Three-Page Link Presentations

• Three-page presentations are specific embeddings of a link into exactly three pages of an open book, optimizing arc count and binding points.
• The methodology transforms spatial link embeddings into planar and circular diagrams, enabling precise upper bounds and invariant estimations.
• Applications range from theoretical knot invariants in torus links to computational document-to-slide synthesis in automated presentation generation.

Searching arXiv for the papers on arXiv and closely related work on “three-page presentation” to ground the article and resolve the topic’s usage across domains. arXiv search query: "^^^^^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^^^^^"three-page presentation\"^^^^^^^^2 OR \2^^^^^^^^"three-page presentations\"" A three-page presentation is, in knot theory, an embedding of a link into exactly three pages of an open book, each page containing a finite collection of pairwise disjoint properly embedded arcs whose union is the link. The associated three-page index PRESERVED_PLACEHOLDER_^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^ is the minimum number of arcs, equivalently the minimum number of binding points on the axis, among all such presentations of a link PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^ (Jang et al., 18 Jul 2025). A complementary diagrammatic formulation replaces the binding axis by a binding circle in a planar diagram and encodes the three pages by outside arcs, inside-over arcs, and inside-under arcs (&&&^^^^2 OR \2^^^^&&&). The term also appears in an unrelated computational setting, where a three-slide presentation is generated from a document by an edit-based workflow; that usage concerns document-to-slides synthesis rather than link embeddings (&&&^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^&&&).

^^^^2 OR \2^^^^. Definition and relation to arc presentations

In the open-book formulation, $\mathbb{R}^3$ is viewed as a book whose binding axis is the $z$-axis and whose pages are half-planes $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$. An arc presentation of a link embeds $L$ into a union of distinct pages $P_1,\dots,P_n$ with exactly one properly embedded arc on each page; the smallest such $n$ is the arc index $\alpha(L)$. A three-page presentation modifies this constraint by fixing the number of pages to be exactly three while allowing several disjoint arcs on the same page (Jang et al., 18 Jul 2025).

This distinction is fundamental. In the classical arc-presentation setting, different arcs must lie on different pages. In the three-page setting, several disjoint arcs may occupy a single page, but the ambient decomposition has only three pages. The invariant $\alpha_3(L)$ therefore measures a different notion of complexity from PRESERVED_PLACEHOLDER_^^^^2 OR \2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^: it records how efficiently a link can be organized when page count is fixed and arc count becomes the optimization target (Jang et al., 18 Jul 2025).

A common misconception is to identify the three-page index with the ordinary arc index. The available results do not support that identification. Instead, the basic relation stated for all links is PRESERVED_PLACEHOLDER_^^^^2 OR \2 OR \2^^^^, since any three-page presentation with PRESERVED_PLACEHOLDER_^^^^2 OR \22^^^^ arcs can be spread into an arc presentation with PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^3 pages (Jang et al., 18 Jul 2025).

2. Planar and circular formulations

Three-page presentations admit a planar encoding that is often more convenient for construction and counting. One draws a planar link diagram PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^4 on PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^5 together with a simple closed curve PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^6, called the binding circle, such that all crossings of PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^7 lie in the interior of PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^8, PRESERVED_PLACEHOLDER_^^^^2 OR \2^^^^9 meets $\mathbb{R}^3$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^ transversely in finitely many points, each arc cut by $\mathbb{R}^3$^^^^2 OR \2^^^^ either lies outside $\mathbb{R}^3$2 or lies inside and is exclusively over-crossing or exclusively under-crossing, and no two arcs of the same type are adjacent along $\mathbb{R}^3$3 (&&&^^^^2 OR \2^^^^&&&).

The three pages are then read off directly from the position and crossing type of the cut arcs. Page ^^^^2 OR \2^^^^ consists of arcs outside $\mathbb{R}^3$4; Page 2 consists of inside-arcs that go over at every crossing; Page 3 consists of inside-arcs that go under at every crossing. The number of intersection points $\mathbb{R}^3$5 equals the total number of arcs in the corresponding three-page presentation (&&&^^^^2 OR \2^^^^&&&).

This circular model is not merely a visualization device. It converts the spatial embedding problem into a diagrammatic optimization problem: choose $\mathbb{R}^3$6 so that the induced arc count is as small as possible while respecting the crossing-type alternation conditions. That reformulation underlies the strongest general upper bounds presently stated in the supplied sources.

3. Existence and basic inequalities

Dynnikov’s fundamental result is that every tame link admits a three-page presentation. An alternative proof for every non-split link proceeds from maximal overpasses in a planar diagram. If a diagram has no overpasses then the link is the trivial knot, which admits a three-page presentation of three arcs. Otherwise, one slides all maximal overpasses onto a single straight line in the plane, nests those that overlap, and then pushes the overpasses into a perpendicular half-plane. The resulting embedding occupies exactly three pages: two pages carry the under-arcs on the two sides of the line, and the third page carries the overpasses (Jang et al., 18 Jul 2025).

This construction yields immediate inequalities. If $\mathbb{R}^3$7 denotes the bridge number, then in any three-page presentation each page contains at least $\mathbb{R}^3$8 arcs, so $\mathbb{R}^3$9. Together with $z$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^, this places the three-page index between two classical invariants that reflect different decompositional constraints (Jang et al., 18 Jul 2025).

These inequalities clarify what the invariant does and does not measure. It is not simply a reparameterization of bridge number or arc index. Rather, it interpolates between diagrammatic overpass complexity and open-book complexity under the fixed-page constraint. This suggests a structural role for $z$^^^^2 OR \2^^^^ in comparing link presentations that privilege different ambient decompositions.

