The triple-crossing number c3(L)c_3(L) is the minimal number of triple crossings — points where three strands pass through one another with a definite top/middle/bottom height ordering — over all diagrams of a knot or link LL. Jabłonowski (2020) proved the lower bound c3(L)≥2gc(L)+r(L)−1c_3(L)\ge 2g_c(L)+r(L)-1 in terms of canonical genus gcg_c and component number rr, and used it to compute c3c_3 exactly for torus knots, closures of positive braids, and a long list of tabulated knots, obtaining additivity of c3c_3 under connected sum for the torus knots and the tabulated knots among them. The subsequent literature (2013–2026) addresses neither general additivity of c3c_3 under connected sum nor a classification of the knots on which the genus lower bound is attained. Both questions — central to determining how far genus controls the triple-crossing number — remain open.

Background

An nn-crossing (multi-crossing) projection of a knot or link is a planar projection in which nn strands of the curve pass straight through a single common point at every crossing. For n=3n=3 one obtains a triple-crossing projection: each crossing has exactly three strands meeting pairwise transversely, together with a height ordering of the strands labeled top, middle, and bottom. Adams (2013) proved that every knot and link admits such a projection, so the triple-crossing number c3(L)c_3(L) — the least number of triple crossings over all triple-crossing diagrams of LL — is a well-defined analogue of the classical double-crossing number c2(L)c_2(L), the ordinary minimal crossing number. Known general bounds include c3(L)≥c2(L)/3c_3(L)\ge c_2(L)/3, improved to c3(L)≥c2(L)/2c_3(L)\ge c_2(L)/2 for alternating links (links admitting a diagram in which over- and under-crossings alternate along every component), the upper bound c3(L)≤c2(L)−2c_3(L)\le c_2(L)-2 for every nontrivial knot or link other than the (2,n)(2,n)-torus links — the links obtained as closures of the two-strand braid, winding twice around the torus core (Adams 2013) — and c3(L)≥β2(L)−1c_3(L)\ge \beta_2(L)-1 for nonsplit links, where β2\beta_2 is the double-crossing braid index, the least number of strands of a classical braid whose closure is LL (Nishida 2019). A Seifert surface for LL is an orientable surface embedded in R3\mathbb{R}^3 whose boundary is LL; the genus g(L)g(L) is the least genus of any such surface, and the canonical genus gc(L)g_c(L) is the least genus of a surface produced by Seifert's algorithm (resolve each crossing of a classical oriented diagram in an orientation-preserving way, cap the resulting disjoint circles with disks, and reconnect them with half-twisted bands) over all classical diagrams. Jabłonowski (2020) proved that c3(L)≥2gc(L)+r(L)−1c_3(L)\ge 2g_c(L)+r(L)-1, where r(L)r(L) is the number of components of LL. This bound is sharp on three kinds of examples: torus knots T(p,q)T(p,q) — knots lying on the surface of a standard torus in R3\mathbb{R}^3, winding pp times around one direction of the torus and qq times around the other — where it yields c3(T(p,q))=(p−1)(q−1)c_3(T(p,q))=(p-1)(q-1); closures of positive braids on pp strands with braid-word length kk — a positive braid is one whose word uses only positively oriented standard generators, and its closure joins corresponding ends to form a link — where c3=k−p+1c_3=k-p+1; and tabulated knots for which the genus lower bound happens to meet the upper bound c2−2c_2-2. The connected sum K1#K2K_1\# K_2 of two knots is formed by deleting a small unknotted arc from each and gluing along the resulting boundary spheres; genus is additive under connected sum, and for torus knots and the tabulated knots above — where both factor values are forced by genus — Jabłonowski's Corollary 7 derives additivity, c3(K1#K2)=c3(K1)+c3(K2)c_3(K_1\# K_2)=c_3(K_1)+c_3(K_2). That proof uses subadditivity c3(K1#K2)≤c3(K1)+c3(K2)c_3(K_1\# K_2)\le c_3(K_1)+c_3(K_2) (elementary, by gluing diagrams along an arc, exactly as for c2c_2) and lower-bounds the composite only through 2g(K1#K2)=2g(K1)+2g(K2)2g(K_1\#K_2)=2g(K_1)+2g(K_2). No other mechanism is known that lower-bounds c3(K1#K2)c_3(K_1\#K_2) by the sum of the factor values for arbitrary knots, which is why general additivity is open — the situation parallels the classical, still-open additivity problem for the double-crossing number c2c_2 itself, settled only for large special classes. The broader setting is the multi-crossing hierarchy, studied by Adams and collaborators through the Kauffman bracket polynomial — a one-variable polynomial invariant of unoriented link diagrams defined by a local skein recursion — and petal diagrams, projections in which all strands pass through one common crossing point; more recent contributions bound multi-crossing numbers by genus (Guha 2022) and by the arc index α(L)\alpha(L), the least number of arcs in an arc presentation of LL on a grid (Ito 2025); within this hierarchy the triple-crossing level is where genus-based lower bounds are strongest and exact values are scarcest.

