🪢Knot Theory Unit 5 – Seifert Surfaces and Genus

What's the Big Idea?

• Seifert surfaces provide a way to study knots and links in three-dimensional space
• Every knot or link bounds a compact, connected, orientable surface called a Seifert surface
• The genus of a knot is the minimum genus of all possible Seifert surfaces for that knot
  • Genus is a non-negative integer that measures the complexity of the knot
• Seifert surfaces and genus play a crucial role in understanding knot invariants and properties
• Constructing Seifert surfaces allows for the calculation of various knot invariants
  • Includes the Alexander polynomial and the knot signature
• The study of Seifert surfaces and genus connects knot theory with other areas of mathematics
  • Relates to topics in algebraic topology, geometric topology, and 4-manifold theory

Key Concepts and Definitions

• Knot: An embedding of a circle $S^1$ into the 3-dimensional space $\mathbb{R}^3$ or the 3-sphere $S^3$
• Link: A collection of knots that may be intertwined or linked together
• Seifert surface: A compact, connected, orientable surface whose boundary is a given knot or link
  • Named after German mathematician Herbert Seifert who introduced the concept in 1934
• Genus: The minimum number of handles required to construct a Seifert surface for a knot
  • Genus is a non-negative integer ($g \geq 0$)
  • The unknot has genus 0, as it bounds a disk
• Orientable surface: A surface that has a consistent notion of "clockwise" and "counterclockwise"
• Knot invariant: A property of a knot that remains unchanged under continuous deformations (isotopies)
  • Examples include genus, crossing number, and various knot polynomials

Historical Background

• The study of knots and links has its roots in the late 19th century
  • Motivated by questions in physics and chemistry related to the behavior of molecules
• In 1934, Herbert Seifert introduced the concept of Seifert surfaces in his paper "Über das Geschlecht von Knoten" (On the genus of knots)
  • Showed that every knot or link bounds a compact, connected, orientable surface
• The genus of a knot was initially defined as the minimum genus of all possible Seifert surfaces for that knot
• In the 1960s, the development of knot polynomials (Alexander, Jones, HOMFLY-PT) provided new tools for studying knots and links
  • Many of these polynomials can be computed using Seifert surfaces
• The study of Seifert surfaces and genus has since become a central topic in knot theory
  • Connections to other areas of mathematics continue to be discovered and explored

Constructing Seifert Surfaces

• Seifert's algorithm: A systematic method for constructing a Seifert surface from a knot diagram
  1. Orient the knot diagram
  2. Smooth each crossing according to the orientation (Seifert smoothing)
  3. Fill in the resulting circles (Seifert circles) with disks
  4. Connect the disks with twisted bands corresponding to the original crossings
• The resulting surface is a Seifert surface for the given knot
  • May not be the minimal genus Seifert surface
• Other methods for constructing Seifert surfaces include:
  • Using a braid representation of the knot (Bennequin surface)
  • Applying Seifert's algorithm to a different knot diagram (after Reidemeister moves)
  • Constructing a branched cover over the 3-sphere branched along the knot
• The process of finding a minimal genus Seifert surface for a knot is a challenging problem
  • Algorithms exist for specific classes of knots (alternating, pretzel, etc.)

Calculating Genus

• The genus of a knot $K$, denoted $g(K)$, is the minimum genus of all possible Seifert surfaces for $K$

• For a Seifert surface $S$ with $n$ Seifert circles and $c$ crossings in the knot diagram, the genus is given by:

  $g(S) = \frac{1}{2}(2 - n + c)$

• This formula provides an upper bound for the genus of the knot

  • The actual genus may be lower if a minimal genus Seifert surface is found

• The genus of a knot is a knot invariant

  • Remains unchanged under continuous deformations (isotopies) of the knot

• Knots with the same genus can have different properties and invariants

  • For example, the trefoil knot and the figure-eight knot both have genus 1

• Calculating the genus of a knot is a difficult problem in general

  • No known algorithm for determining the genus of an arbitrary knot
  • Lower bounds can be obtained using knot invariants (Alexander polynomial, knot signature)

Applications in Knot Theory

• Seifert surfaces and genus are used to define and compute various knot invariants
  • The Alexander polynomial can be computed from a Seifert matrix associated with a Seifert surface
  • The knot signature is defined using the signature of the symmetrized Seifert matrix
• The genus of a knot provides information about its complexity and properties
  • Knots with higher genus are generally considered more complex
  • Genus can be used to distinguish between different knots with the same crossing number
• Seifert surfaces are used in the study of knot concordance and slice knots
  • A knot is slice if it bounds a disk in the 4-ball $B^4$
  • Slice knots have genus 0 in the 4-ball
• The study of Seifert surfaces and genus has connections to other areas of mathematics
  • Relates to the study of 3-manifolds and 4-manifolds
  • Connections to algebraic topology and geometric topology

Common Challenges and Misconceptions

• Finding a minimal genus Seifert surface for a given knot is a challenging problem
  • No known algorithm for determining the genus of an arbitrary knot
  • The Seifert surface obtained from Seifert's algorithm may not be minimal genus
• The genus of a knot is not determined by its crossing number
  • Knots with the same crossing number can have different genera
  • For example, the trefoil knot and the figure-eight knot both have crossing number 4 but different genera
• The genus of a knot is not additive under connected sum
  • The genus of the connected sum of two knots is not always the sum of their individual genera
  • Counterexample: The connected sum of two trefoil knots has genus 1, not 2
• Seifert surfaces are not unique for a given knot
  • Different Seifert surfaces for the same knot may have different genera
  • Finding all possible Seifert surfaces for a knot is an open problem

Further Explorations

• Studying the relationship between the genus of a knot and other knot invariants
  • Investigating connections between genus and crossing number, braid index, bridge number, etc.
• Exploring the properties of minimal genus Seifert surfaces
  • Characterizing minimal genus Seifert surfaces for specific classes of knots
  • Studying the uniqueness and structure of minimal genus Seifert surfaces
• Investigating the behavior of genus under knot operations
  • Connected sum, cabling, mutation, etc.
• Applying Seifert surfaces and genus to the study of 3-manifolds and 4-manifolds
  • Using Seifert surfaces to construct and study 3-manifolds (branched covers, Seifert fibered spaces)
  • Investigating the role of genus in the study of slice knots and knot concordance
• Exploring the connections between Seifert surfaces, genus, and other areas of mathematics
  • Algebraic topology, geometric topology, gauge theory, etc.
• Developing new algorithms and computational methods for studying Seifert surfaces and genus
  • Improving existing algorithms for constructing and simplifying Seifert surfaces
  • Exploring the use of machine learning and AI techniques in the study of knots and surfaces