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On the welded Tube map

Benjamin Audoux

arXiv:1408.5479v3math.GT

TL;DR

The paper asks how welded knots relate to ribbon torus-knots when a ribbon filling is retained or forgotten. It establishes a bijection at the level of filled objects and describes surjectivity and global reversal for the Tube map after passing to knots.

  • Problem

    The Tube map forgets ribbon fillings, and it is not known from the supplied passages whether classical Reidemeister moves and co-orientation reversal generate every filling change.

    Source: §2 · p. 12, §3 · p. 14

  • Method

    The paper encodes ribbon tori by welded tori and compares the diagrammatic moves with changes of filling under the Tube map.

    Source: §2 · pp. 10–11

  • Results

    The map from welded tori to ribbon tori is a bijection, while the Tube map on welded knots is surjective and invariant under global reversal.

    Source: §2 · pp. 10–12

  • Takeaways & Limitations

    Retaining the filling gives a one-to-one correspondence, whereas classical Reidemeister moves on welded tori correspond to changes of filling that preserve the boundary.

    Source: §2 · pp. 11–12

    Whether classical Reidemeister moves and co-orientation reversal generate all possible changes of filling remains an open question in the paper.

    Source: §3 · p. 14

Abstract

from arXiv · show

This note investigates the so-called Tube map which connects welded knots, that is a quotient of the virtual knot theory, to ribbon torus-knots, that is a restricted notion of fillable knotted tori in the 4-sphere. It emphasizes the fact that ribbon torus-knots with a given filling are in one-to-one correspondence with welded knots before quotient under classical Reidemeister moves and reformulates these moves and the known sources of non-injectivity of the Tube map in terms of filling changes.

1. Ribbon, broken and welded objects

The paper defines ribbon torus-knots through ribbon fillings, then describes them by broken torus diagrams and welded knots. The key distinction is whether a filling is retained: diagrammatic moves can change the filling without changing the boundary.

1.1. Ribbon tori and ribbon torus-knots.

  • A ribbon torus is a bi-oriented immersed solid torus with ribbon-disk singularities, and its embedded boundary is a ribbon torus-knot that may admit non-isotopic fillings.Forgetting the interior identifies ribbon tori with the same boundary, so ribbon torus-knots are a quotient of ribbon tori by boundary equivalence.Source: §1.1 · p. 4

1.2. Broken torus diagrams.

  • Projecting near a ribbon disk turns two flying circles into tubes with singular circles, marked by erasing the preimage with the lower projection parameter.Broken torus diagrams record these ordered preimages; the symmetric diagrams considered here have a natural solid-torus filling that can be co-oriented.Source: §1.2 · pp. 4–5

1.3. Welded knots.

  • Broken Reidemeister and virtual moves define broken knots, while virtual diagrams modulo generalized Reidemeister moves and over commute define welded knots.Welded Gauss diagrams encode the cyclic order of lowest crossing encounters and signed predecessor data, and correspond one-to-one with welded tori.Source: §1.2 · p. 5, §1.3 · pp. 7–8

2. The Tube map

The Tube map is built by passing from welded objects through broken torus diagrams to ribbon tori, and it descends to a surjective map on knot classes. Ribbon tori with their fillings correspond bijectively to welded tori, while filling changes explain Reidemeister invariance and global-reversal invariance.

2. The Tube map

  • The composite T = R ◦ B sends welded tori through broken torus diagrams to ribbon tori, and both component maps descend to well-defined maps.The construction thickens a virtual diagram to a broken torus and pushes tubes near singular circles into the fourth dimension to obtain a ribbon torus with filling.Source: §2 · pp. 9–10
  • The inverse construction W records ribbon disks using their cyclic order and signs to form a welded Gauss diagram.It distinguishes essential and contractible preimages and uses the co-orientation to order the essential preimages.Source: §2 · p. 10
  • The map T is a bijection from welded tori to ribbon tori, making welded tori a faithful combinatorial description of ribbon tori.The proof constructs an inverse and establishes its injectivity using extensions of contractible preimages and isotopies of the resulting fillings.Source: §2 · pp. 10–11
  • The Tube map descends to a well-defined surjection from welded knot classes to ribbon torus-knot classes.Surjectivity follows from the bijection on tori.Source: §2 · pp. 11–12
  • Broken Reidemeister moves preserve the boundary torus while allowing different fillings, which makes the Tube map invariant under those moves.The proof exhibits alternative fillings for the same tubes that project to the two sides of each broken move.Source: §2 · p. 11
  • The ∂-equivalence forgets the filling and therefore permits changes that broken Reidemeister moves do not capture, including co-orientation changes.Reversing a welded torus's orientation alone does not generally produce the same broken torus with reversed co-orientation.Source: §2 · pp. 9, 12
  • The Tube map is invariant under global reversal, which simultaneously reverses diagram orientation and crossing signs.The paper denotes this simultaneous operation by glRev.Source: §2 · p. 12

3. A digression on wens

A wen is a portion of a ribbon torus-knot modeled as a Klein bottle cut along a meridional circle. Boundary-fixed isotopies make wens interchangeable and allow them to encode filling changes, Reidemeister I, and torus inversion.

3. A digression on wens

  • A wen is locally a circle making a half-turn inside a 3-ball with an extra time parameter, and its boundary components are fixed for the relevant isotopies.As a portion of a ribbon torus-knot, it is an embedded Klein bottle cut along a meridional circle.Source: §3 · pp. 12–13
  • Any two wens are ∂–isotopic, and gluing two along a boundary circle yields a trivially embedded cylinder up to ∂–isotopy.The proof uses paths of diffeomorphisms of the 3-ball and the contractibility of Diff(B3; ∂B3).Source: §3 · pp. 13–14
  • Filled wens can untwist twisting near ribbon-disk ends, and paired cancellations support the rectification of 3-balls in the proof of Proposition 2.4.The text also uses wens to realize Reidemeister I as a change of filling and gives a broken-torus-diagram isotopy for the move.Source: §3 · p. 14
  • A pair of wens can be created and canceled after one travels around a trivially embedded torus, thereby inverting its orientation in S4.For a knotted torus, the traveling wen changes the neighborhood of each ribbon disk and reverses its sign.Source: §3 · p. 14
  • Whether classical Reidemeister moves and co-orientation reversal generate every possible change of filling remains an open question.The text asks equivalently whether ribbon torus-knots correspond one-to-one with welded knots up to global reversal.Source: §3 · p. 14
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