🪢Knot Theory Unit 6 – The Alexander Polynomial

What's the Alexander Polynomial?

• Polynomial invariant associated with knots and links in knot theory
• Discovered by James Waddell Alexander II in 1923
• Assigns a polynomial with integer coefficients to each oriented knot or link
• Invariant under ambient isotopy meaning it remains unchanged under continuous deformations of the knot that do not involve cutting or passing through itself
• Encodes important topological information about the knot or link
  • Provides insights into the complexity and structure of the knot
  • Helps distinguish between different knot types
• Calculated using a specific algorithm involving the knot diagram and its crossings
• Serves as a powerful tool for studying and classifying knots and links in three-dimensional space

Historical Context

• Developed by American mathematician James Waddell Alexander II in 1923
• Alexander was a pioneer in the field of topology and made significant contributions to knot theory
• Published his groundbreaking paper "Topological Invariants of Knots and Links" which introduced the Alexander polynomial
• Builds upon earlier work on knot invariants by mathematicians such as Peter Guthrie Tait and Carl Friedrich Gauss
• Represents a major milestone in the development of knot theory as a mathematical discipline
• Opened up new avenues for research and sparked interest in the study of knot invariants
• Continues to be a fundamental concept in modern knot theory and has inspired further generalizations and extensions

Key Concepts and Definitions

• Knot: Embedding of a circle into three-dimensional space forming a closed loop
• Link: Collection of intertwined knots where the knots may be interlocked but do not intersect
• Knot diagram: Two-dimensional projection of a knot onto a plane with over/under crossing information at each intersection
• Crossing: Point in a knot diagram where the strands of the knot intersect
  • Positive crossing: Strand passes over the other from left to right
  • Negative crossing: Strand passes under the other from left to right
• Ambient isotopy: Continuous deformation of a knot that does not involve cutting or passing through itself
• Knot invariant: Quantity or property associated with a knot that remains unchanged under ambient isotopy
• Laurent polynomial: Polynomial allowing both positive and negative exponents

Calculating the Alexander Polynomial

• Computed from a knot diagram using a specific algorithm
• Steps involve manipulating the knot diagram and keeping track of crossings
• Begin by labeling the arcs of the knot diagram with variables (usually $t$ or $x$)
• Assign a matrix to each crossing in the diagram based on the labeling
  • Matrix encodes the relationship between the variables at the crossing
• Construct the Alexander matrix by combining the matrices from all crossings
• Calculate the determinant of the Alexander matrix
  • Determinant is a polynomial in terms of the labeling variables
• Normalize the polynomial by dividing by the lowest degree term and evaluating at $t-1$
• Resulting polynomial is the Alexander polynomial of the knot
• Process can be adapted for links by considering each component separately and combining the results

Properties and Characteristics

• Polynomial invariant meaning it remains unchanged under ambient isotopy of the knot
• Coefficients are integers and the polynomial is symmetric
  • Coefficients satisfy $a_i = a_{-i}$ where $a_i$ is the coefficient of the term with exponent $i$
• Degree of the polynomial provides a lower bound on the genus of the knot
  • Genus is a measure of the complexity of the knot related to the minimum number of handles needed to construct a surface bounded by the knot
• Evaluating the Alexander polynomial at specific values yields other knot invariants
  • Example: Alexander polynomial evaluated at $-1$ gives the determinant of the knot
• Satisfies certain skein relations that relate the polynomials of knots differing by specific local changes at a crossing
• Multiplicative under connected sum of knots
  • Connected sum is the operation of joining two knots by cutting them open and gluing the ends together
• Alexander polynomial of the mirror image of a knot is obtained by replacing $t$ with $t^{-1}$ in the original polynomial

Applications in Knot Theory

• Powerful tool for distinguishing between different knot types
  • If two knots have different Alexander polynomials, they are necessarily distinct
  • Converse is not true: knots with the same Alexander polynomial may still be different
• Helps in the classification and tabulation of knots
  • Knots are often organized by their crossing number and Alexander polynomial
• Provides insights into the structure and complexity of knots
  • Degree of the polynomial gives information about the genus and hence the complexity of the knot
• Used in the study of knot concordance and slice knots
  • Concordance relates to the existence of a smooth embedding of the knot in four-dimensional space
  • Slice knots are those that bound a smooth disk in four dimensions
• Plays a role in the computation of other knot invariants and the development of new knot theoretic techniques
• Has applications beyond knot theory in areas such as statistical mechanics and quantum field theory

Examples and Practice Problems

• Unknot (trivial knot): Alexander polynomial is 1
• Trefoil knot: Alexander polynomial is $t^2 - t + 1$
• Figure-eight knot: Alexander polynomial is $t^2 - 3t + 1$
• Cinquefoil knot: Alexander polynomial is $t^4 - t^3 + t^2 - t + 1$
• Whitehead link: Alexander polynomial is $(t^2 - t + 1)(t^2 - t)$
• Practice calculating the Alexander polynomial for simple knot diagrams
  • Start with knots with a small number of crossings and work up to more complex examples
• Verify the properties of the Alexander polynomial using specific knot examples
  • Check symmetry of coefficients, behavior under mirror image, and multiplicativity under connected sum
• Explore the relationship between the Alexander polynomial and other knot invariants for given knots

Connections to Other Knot Invariants

• Related to the Conway polynomial and the Jones polynomial
  • Alexander polynomial can be obtained from the Conway polynomial by a specific substitution
  • Jones polynomial generalizes the Alexander polynomial and captures additional knot information
• Linked to the knot group and the fundamental group of the knot complement
  • Alexander polynomial can be derived from the knot group using the Fox calculus
  • Abelianization of the knot group yields the Alexander module, from which the Alexander polynomial can be extracted
• Connects to the Seifert matrix and the Seifert surface of a knot
  • Seifert matrix encodes linking information between curves on a Seifert surface
  • Determinant of the Seifert matrix gives the Alexander polynomial (up to normalization)
• Relates to the signature and nullity of a knot
  • Signature is the number of positive eigenvalues minus the number of negative eigenvalues of the symmetrized Seifert matrix
  • Nullity is the dimension of the null space of the symmetrized Seifert matrix
• Interacts with the Vassiliev invariants and finite type invariants
  • Alexander polynomial can be expressed as a combination of Vassiliev invariants of a certain degree
• Generalizes to the multivariable Alexander polynomial for links and the twisted Alexander polynomial for knots in manifolds