4. Diagrammatic bounds and the Hopf-link equality case

A reduced, non-split link diagram $z$2 with $z$3 crossings determines a cellular decomposition $z$4 of $z$5 whose $z$6-cells are the crossings, $z$7-cells are the arcs between crossings, and $z$8-cells are the complementary regions. The key construction is to choose a connected contractible subcomplex $z$9 and take the binding circle to be the boundary of a regular neighborhood $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^. Counting intersections of $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$^^^^2 OR \2^^^^ with $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$2 then produces upper bounds for $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$3 (&&&^^^^2 OR \2^^^^&&&).

If $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$4 is any spanning tree of the $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$5-skeleton, then a counting argument gives $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$6, hence $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$7. If $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$8 is enlarged to an extended spanning tree, meaning that it is connected and contractible, contains all $$P_\theta=\{(r,\phi,z)\in\mathbb{R}^3:\phi=\theta,\; r\ge 0\}$$9-cells, and no two $L$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^-cells in $L$^^^^2 OR \2^^^^ share a common $L$2-cell, then each included face removes exactly one intersection point. If $L$3 contains $L$4 faces, the bound becomes $L$5 (&&&^^^^2 OR \2^^^^&&&).

Bound or characterization	Statement	Scope
Spanning-tree bound	$L$6	General
Extended spanning-tree bound	$L$7	With $L$8 added faces
Improved bound	$L$9	Non-split, nontrivial $P_1,\dots,P_n$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^ Hopf link
Equality case	$P_1,\dots,P_n$^^^^2 OR \2^^^^ exactly for split unions of Hopf links	Complete characterization

The improved theorem states that if $P_1,\dots,P_n$2 is a non-split, non-trivial link other than the Hopf link, then

$P_1,\dots,P_n$3

The proof outline given in the source enlarges a spanning tree by two non-adjacent faces, so that the extended spanning tree has $P_1,\dots,P_n$4 faces and the refined count yields the stated inequality. The equality case is also sharp and explicit: for the Hopf link one has $P_1,\dots,P_n$5 and $P_1,\dots,P_n$6, and more generally $P_1,\dots,P_n$7 occurs exactly when $P_1,\dots,P_n$8 is a split union of Hopf links (&&&^^^^2 OR \2^^^^&&&).

5. Torus links and explicit examples

For torus links, the supplied results go beyond general bounds and determine exact values in several families. Assuming $P_1,\dots,P_n$9, the torus link $n$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^ is the closure of the braid $n$^^^^2 OR \2^^^^ on $n$2 strands and has $n$3 components. The exact formula

$n$4

holds for every integer $n$5 (Jang et al., 18 Jul 2025).

More generally, the stated upper bound is

$n$6

When $n$7, the improved bound becomes

$n$8

These are obtained by rewriting the braid word into forms that cluster crossings, introducing kinked arcs, choosing a horizontal binding line, and separating the resulting diagram into three pages according to whether arcs lie above the line, below it, or pass over it (Jang et al., 18 Jul 2025).

The small examples recorded in the sources illustrate how the general theory specializes. For the trefoil knot, the theorem $n$9 gives $\alpha(L)$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^, and an explicit three-page presentation with $\alpha(L)$^^^^2 OR \2^^^^ arcs is described; in the torus-link treatment this is sharpened to the exact identity $\alpha(L)$2 (&&&^^^^2 OR \2^^^^&&&). For the figure-eight knot, the bound gives $\alpha(L)$3, and an explicit drawing with $\alpha(L)$4 arcs has been produced; conjecturally no better presentation exists (&&&^^^^2 OR \2^^^^&&&).

These cases show that the general linear estimates are often close to exact, and in some structured families exact. A plausible implication is that torus-link braid structure is particularly well matched to three-page decompositions, because the constructions exploit repeated crossing patterns and controlled closure operations.

6. Distinct computational usage in document-to-slides generation

In an unrelated literature on automatic presentation generation, a three-page presentation refers literally to a three-slide artifact rather than a link embedded in a three-page book. The system “PPTAgent: Generating and Evaluating Presentations Beyond Text-to-Slides” uses a two-stage, edit-based approach. In Stage I, reference presentations are analyzed to extract slide-level functional types and content schemas; in Stage II, an outline is drafted and editing actions are iteratively generated based on selected reference slides to create new slides (&&&^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^&&&).

The supplied three-slide outline consists of “PPTAgent Framework,” “Two-Stage Edit-Based Workflow,” and “Results & Impact.” The formalization contrasts a conventional formulation,

$\alpha(L)$5

with an agent formulation,

$\alpha(L)$6

The editing actions include operations such as clone_paragraph, replace_span, and del_image, and the workflow uses a code-in-HTML REPL for self-correction. The associated evaluation framework, PPTEval, assesses presentations across Content, Design, and Coherence. The reported metrics in the supplied material are success rate $\alpha(L)$7 across models and average Eval score $\alpha(L)$8 with Content $\alpha(L)$9, Design $\alpha_3(L)$^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^, and Coherence $\alpha_3(L)$^^^^2 OR \2^^^^; the reported ablations include “w/o Outline” decreasing coherence by $\alpha_3(L)$2 and “w/o CodeRender” decreasing SR by $\alpha_3(L)$3 percentage points (&&&^^^^2(Zheng et al., 7 Jan 2025) OR (Yoo, 19 Jan 2026) OR (Jang et al., 18 Jul 2025) OR \2^^^^&&&).

This usage is terminologically distinct from the link-theoretic notion. In the former, “three-page” denotes the length and structure of a slide deck synthesized from a document; in the latter, it denotes an embedding of a link into exactly three pages of an open book and the corresponding minimization problem for arc count. The coexistence of these usages is a matter of terminology rather than shared mathematical or algorithmic content.