Smith Immersions

• Smith Immersions are generic immersions of S1 into S2 with only transverse double points, yielding 4-valent cellular maps.
• The approach utilizes permutation encodings and Frobenius-Burnside methods to classify immersions under specific orientation and checkerboard constraints.
• The framework extends to higher genus surfaces and connects with knot theory by analyzing checkerboard colorings and alternating knot diagrams.

Searching arXiv for the cited paper and closely related immersed-curve/map enumeration work. “Smith Immersions” is not a standard term in the literature on immersed curves or topological maps. Under the interpretation adopted here, it denotes generic immersions of a circle $S^1$ into the $2$-sphere $S^2$ with only transverse double points, i.e. spherical curves with no triple points and no tangencies. In this setting, each immersion determines a connected $4$-valent cellular map, and the classification and enumeration problem reduces to orbit counting for permutation data subject to one-componentness and genus constraints. The systematic treatment of this correspondence, including orientation variants, checkerboard structures, and extensions to higher genus surfaces, is developed in “Maps, immersions and permutations” (Coquereaux et al., 2015).

1. Terminology and geometric-combinatorial correspondence

A generic immersion $f:S^1 \to \Sigma$, where $\Sigma$ is an oriented closed surface of genus $g$, and where only transverse double points are allowed, produces a $4$-valent embedded graph. Its vertices are the double points, its edges are the curve arcs between crossings, and its faces are the connected components of $\Sigma \setminus f(S^1)$. The framework is restricted to cellular immersions, meaning that every face is an open disk (Coquereaux et al., 2015).

For an immersion with $n$ double points, the associated map has

• $2$0 vertices,
• $2$1 edges,
• $2$2 half-edges,

while the number of faces $2$3 is determined by Euler’s formula. The curve is one-component: the Euler traversal runs through all edges in one circuit. In genus $2$4, this is the standard setting of spherical curves.

The paper distinguishes the four orientation variants used in Arnold’s classification:

• OO: oriented circle into oriented sphere,
• UO: unoriented circle into oriented sphere,
• OU: oriented circle into unoriented sphere,
• UU: unoriented circle into unoriented sphere.

It also distinguishes three checkerboard notions. A map or immersion is bicolourable if its faces admit a $2$5-colouring; it is bicoloured if a particular such colouring is chosen; and it is general if no checkerboard constraint is imposed. In genus $2$6, every $2$7-valent planar map is checkerboard, whereas in higher genus checkerboard colourability is not automatic.

This suggests that the central object is not merely the immersed circle itself, but the combinatorial structure of the induced map together with the equivalence relation determined by orientation choices and, when relevant, checkerboard data.

2. Permutation models and genus constraints

The paper uses two permutation encodings, one on $2$8 and one on $2$9, depending on whether general or bicoloured maps are being treated (Coquereaux et al., 2015).

Encoding	Data	Main interpretation
$S^2$0 half-edge model	$S^2$1 of type $S^2$2, $S^2$3 of type $S^2$4	General maps; vertices, edge pairings, and faces via $S^2$5
$S^2$6 edge model	fixed $S^2$7 of type $S^2$8, with $S^2$9	Bicoloured maps; $4$0 and $4$1 encode white and shaded faces
$4$2 cyclic model	$4$3	OO general maps; genus read via an associated $4$4

In the $4$5 model, $4$6 records the cyclic order of the four half-edges at each vertex, and $4$7 is a fixed-point-free involution pairing half-edges into edges. Faces are the cycles of

$4$8

One-componentness is enforced by the condition

$4$9

meaning that $f:S^1 \to \Sigma$0 has exactly two cycles of length $f:S^1 \to \Sigma$1, corresponding to the two traversal directions of the unique circuit.

In the $f:S^1 \to \Sigma$2 bicoloured model, one fixes $f:S^1 \to \Sigma$3 of type $f:S^1 \to \Sigma$4, for example

$f:S^1 \to \Sigma$5

and a map is encoded by permutations $f:S^1 \to \Sigma$6 with

$f:S^1 \to \Sigma$7

Here $f:S^1 \to \Sigma$8 lists edges around white faces clockwise and $f:S^1 \to \Sigma$9 around shaded faces counterclockwise. If

$\Sigma$0

then one-componentness is the condition

$\Sigma$1

Euler’s formula gives

$\Sigma$2

so for a $\Sigma$3-valent map with $\Sigma$4 vertices,

$\Sigma$5

In permutation language, this becomes:

• in the $\Sigma$6 model,

$\Sigma$7

• in the $\Sigma$8 bicoloured model,

$\Sigma$9

The same principle governs the cyclic OO model: for $g$0, one constructs $g$1 and imposes

$g$2

3. Orbit counting, centralizers, and Frobenius-Burnside methods

Enumeration is formulated as a problem of counting orbits of constrained permutations under relabelling groups (Coquereaux et al., 2015). In the $g$3 model, one fixes $g$4 and considers

$g$5

the set of all fixed-point-free involutions $g$6, together with

$g$7

and

$g$8

The acting group is the centralizer

$g$9

of order

$4$0

acting by conjugation.

In the $4$1 bicoloured model, one fixes $4$2 and considers

$4$3

together with the genus-restricted subset $4$4. The relabelling group is

$4$5

the hyperoctahedral group, of order

$4$6

For OO general maps in the cyclic encoding, the relevant group is

$4$7

the subgroup commuting with $4$8 and permuting odd and even labels among themselves.

