Mobius transformations of polygons and partitions of 3-space

Jonathan Simon (University of Iowa); Joshua Tokle (University of Washington); Richard Randell (University of Iowa)

REVIEW

Mobius transformations of polygons and partitions of 3-space

Richard Randell (University of Iowa), Jonathan Simon (University of Iowa), Joshua Tokle (University of Washington)

Geometric Topology · Submitted 21 Feb 2006 · 9 references

Pith's verdict: A real counting theorem with a clean main argument, and the degenerate sphere-arrangement gap is patchable rather than fatal.

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CitationJonathan Simon (University of Iowa), Joshua Tokle (University of Washington), Richard Randell (University of Iowa). "Mobius transformations of polygons and partitions of 3-space." arXiv:math/0602466 (2006). https://pith.science/paper/TJCAO3TU

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Editor's note

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Here is the bottom line: this paper proves something new and likely true. For a fixed n-edge polygon K, polygonal inversion produces only polynomially many knot types, O(n^6), while the total number of n-edge knot types is exponential. So no n-edge polygon is universal for n >= 72 under a single inversion, and n >= 75 for the whole Mobius group. The main idea is simple and elegant: encode the reachable knot types by the complementary domains of an arrangement of at most n(n-3)/2 round spheres. What is good: the operation itself, inverting vertices and reconnecting with straight lines, is introduced clearly, and the 7-edge figure-eight example that can be inverted to all knot types representable with 7 or fewer edges is a nice concrete illustration. The exponential lower bound is quoted from Ernst-Sumners and Negami, not fitted, and the polynomial side is derived in the paper. The exact formula in Section 5, via Ziegler and Zivaljevic, is more than the main argument needs, but it is a legitimate addition and gives sharpness for generic arrangements. The citation pattern looks fine; none of the load-bearing counting is self-referential. The soft spot is exactly what the stress-test note identified. Proposition 3's induction assumes each new circle intersects the previous circles in at most 2(k-1) points. That is false when two distinct spheres induce the same circle on the next sphere, so circles in the induced arrangement can coincide. The paper notes that the spheres need not be mutually transversal but does not supply the degenerate-case argument. Since Theorem 4 is the load-bearing step for the O(n^6) bound, this is a real gap in the written proof, and a referee should ask for a patch. That said, the gap is local and I am fairly confident it closes. Delete duplicate circles and the induction runs; the region bound is monotone in the number of circles, so the theorem goes through. The authors should also double-check the parity remark in Section 6, but that is minor. The threshold comparisons at n=72 and n=75 appear correct. Verdict: this deserves a serious referee. The result is meaningful for geometric knot theory, the proof is mostly clean, and the missing piece is a one-paragraph robustness argument. I would send it out, and I would cite it if I worked on polygonal knots or knot energies.

Referee report

Significance

If the claims hold, the paper establishes a clear separation between two natural quantities: the number of n-segment knot types grows exponentially, while the polygonal-inversion orbit of any one polygon has only polynomial size. The bounds are explicit and parameter-free, and the concrete thresholds n = 72 and n = 75 are falsifiable statements. The elementary sphere-arrangement bound in Theorem 4 is of independent interest, and the exact formula in Theorem 5 is a nice application of known subspace-arrangement theory. The paper builds on standard external lower bounds (Ernst–Sumners, Negami) rather than on self-citations, and the comparison between upper and lower bounds is clean. The examples in Section 2 are instructive and provide useful intuition, and the computational confirmation via knot-energy software is a reasonable auxiliary check.

Major objections (1)

[Section 5, Proposition 3 and Theorem 4] The proof of Proposition 3 asserts that the last circle C_k meets the union of the previous circles in at most 2(k−1) points. This is false when C_k coincides with one of the earlier circles, which occurs for non-transversal sphere arrangements, for example when two distinct spheres S_i and S_j induce the same circle on S_k. In that case the intersection of C_k with the previous union is a whole circle, not a finite set. Since Theorem 4's induction passes through Proposition 3, the O(n^6) upper bound and Corollary 6 depend on this step. The manuscript explicitly acknowledges that spheres need not be mutually transversal but does not supply the degenerate-case argument. The gap appears patchable: after deleting duplicate circles, the induction for distinct circles applies, and a coincident circle adds no new complementary regions. The authors should add this argument or explicitly restrict to distinct circles and justify the reduction before the paper is accepted.

