The 2014 low-dimensional topology problem list

Research note, 18 September 2026.

Our review of the problem list edited by Tomotada Ohtsuki has produced a counterexample to the printed Question 9.2 and a complete proposed classification for Problem 8.1. Questions 7.2 and 7.3 remain under investigation. The two proposed results have internal checks but no external review or established priority.

Problem 8.1: which diagrams have the largest state counts?

At every real crossing of a virtual knot diagram, choose one of the two smoothings. The resulting state is a collection of circles. Problem 8.1 asks which diagrams attain the given upper bounds for the numbers of states with one, two, or three circles.

Construct the diagram’s linked-chord graph G: each real crossing gives a chord in its Gauss diagram, and two graph vertices are adjacent when the corresponding chord endpoints alternate around the circle. Let P_m be a path on m vertices, C_m a cycle on m vertices, and I an isolated vertex. Write P_0 for the empty graph.

For a diagram with r real crossings, the proposed classification is

$$s_1=\frac{2^{r+1}-(-1)^{r+1}}3 \quad\Longleftrightarrow\quad G\cong P_r;$$

$$s_2=2^{r-1} \quad\Longleftrightarrow\quad G\cong P_{r-1}\sqcup I \quad\text{or}\quad G\cong C_r;$$

$$s_3=3\cdot2^{r-3} \quad\Longleftrightarrow\quad G\cong P_{r-3}\sqcup3I \quad\text{or}\quad G\cong C_{r-2}\sqcup2I.$$

The second line requires r >= 1; the third requires r >= 3. A cycle must have at least three vertices. These conditions concern all Gauss diagrams with those linked-chord graphs, regardless of crossing signs or chord arrows. They do not classify knot types or solve the separate minimization problems 8.2 and 8.3.

The source’s Figure 6 provides examples from these families. The proposed proof establishes necessity as well as sufficiency. It uses the known extended Cohn-Lempel equality to translate state circles into binary-matrix nullities, followed by a positive three-term recursion and an analysis of its equality cases. The graph classification holds even for finite simple graphs that are not linked-chord graphs.

Exact checks cover all 33,868 labeled simple graphs through six vertices and all 1,253 unlabeled graphs through seven vertices. A separate direct smoothing calculation checks 697,335 states. These checks support the proof; they are not its replacement for arbitrary crossing number.

Read the full proposed proof.

Question 9.2: an infinite counterexample

Takahiro Matsushita asks whether a connected graph with bounded degrees, distinct open neighborhoods, and no pair of vertices with exactly one common neighbor must be finite.

The answer to the printed question is no. Put two vertices in each integer-indexed column. Join the two vertices in each column and join all vertices in consecutive columns. Every vertex has degree five. For distinct vertices in the same column, adjacent columns, columns two apart, or columns at least three apart, the common-neighbor counts are respectively 4, 2, 2, and 0. Every neighborhood has five vertices, so no two neighborhoods are equal.

This graph is infinite and connected. Its bipartite double cover also works, removes all triangles, and has distinct open and closed neighborhoods.

Read the counterexample and bipartite strengthening.

Questions 7.2 and 7.3: canonical cabling on minimal grids

These questions ask whether Lee and Takioka’s canonical cabling produces the same number of vertical segments from every minimal grid of a knot, and whether each minimal grid minimizes that output size among all companion grids.

Dynnikov and Prasolov’s theorem fixes the two associated Thurston-Bennequin numbers on every minimal grid. Consequently, if the canonical output size depends only on those two numbers, Question 7.2 has an affirmative answer. A suitable monotonicity property would also settle Question 7.3.

Those hypotheses have not been established for the exact canonical algorithm. The checked trefoil examples illustrate why writhe alone is insufficient: their writhes are three and four, but their corner counts compensate to give the same pair of Thurston-Bennequin numbers, and the source reports the same canonical cable size of 15.

The full algorithm is the missing input for this route. Neither question is claimed solved.

Read the reduction and checked trefoil examples.

Answers already in the literature

Conjectures 1.3 and 1.4 are covered by Victoria Lebed’s Cohomology of finite monogenic self-distributive structures (2015), arXiv:1503.07030.

Question 10.1 has an affirmative answer in Stefan Friedl, Takahiro Kitayama and Matthias Nagel’s Representation varieties detect essential surfaces (2016), arXiv:1604.00584. Every connected essential surface is detected by an ideal point of a rational curve in an SL(n)-character variety for some n. This is not a statement about one fixed small n.

These are credited literature results, not lab discoveries.

Sources

• T. Ohtsuki, editor, Problems on Low-dimensional Topology, 2014, supplied prob14.pdf, Sections 7-10.
• L. Traldi, Binary nullity, Euler circuits and interlace polynomials, arXiv:0903.4405, Theorem 4. https://arxiv.org/abs/0903.4405
• I. Dynnikov, M. Prasolov, Bypasses for rectangular diagrams. Proof of Jones’ conjecture and related questions, arXiv:1206.0898v2, Corollary 3. https://arxiv.org/abs/1206.0898
• H. J. Lee, H. Takioka, On the arc index of cable links and Whitehead doubles, DOI:10.1142/S0218216516500413. https://scholar.dgist.ac.kr/handle/20.500.11750/2269
• V. Lebed, Cohomology of finite monogenic self-distributive structures. https://arxiv.org/abs/1503.07030
• S. Friedl, T. Kitayama, M. Nagel, Representation varieties detect essential surfaces. https://arxiv.org/abs/1604.00584