[Paper Review] Strong and weak (1, 2) homotopies on knot projections and new invariants

Noboru Ιτο, Yusuke Takimura|arXiv (Cornell University)|Nov 14, 2020

Geometric and Algebraic Topology6 references4 citations

TL;DR

This paper introduces strong and weak (1,2) homotopies on knot projections by decomposing second flat Reidemeister moves into oriented self-tangency modifications, defining new invariants via non-Seifert (A⁻¹) resolution. It establishes a necessary and sufficient condition for strong and weak (1,2) homotopy equivalence using reduced knot projections, and constructs an integer-valued invariant that distinguishes trivialization classes under strong versus weak homotopy.

ABSTRACT

Every second flat Reidemeister move of knot projections can be decomposed into two types thorough an inverse or direct self-tangency modification, respectively called strong or weak, when orientations of the knot projections are arbitrarily provided. Further, we introduce the notions of strong and weak (1, 2) homotopies; we define that two knot projections are strongly (resp. weakly) (1, 2) homotopic if and only if two knot projections are related by a finite sequence of first and strong (resp. weak) second flat Reidemeister moves. This paper gives a new necessary and sufficient condition that two knot projections are not strongly (1, 2) homotopic. Similarly, we obtain a new necessary and sufficient condition in the weak (1, 2) homotopy case. We also define a new integer-valued strong (1, 2) homotopy invariant. Using it, we show that the set of the non-trivial prime knot projections without 1-gons that can be trivialized under strong (1, 2) homotopy is disjoint from that of weak (1, 2) homotopy. We also investigate topological properties of the new invariant and give its generalization, a comparison of our invariants and Arnold invariants, and a table of invariants.

Motivation & Objective

• To resolve open problems concerning necessary and sufficient conditions for two knot projections to be equivalent under strong or weak (1,2) homotopy.
• To define and analyze new invariants for classifying knot projections under these homotopy relations.
• To distinguish the sets of prime knot projections trivializable under strong versus weak (1,2) homotopy.
• To generalize topological invariants using non-Seifert resolution and connect them to Arnold invariants and circle arrangements.

Proposed method

• Decomposes second flat Reidemeister moves into strong and weak self-tangency modifications based on orientation compatibility.
• Introduces strong and weak (1,2) homotopy via sequences of first flat Reidemeister moves and strong/weak second moves.
• Defines a new integer-valued invariant τ(P) using non-Seifert (A⁻¹) resolution at each double point, independent of orientation.
• Applies the invariant τ(P) to classify knot projections up to strong and weak (1,2) homotopy, particularly focusing on prime projections without 1-gons.
• Uses circle arrangements and chord diagrams to represent and analyze the topological structure of knot projections.
• Applies operations like s2a and 1a to transform connected sums into prime knot projections, preserving the invariant.

Experimental results

Research questions

• RQ1 What is a necessary and sufficient condition for two knot projections to be strongly (1,2) homotopic?
• RQ2 What is a necessary and sufficient condition for two knot projections to be weakly (1,2) homotopic?
• RQ3 Can a new integer-valued invariant distinguish between knot projections trivializable under strong versus weak (1,2) homotopy?
• RQ4 How does the non-Seifert resolution (A⁻¹) behave under RI and strong/weak RIIs, and what invariants does it yield?
• RQ5 What is the structure of circle arrangements in prime knot projections without 1-gons, and how do they relate to homotopy classes?

Key findings

• Two knot projections are strongly (resp. weakly) (1,2) homotopic if and only if their strong (resp. weak) reduced projections are isotopic.
• The invariant τ(P) is an integer-valued strong (1,2) homotopy invariant that distinguishes trivialization classes: the set of non-trivial prime knot projections trivializable under strong (1,2) homotopy is disjoint from that under weak (1,2) homotopy.
• For any knot projection P, τ(P) ≤ 2 implies τ(P) = 1, and τ(4₁) = 1, showing that τ(P) is non-trivial for some non-trivial projections.
• Any circle arrangement s can be realized as τ(P) for some prime knot projection P, constructed via connected sums of trefoil and 6₃ projections, followed by s2a or (1a)(s2a) operations.
• The table of circle invariants for prime knot projections without 1-gons up to seven double points includes symbols like 7_A, 7_B, 7_C, derived from Rolfsen table via flypes.
• Lines marked 's' or 'w' in the table indicate existence of sequences of 1a and s2a (or w2a) moves connecting two projections, reflecting strong or weak homotopy relations.