Minor objections (6)

[Section 5, Proposition 3] Please state explicitly that the collection of circles is either distinct or that repeated circles have been removed; as written, the phrase 'collection of k round circles' could be read as allowing coincident circles, making the claimed intersection bound false.
[Section 6, Theorem 7 and Remark] The bound (1/12)(√2^n − 4) is derived by substituting q = n/2 into the Ernst–Sumners formula, but for odd n the crossing number q is not an integer. The remark acknowledges this but does not display the adjusted bound; please spell out that the same conclusion, including the n = 75 case, still follows with the corrected odd-n estimate.
[Section 5, around Theorem 5] The presentation of the exact formula is very compressed: a reader not already familiar with Ziegler–Živaljević will have difficulty verifying the direct-sum formula and the example. Please add a sentence explaining the notation d(p), P_{<p}, and the join construction, or clearly label Theorem 5 as a quoted result from [9] with the necessary context.
[Throughout] There are several typographical errors, including 'M ¨OBIUS' in the title, 'transfmor' in the introduction, and inconsistent rendering of the mirror-image notation; a careful proofreading pass is needed.
[Sections 1 and 2.2] The abstract refers to transforming the figure-eight to 'a trefoil' or 'an unknot', while Section 2.2 lists both left-handed and right-handed trefoils; please reconcile these descriptions for precision.
[Section 7, Theorem 8] The proof would be clearer if it explicitly noted that ρ is an involution, so that (μ∘ρ)(ρ̂(K)) = μ̂(K); currently this identity is implicit.

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The paper

The paper claims

A fixed n-edge polygon can be inverted into at most polynomially many knot types, while all n-edge knot types grow exponentially, so no large polygon is universal under inversion.

The reading

This paper studies a natural operation on polygonal knots: invert all vertices through a sphere, then reconnect them with straight segments. The authors prove that from a fixed n-edge knot, a single inversion can produce only polynomially many—at most about $n^6/24$—knot types, whereas the number of knot types representable by n-edge polygons grows exponentially in n. The reason is that the knot type of an inverted polygon can change only when the inversion center crosses one of the round spheres determined by four vertices of the original knot; the number of distinct knot types is therefore bounded by the number of regions into which at most $n(n-3)/2$ spheres divide space. The paper counts those regions: any $m$ round spheres cut $\mathbb{R}^3$ into at most $2\binom{m}{3}+2m$ complementary domains. Consequently no 72-edge polygon can reach all n-edge knot types by a single inversion, and no 75-edge polygon can do so using the full Möbius group.

What carries the argument

The engine of the proof is a 'danger sphere' construction. For each pair of non-adjacent edges of $K$, the four endpoints determine a unique round sphere or plane; if the inversion center lies off that sphere, the corresponding two edges of the inverted polygon cannot intersect. Therefore the collection of at most $n(n-3)/2$ such spheres cuts $\mathbb{R}^3$ into complementary domains, and within each domain the knot type of $\hat\rho(K)$ is constant, with a crossing of a sphere the only possible way the type can change. The paper proves that $m$ round spheres, even with degenerate intersections, divide space into at most $2\binom{m}{3}+2m$ regions by induction: each new sphere is cut into at most $k^2-k+2$ regions by the circles from earlier spheres. For the exact count, the paper uses the poset of connected components of sphere intersections to compute the cohomology of the complement, showing the generic transversal case attains the maximum.

Extended reading notes

Core claim

The paper establishes that polygonal inversions acting on a given n-edge knot $K$ can yield only polynomially many knot types, while there are exponentially many n-segment knot types. More precisely, a single inversion can realize at most $\frac{1}{24}(n^6-9n^5+21n^4+9n^3-22n^2-96n)$ knot types, and the total number of n-edge knot types is at least $\frac{1}{12}(\sqrt{2}^{\,n}-4)$. These bounds imply that for $n\ge 72$ no single inversion can reach every n-edge knot type, and for $n\ge 75$ the same obstruction holds when arbitrary compositions of inversions are allowed. The paper also exhibits a 7-segment figure-eight knot that is universal among all knots realizable with at most 7 segments, showing that such universality is a small-$n$ phenomenon rather than a general one.

