Touseef Haider

An intro to my research:

One problem we worked on is a subproblem of knot equivalence. This problem states that: given two knot diagrams, do they represent equivalent knots? This problem has been decidable since the 1960s and 1970s. One common approach to this problem involves using Reidemeister moves, which are a set of diagrammatic moves to "simplify" the diagram so one can identify if the two diagrams represent the same knot. However, if the diagrams represent different knots, the process of applying Reidemeister moves can run indefinitely. So, one has to come up with an upper bound on the number of Reidemeister moves required. One known bound, by the work of Coward and Lackenby, exists; however, this bound is huge and is expressed as a tower of exponentials in terms of crossings. A natural question is whether this bound can be improved. By restricting this problem to alternating knots, we showed that the Alternating knot equivalence problem has a polynomial-time algorithm in terms of crossing number.

Polynomial algorithm for alternating link equivalence (with A. Tsvietkova): arXiv

One of our projects focused on computing the meridian length for an infinite family of links that are closures of 4-braids with word \( \left( \sigma_{2}^{-1} \sigma_{1} \sigma_{3} \sigma_{2}^{-1} \right)^{n} \) for odd \( n \geq 3 \). While software like SnapPy makes it possible to compute link invariants, it is not well-suited for handling the computation of hyperbolic invariants for infinite families of links, as the time complexity grows exponentially as the number of crossings increases, making it impractical. Using methods from Thistlethwaite and Tsvietkova, we derived an exact algebraic expression for the meridian length of this infinite family. Moreover, in our work, we developed a generalized formula that provides an exact value for the meridian length of the infinite family.

Publications and Preprints

• AI Experiments of 2-component link (in progress)
  (with S. Jayasekara, S. Kumar, R. Tripathi)
• A Summer(2025) of Lean4 (in progress)
  (with J. Ahmed, I. Alam, J. Alfredo, T. Ansari, M.H. Shaad, M. Huzaifa, M.D. Khan, M.S. Siddiqui, R. Trujilo)
• Number of Flypes and Exact Meridian Length From Alternating Link Diagrams (PhD thesis)
  View thesis on ProQuest
• Polynomial algorithm for alternating link equivalence(accepted in Proceedings of the AMS)
  with A. Tsvietkova
  arXiv

Talks and Poster Presentations

• AMS Fall 2024 Eastern Sectional Meeting, University at Albany, Albany, NY. (10/2024)
• 41st Workshop in Geometric Topology, Calvin University, Grand Rapids, MI. (06/2024)
• Knots informed by random models and experimental data, BIRS Workshop, Banff, Canada. (04/2024)
• Poster Presentation, Okinawa Institute of Science and Technology (OIST), Japan. (04/2019)

Participation in Workshops and Conferences

• Braid Reunion Workshop, ICERM, Brown University, Providence, RI. (07/2024)
• Unknot V, Seattle University, Seattle, WA. (07/2024)
• 9th Annual Graduate Student Conference in GTA, Temple University, Philadelphia, PA. (05/2024)
• The Nearly Carbon Neutral Geometric Topology Conference (Online). (06/2023)
• Geometric Topology in New York, Columbia University, NYC, NY. (06/2023)
• Computational Problems in Low-dimensional Topology-III, Rutgers University, Newark, NJ. (04/2023)
• The South Central Topology Conference II, Texas State University, San Marcos, TX. (02/2023)
• Computational Problems in Low-dimensional Topology-II, OIST, Okinawa, Japan. (04/2019)
• School on Low-dimensional Geometry and Topology, Institute Henri Poincaré, Paris, France. (06/2018)
• Geometry and Topology of 3-Manifolds, OIST, Okinawa, Japan. (05/2018)
• Computational Problems in Low-dimensional Topology-I, OIST, Okinawa, Japan. (04/2018)
• Quantum Information Summer School, Habib University, Karachi, Pakistan. (06/2016)