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arxiv: 1912.03697 · v2 · submitted 2019-12-08 · 🧮 math.GT · math.QA

An infinite family of knots whose hexagonal mosaic number is only realized in non-reduced projections

Hugh Howards , Jiong Li , Xiaotian Liu , Anna Paulec This is my paper

Pith reviewed 2026-05-24 15:00 UTC · model grok-4.3

classification 🧮 math.GT math.QA MSC 57M25
keywords hexagonal mosaicsmosaic numberknot diagramsnon-reduced projectionsflypesalternating knotsinfinite families
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The pith

An infinite family of knots achieves its hexagonal mosaic number only in non-reduced projections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an infinite family of knots with the property that, for every r at least 3, some member of the family embeds on a hexagonal r-mosaic yet cannot do so with any diagram that realizes the knot's crossing number. This shows that the smallest hexagonal mosaic embedding for these knots must use a diagram with superfluous crossings. The result extends an earlier finding for rectangular mosaics and is supported by a new systematic procedure for enumerating all flypes of a diagram. A reader would care because it demonstrates that mosaic number and crossing number are not always simultaneously minimized in hexagonal grids.

Core claim

The authors establish an infinite family of knots such that for any given r greater than or equal to 3 the family contains a knot which can be embedded on a hexagonal r-mosaic but cannot fit on a hexagonal r-mosaic in an embedding that achieves its crossing number.

What carries the argument

The infinite family of knots constructed to force non-reduced projections on every hexagonal r-mosaic, together with the new flype enumeration tool that generates all minimal-crossing diagrams of prime alternating knots.

If this is right

  • For every r at least 3 there exists at least one knot whose hexagonal mosaic number is realized only by a non-reduced diagram.
  • The hexagonal mosaic number can exceed the value implied by the crossing number alone.
  • The flype tool provides a complete enumeration of minimal-crossing diagrams for any prime alternating knot.
  • The same separation between mosaic number and crossing number holds in the hexagonal case as was previously shown for rectangular mosaics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Algorithms that search for mosaic embeddings may need to consider diagrams with extra crossings even when the crossing number is known.
  • The construction technique could be adapted to produce similar families in other regular tilings or grid types.
  • The flype enumeration method might be used to certify that a given mosaic embedding is minimal for other knot invariants.

Load-bearing premise

The specific knots in the family have no reduced diagram that fits inside a hexagonal r-mosaic.

What would settle it

Exhibit one knot from the family together with a reduced diagram that tiles a hexagonal r-mosaic, or show that the flype tool misses a minimal-crossing diagram for a known prime alternating knot.

Figures

Figures reproduced from arXiv: 1912.03697 by Anna Paulec, Hugh Howards, Jiong Li, Xiaotian Liu.

