Canonical cabling on minimal grids: Questions 7.2 and 7.3

FPRD Lab research note, 18 September 2026

Status: A conditional reduction and checked input examples. Neither question is resolved in this note. The full Lee-Takioka canonical cabling algorithm was not obtained, and no substitute algorithm has been used to claim a result.

1. Exact questions

In T. Ohtsuki (ed.), Problems on Low-dimensional Topology, 2014, pp. 10-11, Lee and Takioka associate a canonical (p,q)-cable grid G^(p,q) to a companion grid G. Let

$$C_{p,q}(G)=\alpha(G^{(p,q)})$$

be its number of vertical segments. This is the size of a prescribed output diagram, not automatically the arc index of the cable link. Their canonical arc index is

$$\alpha_c(K^{(p,q)})=\min_{G\text{ a grid of }K} C_{p,q}(G).$$

Question 7.2 asks whether C has the same value on every minimal grid of a fixed knot. Question 7.3 asks whether every minimal companion grid realizes the minimum of C over all companion grids.

A pair of minimal grids with different canonical output sizes would refute both questions. Conversely, an affirmative answer to 7.2 alone would not settle 7.3: a nonminimal companion grid might give a smaller canonical cable.

2. A known theorem fixes two numerical coordinates on all minimal grids

Every rectangular diagram G determines two associated Legendrian diagrams, for its knot and its mirror. Write their Thurston-Bennequin numbers as t_+(G) and t_-(G), using a consistent convention to label the two. Dynnikov and Prasolov prove that a minimal rectangular diagram maximizes the Thurston-Bennequin number for its link type. Apply this also to its rotated, crossing-switched diagram, which is minimal for the mirror.

Thus, for every minimal grid G of K,

$$(t_+(G),t_-(G))=(T_+(K),T_-(K)),$$

where T_+(K) and T_-(K) are the maximal Thurston-Bennequin numbers of K and its mirror. The same paper gives the identity

$$n(G)=-t_+(G)-t_-(G), \tag{1}$$

where n(G) is the grid size. In particular,

$$\alpha(K)=-T_+(K)-T_-(K).$$

Source: I. Dynnikov and M. Prasolov, Bypasses for rectangular diagrams. Proof of Jones’ conjecture and related questions, arXiv:1206.0898v2. Corollary 3, PDF p. 25, proves the maximality assertion. Equation (2), PDF p. 9, gives (1); Section 2.2 specifies the grid-to-Legendrian conversion. This is an imported theorem, not a lab result.

For an arbitrary companion grid H, set

$$a=T_+(K)-t_+(H)\ge0,\qquad b=T_-(K)-t_-(H)\ge0.$$

Subtracting (1) for a minimal grid from (1) for H gives the useful exact identity

$$n(H)-\alpha(K)=a+b. \tag{2}$$

Thus the excess companion grid size is exactly the sum of these two deficits. This is an elementary consequence of the cited theorem and identity, not a new assertion about the cable algorithm.

3. Conditional reduction of both questions

Proposition. Fix p,q. Suppose the authors’ canonical output size has the form

$$C_{p,q}(G)=\Phi_{p,q}(t_+(G),t_-(G)) \tag{3}$$

for all relevant companion grids. Then Question 7.2 has an affirmative answer. If, in addition, Phi is nonincreasing as either coordinate increases, then Question 7.3 also has an affirmative answer.

Proof. All minimal grids have the same input pair (T_+,T_-), so (3) gives equal output sizes. For any other grid H, both its coordinates are at most the corresponding T values. The monotonicity assumption implies

$$C_{p,q}(G_{\min}) =\Phi_{p,q}(T_+,T_-) \le \Phi_{p,q}(t_+(H),t_-(H)) =C_{p,q}(H).$$

Taking the minimum over H proves the second assertion.

A formula also involving n(G) still fits this reduction when n is eliminated using (1). More generally, monotonicity is only required for the realized pairs being compared.

Missing premise: Neither factorization (3) nor its monotonicity has been established for the Lee-Takioka algorithm. Its output could retain dependence on corner order, distinguished vertices, or other information not captured by the two numbers. This proposition therefore does not settle either printed question.

It does identify a much sharper next calculation than an unstructured grid census: derive the exact output-size formula from the authors’ construction, express its writhe/corner terms in these coordinates, and check what information remains.

4. Exact trefoil controls recovered from Figure 5

Figure 5 on page 10 of the supplied problem list displays two size-five grids G1,G2 for the right-handed trefoil, followed by a size-six grid G3. Its stated (3,16)-canonical cable sizes are respectively 15,15,18. The first two are minimal: the source states that a nonsplit alternating knot’s arc index is its crossing number plus two. The third is not minimal, so these numbers do not refute Question 7.2.

The following coordinates were recovered by reading the figure. Rows are numbered bottom-to-top, columns left-to-right, both starting at zero. Each row contains one O and one X; horizontal edges are oriented O to X, and vertical edges X to O.

G1: O = (4, 3, 2, 1, 0)
    X = (1, 0, 4, 3, 2)

G2: O = (0, 4, 3, 2, 1)
    X = (2, 1, 0, 4, 3)

G2 is a one-step cyclic row translation of G1. An exact checker validates that both are connected grids and that the move gives the stated second grid.

The source’s intermediate braid exponents are 7 and 4, agreeing with the checked arithmetic 16-3w. Their difference is not a difference in final canonical cable size.

This is a concrete warning against treating different writhe values on minimal grids as a counterexample. Here the corner counts compensate exactly in both Thurston-Bennequin numbers, and the source reports equal canonical output sizes.

The checker also takes the closure of G1 under cyclic row/column shifts and adjacent commutations with disjoint, noninterleaving endpoint sets. This closure contains five grids and ten directed move instances. All five have the same pair (1,-6), while the writhe is three on three grids and four on two grids. Parent/move records are saved so each grid can be traced back to the source example.

This is not claimed to enumerate every minimal trefoil grid. Most importantly, the checker performs zero canonical cabling evaluations. The two cable sizes above are explicitly source-reported values, not newly computed outputs.

5. Literature access and the next mathematical obligation

The later published paper was identified as:

Hwa Jeong Lee and Hideo Takioka, On the arc index of cable links and Whitehead doubles, Journal of Knot Theory and Its Ramifications 25(7) (2016), 1650041, 23 pages. DOI: 10.1142/S0218216516500413.

The DGIST institutional record gives its abstract and bibliographic information. The available preview did not supply the complete algorithm. Neither the uploaded problem list nor the retrieved Library material contained its full definition. Publisher retrieval attempts did not return the full text in this session.

The exact next obligation is to obtain and read that algorithm, including all parameter branches, then derive or refute (3) and the required monotonicity. A full-text copy would remove the present source-access obstacle. No claim is made here about whether these questions remain open in the literature.

References

1. T. Ohtsuki (ed.), Problems on Low-dimensional Topology, 2014, supplied PDF, pp. 9-11, Figure 5 and Questions 7.2-7.3.
2. I. Dynnikov, M. Prasolov, Bypasses for rectangular diagrams. Proof of Jones’ conjecture and related questions, arXiv:1206.0898v2. https://arxiv.org/abs/1206.0898 . Relevant full-text sections checked.
3. H. J. Lee, H. Takioka, On the arc index of cable links and Whitehead doubles, DOI: 10.1142/S0218216516500413. Institutional record: https://scholar.dgist.ac.kr/handle/20.500.11750/2269 . Bibliography and abstract checked, full algorithm not obtained.