Problem Statement

The problem left open by Jabłonowski (2020) and untouched by the subsequent literature has two linked parts with one common objective — determining how far genus controls the triple-crossing number:

(1) Additivity under connected sum. Is c3(K1#K2)=c3(K1)+c3(K2)c_3(K_1\# K_2)=c_3(K_1)+c_3(K_2) for all knots K1K_1 and K2K_2? Subadditivity is elementary (glue minimal triple-crossing diagrams along an arc of the unbounded face, as in the proof of Corollary 7 of the source paper), so the open content is the reverse inequality c3(K1#K2)≥c3(K1)+c3(K2)c_3(K_1\# K_2)\ge c_3(K_1)+c_3(K_2): no argument for it is known except genus additivity, which works only when both factors' triple-crossing numbers are themselves forced by the genus bound. A single certified counterexample — a pair of knots whose composite provably has fewer triple crossings than the sum — would settle the question negatively.

(2) Sharpness and the equality class. Determine for which knots KK the genus-based lower bound is attained, i.e. c3(K)=2gc(K)c_3(K)=2g_c(K), and for which links c3(L)=2gc(L)+r(L)−1c_3(L)=2g_c(L)+r(L)-1, beyond the known families of torus knots, closures of positive braids, and tabulated knots on which the bound coincides with the upper bound c2−2c_2-2. In particular: is the bound attained by any knot with gc(K)>g(K)g_c(K)>g(K), or by any infinite family not arising from positive braids?

The two parts are jointly posed, linked by the objective of determining how far genus controls the triple-crossing number. Within (2), the main objective is the equality-class characterization — the knots KK with c3(K)=2gc(K)c_3(K)=2g_c(K) and the links LL with c3(L)=2gc(L)+r(L)−1c_3(L)=2g_c(L)+r(L)-1 beyond the known families — while the two questions posed after "In particular" are sub-goals of (2) that do not by themselves settle it.

A positive resolution of (1) together with (2) would complete the source paper's program: exact triple-crossing numbers for all connected sums of equality-class factors, and a characterization of the knots whose triple-crossing number is computable from genus data alone.

References

  1. Michał Jabłonowski, Triple-crossing number, the genus of a knot or link and torus knots, Topology and its Applications 285 (2020) 107389, DOI 10.1016/j.topol.2020.107389, https://arxiv.org/abs/2003.12851
  2. Colin Adams, Triple crossing number of knots and links, Journal of Knot Theory and Its Ramifications 22 (2013) 1350006, DOI 10.1142/S0218216513500065, https://arxiv.org/abs/1207.7332
  3. Colin Adams, Orsola Capovilla-Searle, Jesse Freeman, Daniel Irvine, Samantha Petti, Daniel Vitek, Ashley Weber, Sicong Zhang, Multi-crossing number for knots and the Kauffman bracket polynomial, Mathematical Proceedings of the Cambridge Philosophical Society 164(1) (2018) 147–178, DOI 10.1017/S0305004116000906, https://arxiv.org/abs/1407.4485
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