Figure 1.1
Figure 1.1. Figure 1.1: The trefoil embedded in a hexagonal mosaic with r = 2. 1 arXiv:1912.03697v1 [math.GT] 8 Dec 2019 [PITH_FULL_IMAGE:figures/full_fig_p001_1_1.png] view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Here we see the hexagonal tiles up to rotation. Numbers to the left of the tile are given so that we can refer to specific tiles by name when convenient. In this paper we extend a result of result of Ludwig, Evans, and Paat [16] from square mosaics to the hexagonal setting. We give an infinite family of knots such that for any given r ≥ 3, the family contains a knot which can be embedded on a hexagonal r… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: The tiles inside the green curve - all but the outer two rings of tiles - are called the central tiles. The ring of tiles that is second farthest out, between the red and green curves, is called the penultimate corona (the second corona here). The tiles inside the red curve (the central tiles plus the penultimate tiles) are called the interior tiles, and the tiles outside the red curve (the r − 1 st coro… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Here we see the two ways of connecting up a collection of tiles through the boundary. The interior tiles on the two mosaics are identical and only the boundary tiles are different. The link on the left is called L4. only knots one can get are the trefoil and figure eight knot, but hexagonal 3-mosaics already yield knots up to crossing number 19 and hexagonal 5-mosaics contain knots up through crossing nu… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Here we see L3 on the left and K3 on the right. Because a hexagonal 3-mosaic only contains one central tile and that tile only intersects one component of L3, K3 is not built in the exact same way as the other examples where all the smoothings to from Kr from Lr can be done on central tiles(r ≥ 4). tiles of a mosaic are all saturated, all of the tiles in the penultimate corona are exactly one of these th… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: The same strategy is used for Kr for all r ≥ 4. Here the algorithm is detailed in general. We can take the link Lr, r ≥ 4 and form an alternating knot Ar from it by removing r − 2 of the existing central tiles (all of which are equivalent to tile 26 in [PITH_FULL_IMAGE:figures/full_fig_p005_3_3.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Here we see a 4-mosaic non-alternating projection of K4 on the left and a 5-mosaic alternating projection on the right. Theorem 6.12 shows that there is no reduced alternating 4-mosaic projection of K4. two different components we reduce the number of components of the link by one (if both arcs of the crossing were from a single component smoothing might increase the number of components or leave the num… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Here L5 (top left) is converted into an alternating knot A5 (top right) by smoothing crossings, then the second highest horizontal strand is changed so that it can be lifted up (bottom left), and the crossings above it are changed to form K5 (bottom right). The same general strategy is used for Kr for all r ≥ 4 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: that contains the same arcs of the link, but also include arcs of the complement. Looking [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The complement is in blue for green arcs of a mosaic. When the link intersects the tile in a single arc II there are usually two choices for the complement (pictured in the first three pairs). Note that the complement is trivial on the 11 tiles that contain 3 arcs of the link since they already hit all 6 connection points for the hexagonal tile. The complement is completely contained on the interior of t… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Given a link L the arcs of the complement are drawn in blue and labeled a1 and a2 and the loop of the complement is labeled c1. We move c1 from the complement to the link and add in arc a1 to form L 1 , then we add a2 to L 1 to form L 2 = L 0 . Finally we show Lb which will be constructed out of L 0 in the proof of Theorem 6.8. The complement is always drawn in blue, but as arcs are changed or moved from… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: The dual graph for the projection of a figure eight knot. It has exterior degree ∆ = 3. 2 3 4 1 5v v v v v 6 7 8 9 10 V V V V V 2 3 4 5v v v v 6 7 8 9 10 V V V V V [PITH_FULL_IMAGE:figures/full_fig_p012_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: The standard alternating embedding of K3 is pictured left. The dual graph is pictured center. Because the degree of the exterior vertex (v1 here) gets quite large for Kr as r increases, we usually draw the start of the edges running to the exterior vertex in gray, but omit the actual vertex to simplify the picture as seen on the right for K3. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: A flype takes us from left to center, but an isotopy moves the crossing back to the original side of the tangle yielding the picture on the right. This is equivalent to flipping over the entire link projection and is considered a trivial flype. 2 3 4 1 5 v v v v v 2 3 4 1 5 v v v v I v / [PITH_FULL_IMAGE:figures/full_fig_p013_5_3.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Because flypes are simple moves the influence on the dual graph is always predictable. For every possible flype for Kr, r ≥ 4 a subgraph of the dual graph like the one on the left is always replaced by a sugbraph like the one on the right - the subgraphs are isomorphic to each other, but the degrees of some vertices do change. K3 is different from the other Kr’s with one slightly more complicated flype a… view at source ↗
Figure 4
Figure 4. Figure 4: ) [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: In the first mosaic, on the left, we have the set of arcs A = {a1, a2, . . . a6}, In the second we take the subset E = {a1, a2, a3} to get rid of the nested arcs in A. In the third we change a2 and a3 which meet in a central tile into a 0 2 and a 0 3 finally on the left we restrict to our final set of arcs D and rename them d1 and d2. D has the desirable properties that it has no nested arcs, every tile … view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: On the left we see (most of) the dual graph for L5 in its standard alternating projection. The exterior vertex is not drawn. All the gray edges would connect up to the exterior vertex. On the right we see the dual graph for A5 (as well as K5 in its non-alternating projection) formed from the graph on the left by deleting sets of three edges near the center of the graph and replacing them with single edge… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: The dual graph for Lr is easy to visualize based on the dual graph for L5 shown on the left in [PITH_FULL_IMAGE:figures/full_fig_p023_5_4.png] view at source ↗
Figure 6.3
Figure 6.3. Figure 6.3: The knot K5 in our standard alternating projection and its dual graph (aside from v1) are shown on the left. Four flypes are then performed to get to the picture on the right. Two flypes contain v1 corresponding to the outer 4-cycles v1, f1, e2, f2, v1, m0, i1, m1, and two contain v2 corresponding to the outer 4-cycles v2, b1, a2, f = b2, and v2, b3, a4, b4. Note that the first type of flype lowers the e… view at source ↗
Figure 1
Figure 1. Figure 1: ) [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
read the original abstract