Load-bearing premise

The polynomial bound rests on the assertion that any collection of $m$ round spheres—including degenerate ones that are tangent or coincide—divides $\mathbb{R}^3$ into at most $2\binom{m}{3}+2m$ regions; that step is stated for all collections, but the written induction does not spell out the degenerate cases.

Editorial extensions

If this is right

For any fixed n-edge polygon $K$, the set of knot types obtainable by single inversions has size at most the degree-six polynomial in Corollary 6, so as $n$ grows it is negligible compared with the exponential family of all n-edge knots.
No polygon with 72 or more edges can be universal under single inversions; with 75 or more edges, none can be universal even when arbitrary compositions of inversions are allowed.
Because composing Möbius transformations at most doubles the number of knot types, the action of the whole Möbius group on a polygon is not substantially more powerful than a single inversion for producing new knot types.
The exact count of complementary domains shows that the worst-case number of regions is attained by generic transversal sphere arrangements, so degenerate configurations do not create additional knot types beyond the bound.
The 7-segment figure-eight that realizes every knot type realizable with at most seven edges is a small-$n$ exceptional case, not a pattern that continues to larger edge counts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $n=72$ threshold is almost certainly not the true cutoff: the exponential lower bound used is very crude, so stronger counts of n-edge knots would likely push the non-universality threshold much lower.
The same sphere-arrangement mechanism should apply to any operation on polygons that depends on a parameter point in $\mathbb{R}^3$ and changes only when four vertices become cospherical with it; the $O(n^6)$ chamber bound would then be generic.
A concrete algorithmic consequence is that searches for low-energy Möbius-distorted polygons should start from many distinct polygons rather than relying on inversions of a single one, since most knot types are unreachable from any fixed starting polygon.
A systematic tabulation of reachable knot types per polygon for small $n$ could reveal how quickly the maximal reachable-set size grows and test how soon the polynomial bound becomes dominant.

Original abstract

The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type. For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group. The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3. We show the number of domains is at most polynomial in the number of spheres. In the analysis, we obtain an exact formula for the number of complementary domains. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,475 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.72 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.96 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K. Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon. This may be a different knot type. For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group. The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3. We show the number of domains is at most polynomial in the number of spheres. In the analysis, we obtain an exact formula for the number of complementary domains. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The image of a polygonal knot K under a spherical inversion of R^3 (union infinity) is a simple closed curve made of arcs of circles, having the same knot type as the mirror image of K.	background 0.90	0.20	0.05	asserted 1.00
s1	Suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon.	method 0.87	0.27	0.02	asserted 0.50
s2	This may be a different knot type.	background 0.46	0.19	0.05	hedged possibility 1.00
s3	For example, a certain 7-segment figure-8 knot can be transformed to a figure-8 knot, right and left handed trefoils, or an unknot, by selecting different inverting spheres.	result 0.76	0.64	0.05	asserted 0.92
s4	Which knot types can be obtained from a given original polygon K under this process?	gap 0.54	0.07	0.04	unclear 0.95
s5	We show that for large n, almost all n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Mobius group.	result 0.99	0.94	0.72	asserted 1.00
s6	The number of knot types arising from an n-vertex polygon is bounded by the number of complementary domains of a certain system of n(n-3)/2 round 2-spheres in R^3.	result 0.91	0.86	0.18	asserted 1.00
s7	We show the number of domains is at most polynomial in the number of spheres.	result 0.98	0.93	0.11	asserted 1.00
s8	In the analysis, we obtain an exact formula for the number of complementary domains.	result 0.85	0.94	0.07	asserted 1.00
s9	On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n.	result 0.65	0.38	0.18	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Inverting a polygon through a sphere to get a circle–arc polygon

**Figure 2.** For this inversion of the upper trefoil K, ρ(K) is a LH trefoil, and ˆρ(K) is a RH trefoil

**Figure 3.** With the same polygon K and a different inversion sphere, we have ρ(K) is a LH trefoil, and ˆρ(K) is an unknot

**Figure 4.** Then Ke = ρ(ˆρ(K)), so Ke is the same knot type as ˆρ(K) ∗ . Seen this way, our question of which knot types can arise as Kb is a special case of the problem of determining which knot types can arise if one replaces the edges of a polygonal knot by a some set of circle-arcs, passing in this way from Euclidean polygons to non-Euclidean polygons. We originally were led to the study of inverting polygons from questions about knot energies, in particular to find polygons that look very different geometrically yet have the same, or at least close, polygonal knot energies. Suppose K is…

Figures reproduced from arXiv:math/0602466 by the authors.