Burnside’s lemma yields the number of inequivalent immersions represented by a subset $4$9:

$\Sigma \setminus f(S^1)$0

For whole conjugacy classes, the paper uses Frobenius’ double-coset formula. If $\Sigma \setminus f(S^1)$1 is finite, $\Sigma \setminus f(S^1)$2 are subgroups, and $\Sigma \setminus f(S^1)$3, then the orbits of the adjoint action of $\Sigma \setminus f(S^1)$4 on the conjugacy class $\Sigma \setminus f(S^1)$5 are in bijection with double cosets $\Sigma \setminus f(S^1)$6, and

$\Sigma \setminus f(S^1)$7

where the sum runs over conjugacy classes $\Sigma \setminus f(S^1)$8 of $\Sigma \setminus f(S^1)$9.

Several subset sizes are given explicitly:

• $n$0,
• $n$1,
• $n$2,
• $n$3.

The asymptotic observation that almost all maps are asymmetric implies that the number of inequivalent maps is close to the raw set size divided by the relabelling group size. For example,

$n$4

by Stirling’s approximation.

4. Orientation classes, checkerboard symmetries, and Arnold-type relations

The counting problem depends on which source and target symmetries are identified. The paper analyzes the commuting involutions $n$5, $n$6, and $n$7, corresponding respectively to checkerboard colour swap, mirror, and orientation reversal, and organizes the resulting orbits into Arnold-type symmetry classes (Coquereaux et al., 2015).

The basic identifications are:

• source orientation: $n$8 versus $n$9,
• target orientation: mirror $2$00, realized by conjugation by $2$01,
• checkerboard colour swap $2$02, given in the bicoloured setting by

$2$03

These symmetries lead to exact relations among the cardinalities of the various oriented and unoriented, coloured and colourable classes. The paper states the following identities as Theorem 4:

$2$04

In genus $2$05, one has $2$06 and $2$07 because checkerboard colourability is automatic.

The paper also gives group-action realizations of these distinctions. For UOc, after restricting $2$08 to a left coset

$2$09

orbits under $2$10 correspond to UOc, while orbits under $2$11 correspond to OOc. For OO in the cyclic encoding, $2$12 acts on the class $2$13, with orientation reversal $2$14 and mirror $2$15.

A plausible implication is that the orientation-sensitive and checkerboard-sensitive counts are best understood not as separate enumeration problems, but as quotients of a common permutation model by different involutive symmetries.

5. Enumerative results and explicit examples

For spherical curves, the paper gives exact counts up to $2$16, and values for $2$17 in some cases are listed but marked as awaiting confirmation (Coquereaux et al., 2015). For $2$18, the genus-$2$19 counts are:

• OO: $2$20,
• UO: $2$21,
• OU: $2$22,
• UU: $2$23,
• UOc: $2$24.

These sequences agree with known OEIS entries and with Arnold’s counts for spherical curves, while extending previous tables.

For OO, the total number of immersions over all genera, computed via the $2$25 method, is given for $2$26 by

$2$27

The paper also tabulates genus decompositions, including for example the OO genus-$2$28 sequence

$2$29

Three concrete permutation examples illustrate the framework.

For UO with $2$30, take

$2$31

Then

$2$32

so one-componentness holds. Moreover,

$2$33

has cycles $2$34, hence $2$35. Since

$2$36

one gets $2$37. This is the unique UO spherical immersion at $2$38.

For UO with $2$39, take

$2$40

Then

$2$41

and $2$42 decomposes into two $2$43-cycles, so the immersion is one-component. A direct check gives

$2$44

hence $2$45. This corresponds to one of the two UO spherical immersions at $2$46.

For UOc with $2$47, fix

$2$48

and choose

$2$49

Here

$2$50

and since $2$51 and $2$52,

$2$53

so $2$54. The paper identifies this as a bi-coloured alternating diagram of a spherical curve.

6. Higher genus, algorithmic realization, and links to knot theory

The permutation frameworks extend directly to immersions in closed oriented surfaces of genus $2$55 (Coquereaux et al., 2015). The genus condition is imposed by

• $2$56 in the $2$57 model,
• $2$58 in the $2$59 bicoloured model,
• $2$60 in the cyclic OO model.

The principal structural difference from genus $2$61 is that checkerboard colourability is no longer automatic. Consequently, the $2$62 method counts bi-coloured and bicolourable immersions separately, and in higher genus one has

$2$63

in general.

The algorithmic workflow is explicit:

• fix $2$64 and $2$65;
• enumerate admissible $2$66 or $2$67 data;
• enforce one-componentness and genus constraints;
• compute orbits under the appropriate group, namely $2$68, $2$69, $2$70, or after gauge fixing, $2$71 or $2$72;
• use the identities of Theorem 4 to pass between OO, UO, OU, UU and their coloured or colourable variants.

The implementation described in the paper combines brute-force orbit enumeration in Mathematica and Magma up to $2$73 (and $2$74 in some spherical cases) with faster total counts via Frobenius double cosets, including OO totals up to $2$75.

The connection with knot theory comes through checkerboard structures. Bicoloured maps correspond to alternating knot diagrams, and the two checkerboard colourings correspond to interchange of over- and under-crossings. Within this framework, the permutation model can be used to detect bicolourability, filter out kinks through $2$76-cycles in $2$77 or $2$78, analyze prime versus composite diagrams via graph reducibility, and in principle support flype-equivalence reductions for alternating knot census.

The paper also notes a link with matrix integrals generating maps, where faces and components correspond to powers of $2$79 and $2$80, respectively. This suggests that the enumeration of immersed curves sits naturally at the interface of map combinatorics, low-dimensional topology, and algebraic enumeration.

A final caveat concerns scope. The counts are for cellular immersions, so noncellular immersions require additional surgery. For genus $2$81, the counts are given up to stable geotopy, in the sense stated in the paper: the minimal genus achieved by the curve is the essential datum, while handles may be added without changing the class.