We give an infinite family of knots such that for any given $r \geq 3$, the family contains a knot which can be embedded on a hexagonal $r$-mosaic, but cannot fit on a hexagonal $r$-mosaic in an embedding that achieves its crossing number. This extends the rectangular mosaic result of Ludwig, Evans, and Paat. We also introduce a new tool for systematically finding all possible flypes for the diagram of any link thus making it easier to find all possible minimal crossing embeddings of prime, alternating knots.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper constructs an infinite family of knots such that, for every r ≥ 3, at least one member embeds on a hexagonal r-mosaic but none of its minimal-crossing diagrams do. It also introduces a flype-based enumeration procedure claimed to generate all minimal-crossing diagrams of any prime alternating knot.

Significance. If the construction and the completeness of the enumeration hold, the result supplies the first explicit infinite family separating hexagonal mosaic number from crossing number, extending the rectangular-mosaic examples of Ludwig-Evans-Paat. The flype tool is presented as a systematic aid for exhaustive diagram enumeration and could be reusable for other alternating-knot problems.

major comments (2)
  1. [flype enumeration procedure] The completeness argument for the flype enumeration procedure (introduced after the abstract and used to verify the family) rests on a finite case analysis of flype configurations. It is not shown that every configuration arising in the infinite family constructed later is covered by this analysis; omission of even one minimal diagram would falsify the claim that no minimal-crossing embedding fits on the r-mosaic.
  2. [construction of the infinite family] The existence statement for the family requires, for each r, both an explicit r-mosaic embedding of the chosen knot and a verification that every minimal-crossing diagram fails to embed. The manuscript presents the family and invokes the enumeration tool, but does not supply an independent check that the tool output is exhaustive for the specific diagrams of the family members.
minor comments (2)
  1. Notation for mosaic tiles and boundary conditions should be defined once in a preliminary section rather than reintroduced when the family is presented.
  2. The abstract states that the tool makes it 'easier' to find all minimal embeddings; a precise statement of what the tool guarantees (exhaustive enumeration versus heuristic search) would clarify its scope.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on the completeness of the flype enumeration and the verification for the infinite family. We address each point below and indicate where revisions will be made for added clarity.

read point-by-point responses

  1. Referee: The completeness argument for the flype enumeration procedure (introduced after the abstract and used to verify the family) rests on a finite case analysis of flype configurations. It is not shown that every configuration arising in the infinite family constructed later is covered by this analysis; omission of even one minimal diagram would falsify the claim that no minimal-crossing embedding fits on the r-mosaic.

    Authors: The finite case analysis enumerates all possible flype configurations that can occur in any prime alternating diagram, based solely on the local crossing and tangle structures permitted by alternating projections. This classification is universal and does not depend on the global knot or the particular infinite family. Every knot in the family is prime and alternating, and its minimal-crossing diagrams arise exclusively from flypes within the enumerated configurations; no new configurations are introduced by the construction. revision: no

  2. Referee: The existence statement for the family requires, for each r, both an explicit r-mosaic embedding of the chosen knot and a verification that every minimal-crossing diagram fails to embed. The manuscript presents the family and invokes the enumeration tool, but does not supply an independent check that the tool output is exhaustive for the specific diagrams of the family members.

    Authors: We agree that the manuscript would benefit from an explicit statement confirming that the enumerated diagrams for the family members have been checked against the mosaic constraints. In the revision we will insert a short paragraph (or subsection) that, for each r, lists the output of the enumeration tool for the relevant knot and records the geometric reason each diagram fails to embed on the given hexagonal r-mosaic. The explicit mosaic embeddings themselves are already constructed in the text. revision: yes

Circularity Check

0 steps flagged

Explicit construction with internal case analysis; no reduction to inputs

full rationale

The paper constructs an explicit infinite family of knots and introduces a flype enumeration tool whose completeness argument is a finite case analysis presented within the manuscript. The central claim (mosaic number realized only in non-reduced projections) follows from verifying the constructed examples against this enumeration rather than from any fitted parameter, self-referential definition, or load-bearing self-citation chain. No quoted step equates a derived quantity to its own input by construction, satisfying the requirement that circularity be exhibited only via explicit reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions from knot theory and mosaic theory; no free parameters, new entities, or ad-hoc axioms are visible from the abstract.

axioms (1)
  • standard math Standard definitions of knot diagrams, crossing number, mosaic embeddings, and flypes in S^3.
    The result is stated in terms of these established concepts.

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