse intersection Floer complex. The $3$91-operations

$3$92

count rigid perturbed pseudo-holomorphic $3$93-gons. The main composition theorem identifies morphisms between bordered Floer invariants with morphisms in the Fukaya category: $3$94 and algebraic composition corresponds to Fukaya-category composition by $3$95 (Cohen et al., 10 Jan 2025).

An analogous classification exists for $3$96-ended tangles. Bar-Natan’s universal invariant of an oriented $3$97-ended tangle $3$98 is reduced to a complex $3$99 over the algebra

$H_1$00

and over a field every bigraded $H_1$01-complex is homotopy-equivalent to a unique multicurve $H_1$02 on the $H_1$03-punctured sphere, consisting of immersed loops or arcs with local systems on loop components. The resulting immersed-curve invariant is

$H_1$04

From $H_1$05, one further constructs two mapping-cone invariants $H_1$06 and $H_1$07, with $H_1$08, giving immersed-curve models for reduced and unreduced Khovanov homology. Their gluing theorems identify the homology of links obtained by gluing tangles with wrapped Floer homology groups of the corresponding multicurves: $H_1$09 and similarly for $H_1$10 and $H_1$11 (Kotelskiy et al., 2019).

In this Floer-theoretic sense, an immersed curve invariant is not primarily a number. It is an object in a geometric model for an algebraic theory, and its invariance is up to homotopy of the underlying curves together with equivalence of local systems.

6. Applications, obstructions, and scope of the notion

The Floer-theoretic formalism supports applications to $H_1$12-manifolds and concordance. For the knot complement of the mirror of $H_1$13, the immersed multicurve model detects that two distinct slice disks $H_1$14 induce different maps

$H_1$15

because the corresponding bounding chains determine different intersection points in $H_1$16. The same formalism yields Whitehead-double satellite obstructions and splice-cobordism distinctions, and it packages secondary $H_1$17-invariants through cones $H_1$18 that are again represented by immersed curves in low-complexity cases (Cohen et al., 10 Jan 2025).

Immersed curves also appear in sliceness detection for knots in homology spheres. If $H_1$19 is a nontrivial knot smoothly slice in $H_1$20 and $H_1$21, then the dual knot $H_1$22 is slice in a contractible $H_1$23-manifold $H_1$24 with boundary $H_1$25. For the positively clasped Whitehead pattern $H_1$26, the satellite

$H_1$27

is slice in $H_1$28 but can be shown not to be slice in $H_1$29. The obstruction uses the Heegaard Floer invariants

$H_1$30

which satisfy $H_1$31 for any embedded surface $H_1$32 bounding $H_1$33. If $H_1$34 is slice in $H_1$35, then $H_1$36 for all $H_1$37. In the immersed-curve model, each $H_1$38 corresponds to an intersection point $H_1$39, and

$H_1$40

the Alexander-height coordinate of $H_1$41 in the covering strip. The key lemma states that any generator lying in an acyclic summand of the simplified knot Floer complex of $H_1$42 produces two surviving intersection points of different Alexander heights after pairing with the skewed Whitehead pattern; consequently at least one $H_1$43 is nonzero, obstructing sliceness in the collar (Mcconkey et al., 29 Jun 2026).

The broader literature therefore uses the same phrase for different invariant-theoretic regimes. In the cited works, the preserved relation may be Heisenberg-rigid motion (Chiu et al., 2013), regular homotopy of generic immersions (Lanzat et al., 2011), avoidance of discriminant strata of specified multiplicity (Vassiliev, 2014), isometry or conformal motion of surfaces (Shaikh et al., 2018, Lone, 2019), or homotopy/equivalence of immersed multicurves with local systems in Fukaya-type categories (Cohen et al., 10 Jan 2025, Kotelskiy et al., 2019). A plausible implication is that “immersed curve invariant” functions less as the name of a single theory than as a family of theories unified by the representation of geometric or topological data on immersed $H_1$44-manifolds, with the governing notion of equivalence supplied by the ambient geometry or the relevant homological formalism.