Computational experimentation is an essential apparatus in low-dimensional topology. While smooth 4-manifold topology and knot concordance ultimately rely on rigorous geometric and algebraic proofs, computation allows us to test conjectures against large tables of knots, compute complex invariants that are prohibitive by hand, and isolate promising candidates for theoretical analysis. Rather than constituting separate research topics, computational tasks integrate directly into our theoretical projects—serving as the experimental engine for studying concordance, sliceness obstructions, and knotted surfaces.

Students interested in computational work will primarily use Python, SageMath, and specialized topological packages such as SnapPy and Spherogram. These workflows do not require prior experience in advanced topology; they require solid programming fundamentals, an understanding of basic data structures (such as graphs, lists, and matrices), and the ability to translate diagrammatic topological rules into algorithmic steps.

Potential Computational Tasks

Task 1: Building Automated Invariant Pipelines for Sliceness Obstructions

When investigating whether an infinite family of knots contains slice candidates, checking classical obstructions manually for each parameter becomes intractable. This task involves developing Python and SageMath scripts to automate the calculation of classical sliceness obstructions across parametric families (such as pretzel knots $P(p, q, r)$, twist knots, or weaving links $W(p, q)$):

  • Fox–Milnor Verification: Implement an algorithmic check to determine whether the symmetrized Alexander polynomial $\Delta_K(t) \in \mathbb{Z}[t, t^{-1}]$ factors in the form $f(t)f(t^{-1})$ for some polynomial $f(t) \in \mathbb{Z}[t]$ with $\vert{}f(1)\vert{} = 1$.
  • Levine–Tristram Signature Scanning: Compute Seifert matrices $V$ for a given knot diagram, construct the Hermitian matrices $(1 - \omega)V + (1 - \bar{\omega})V^T$ for unit complex numbers $\omega = e^{i\theta}$, and evaluate the signature jumps across the unit circle.
  • Arf Invariant Calculation: Extract the mod 2 intersection form and quadratic refinement from the Seifert matrix to evaluate $Arf(K) \in {0, 1}$ algorithmically, cross-checking the result against $\Delta_K(-1) \pmod 8$.

Task 2: Combinatorial Diagram Manipulation and Pass Move Calculus

Geometric operations such as band pass moves and band surgeries alter knot diagrams locally while preserving or systematically changing specific invariants. This task focuses on implementing diagrammatic calculus directly on combinatorial representations of knots:

  • Parsing Diagram Encodings: Write parser functions to translate between standard knot notations, including Planar Diagram (PD) codes, Dowker–Thistlethwaite (DT) codes, and braid words.
  • Simulating Local Diagrammatic Moves: Develop routines that identify pairs of oppositely oriented parallel strands in a PD code and execute a formal pass move by rewiring the underlying crossing connectivity.
  • Tracking Invariant Invariance: Construct test suites to verify that pass move rewiring preserves the Arf invariant and Alexander polynomial parity while monitoring how the 4-ball genus bounds and unknotting numbers fluctuate.

Task 3: High-Throughput Geometric Computations with SnapPy and Spherogram

SnapPy provides an extensive Python interface for studying 3-manifold triangulations and link exteriors, while its Spherogram module enables direct manipulation of link diagrams and planar graphs. Computational tasks using these tools include:

  • Batch Processing of Link Exteriors: Use SnapPy’s Link and Manifold classes to construct triangulations of link complements, verify hyperbolicity, and compute geometric invariants such as hyperbolic volume, cusp shapes, and symmetry groups.
  • Seifert Matrix and Form Extraction: Interface with Spherogram to construct spanning surfaces algorithmically, extract canonical Seifert matrices for multi-crossing diagrams, and export coordinate data for linear algebra routines in NumPy or SageMath.
  • Concordance and Ribbon Link Filtering: Utilize SnapPy’s band-attachment utilities to test whether designated knots can be reduced to trivial links through sequences of ribbon-admissible band moves.

Task 4: Tri-Plane Diagram Verification for Knotted Surfaces

Studying knotted surfaces in $S^4$ via bridge trisections requires tracking three interdependent planar link diagrams that satisfy strict boundary compatibility conditions. This task focuses on data modeling and structural validation for higher-dimensional diagrammatics:

  • Data Structures for Tri-Planes: Design a structured representation (such as a triple of linked PD code objects) that encodes the three planar projections of a knotted surface.
  • Compatibility and Boundary Checks: Write verification scripts that test whether the trivial arc endpoints in the three half-spaces align correctly and satisfy the required Euler characteristic and surface topology constraints.
  • Crossing Complexity Minimization: Implement combinatorial search heuristics to apply tri-plane Reidemeister moves algorithmically, searching for representations that minimize bridge index or global crossing number.

Computing Environment and Getting Started

All code developed for these tasks will be written in Python 3 within Jupyter notebooks or modular script repositories. Students can configure a local environment on macOS using Homebrew (brew) and pip to install SageMath, SnapPy (along with the complete SnapPy documentation), and standard scientific libraries (NumPy, SciPy, Matplotlib).

For an all-in-one containerized setup preconfigured with SageMath, SnapPy, Regina, and related topological tools, students can run the 3-manifolds/sagedocker image (computop/sage on Docker Hub).

Alternatively, preliminary experiments can be run directly in the browser via JupyterLite without requiring any local software installation.

For a comprehensive clearinghouse of software for low-dimensional topology and geometry, consult Nathan Dunfield’s computop.org.

For visual and interactive diagram manipulation—particularly for exploring knotted surfaces and cross-sectional movies in 4-space—students can use Frank Swenton’s KLO (Knot-Like Objects).