Geodesic
Bridge trisections and Seifert solids
Joseph, Jason1 ; Meier, Jeffrey2 ; Miller, Maggie3 ; Zupan, Alexander4
1Department of Mathematics, North Carolina School of Science and Mathematics, Morganton, NC, United States
2Department of Mathematics, Western Washington University, Bellingham, WA, United States
3Department of Mathematics, University of Texas at Austin, Austin, TX, United States
4Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, United States
Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1501-1522
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We adapt Seifert’s algorithm for classical knots and links to the setting of triplane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard diagram. The Seifert solids produced can be assumed to have exteriors that can be built without 3-handles; in contrast, we give examples of Seifert solids (not coming from our construction) whose exteriors require arbitrarily many 3-handles. We conclude with two classification results. The first shows that surfaces admitting doubly standard shadow diagrams are unknotted. The second says that a b-bridge trisection in which some sector contains at least b − 1 patches is completely decomposable, thus the corresponding surface is unknotted. This settles affirmatively a conjecture of the second and fourth authors.

DOI : 10.2140/agt.2025.25.1501
Keywords: knot theory, surface, trisection, bridge trisection, 2-knot, Seifert surface, Seifert solid

Joseph, Jason  1   ; Meier, Jeffrey  2   ; Miller, Maggie  3   ; Zupan, Alexander  4

1 Department of Mathematics, North Carolina School of Science and Mathematics, Morganton, NC, United States
2 Department of Mathematics, Western Washington University, Bellingham, WA, United States
3 Department of Mathematics, University of Texas at Austin, Austin, TX, United States
4 Department of Mathematics, University of Nebraska–Lincoln, Lincoln, NE, United States 
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Joseph, Jason; Meier, Jeffrey; Miller, Maggie; Zupan, Alexander. Bridge trisections and Seifert solids. Algebraic and Geometric Topology, Tome 25 (2025) no. 3, pp. 1501-1522. doi: 10.2140/agt.2025.25.1501

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