References

Works cited (9)

[1]

J. A. Calvo, Geometric knot spaces and polygonal isotopy , J. Knot Theory and its Ramiﬁcations 10 (2001), 245-267

View on Pith 2001
[2]

C. Ernst and D.W. Sumners, The growth of the number of prim e knots, Proc. Camb. Phil. Soc. 102 (1987), 303-315

View on Pith 1987
[3]

G. T. Jin and H. S. Kim, Polygonal knots, J. Korean Math. So c. 30 (1993), 371-383

View on Pith 1993
[4]

S. Negami, Ramsey theorems for knots, links, and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), 527-541

View on Pith 1991
[5]

R. Randell, Invariants of piece-wise linear knots, in Kn ot Theory, Banach Center Publications Vol. 42, V.F.R. Jones, J. Kania-Bartoszy´ nska, J. H. Przytycki, P. Traczyk, V.G. Turaev, eds, pp. 307-319

View on Pith
[6]

E. Rawdon and J. Simon, Polygonal approximation and ener gy of smooth knots, J. Knot heory and its Ramiﬁcations (to appear), preprint 200 3 Mathematics ArXiv GT/0305414

View on Pith arXiv
[7]

W. Thurston (and S. Levy Ed.), Three-Dimensional Geomet ry and Topology, Prince- ton University Press, NJ, 1997

View on Pith 1997
[8]

Y.-Q. Wu, software MING, http://www.math.uiowa.edu/ ∼ Wu/ming/ming.html

View on Pith
[9]

Ziegler, G¨ unter and ˇZivaljevi´ c, Rade, Homotopy types of subspace arrangement s via diagrams of spaces, Math. Ann. 295, 527-548 (1993). Department of Mathematics, University of Iow a,Iow a City IA 52242

View on Pith 1993

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. A. Calvo, Geometric knot spaces and polygonal isotopy , J. Knot Theory and its Ramiﬁcations 10 (2001), 245-267

View on Pith 2001

[2] [2]

C. Ernst and D.W. Sumners, The growth of the number of prim e knots, Proc. Camb. Phil. Soc. 102 (1987), 303-315

View on Pith 1987

[3] [3]

G. T. Jin and H. S. Kim, Polygonal knots, J. Korean Math. So c. 30 (1993), 371-383

View on Pith 1993

[4] [4]

S. Negami, Ramsey theorems for knots, links, and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), 527-541

View on Pith 1991

[5] [5]

R. Randell, Invariants of piece-wise linear knots, in Kn ot Theory, Banach Center Publications Vol. 42, V.F.R. Jones, J. Kania-Bartoszy´ nska, J. H. Przytycki, P. Traczyk, V.G. Turaev, eds, pp. 307-319

View on Pith

[6] [6]

E. Rawdon and J. Simon, Polygonal approximation and ener gy of smooth knots, J. Knot heory and its Ramiﬁcations (to appear), preprint 200 3 Mathematics ArXiv GT/0305414

View on Pith arXiv

[7] [7]

W. Thurston (and S. Levy Ed.), Three-Dimensional Geomet ry and Topology, Prince- ton University Press, NJ, 1997

View on Pith 1997

[8] [8]

Y.-Q. Wu, software MING, http://www.math.uiowa.edu/ ∼ Wu/ming/ming.html

View on Pith

[9] [9]

Ziegler, G¨ unter and ˇZivaljevi´ c, Rade, Homotopy types of subspace arrangement s via diagrams of spaces, Math. Ann. 295, 527-548 (1993). Department of Mathematics, University of Iow a,Iow a City IA 52242

View on Pith 1993