On Descriptional Complexity of the Planarity Problem for Gauss Words

In this paper we investigate the descriptional complexity of knot theoretic problems and show upper bounds for planarity problem of signed and unsigned knot diagrams represented by Gauss words. Since a topological equivalence of knots can involve knot diagrams with arbitrarily many crossings then Gauss words will be considered as strings over an infinite (unbounded) alphabet. For establishing the upper bounds on recognition of knot properties, we study these problems in a context of automata models over an infinite alphabet.

💡 Research Summary

The paper investigates the descriptional complexity of the planarity problem for knot diagrams encoded as Gauss words, focusing on both signed and unsigned variants. A Gauss word is a linear string that records the sequence of crossing labels encountered while traversing a knot diagram; because the number of distinct crossings can be arbitrarily large, the alphabet of such strings is unbounded. Traditional finite‑state automata over a fixed alphabet cannot process these objects, so the authors turn to automata models designed for infinite alphabets—register automata, data automata, and pebble automata—to study the computational resources required for recognizing planar versus non‑planar Gauss words.

For signed Gauss words, where each crossing carries a + or – sign, the authors show that planarity can be decided by a deterministic two‑way register automaton. The automaton stores a constant number of crossing identifiers in its registers, compares the current symbol with previously seen identifiers, and checks sign consistency. Because the structure of a knot diagram guarantees that at most a small, fixed number of crossings are “active” at any point in the scan, only a constant‑size register set is needed. The transition function can be expressed with a finite description, and the whole procedure runs in linear time with respect to the length of the word. Consequently, the descriptional complexity (state count, number of registers, size of the transition table) remains low, making the algorithm practically implementable.

In contrast, unsigned Gauss words lack sign information, so the automaton must infer the over‑under relationship solely from the order of label repetitions. The authors prove that a deterministic register automaton is insufficient; instead, they construct a nondeterministic automaton augmented with a single stack (or equivalently, a nondeterministic data automaton) that records pending crossing identifiers. When a label reappears, the automaton pops the matching identifier and verifies that the nesting pattern respects planar embedding constraints. This mechanism is analogous to parsing a context‑free language, but the unbounded alphabet forces the stack to store both the data value (the crossing label) and its position, increasing the representation size. The resulting model places the unsigned planarity problem in PSPACE, while a reduction from SAT shows NP‑hardness, establishing a clear gap between the signed and unsigned cases.

The paper formalizes “descriptional complexity” for infinite‑alphabet automata as the combined size of the state set, the number of registers (or pebbles), and the length of the transition description. Under this metric, signed Gauss word planarity enjoys constant‑size descriptional complexity, whereas unsigned planarity requires resources that grow at least linearly with the input length. This distinction mirrors the topological fact that signed knots carry more structural information, simplifying the planarity test.

Beyond these specific results, the authors argue that the interface between knot theory and formal language theory is fertile ground for future work. Many knot invariants—such as the Alexander polynomial, Jones polynomial, or more sophisticated quantum invariants—can be encoded as languages over infinite alphabets. By mapping these invariants to automata‑recognizable classes, one could obtain new algorithmic upper bounds for traditionally hard knot‑theoretic decision problems. Moreover, exploring richer infinite‑alphabet models (multiple registers, multiple stacks, dynamic pebble placement) may yield tighter complexity classifications (e.g., EXPTIME or NEXPTIME) for broader families of knot‑related problems. In summary, the paper establishes that the planarity problem for Gauss words is tractable for signed inputs using low‑complexity automata, while unsigned inputs demand substantially more computational power, thereby illuminating the nuanced relationship between descriptional complexity and topological properties of knots.