A homological definition of the Jones polynomial

By Stephen Bigelow. 2002

49 citations recorded by the source corpus.

We give a new definition of the Jones polynomial. Let L be an oriented knot or link obtained as the plat closure of a braid β ∈ B2n. We define a covering space ˜ C of the space of unordered n-tuples of distinct points in the 2n-punctured disk. We then describe two n-manifolds ˜ S and ˜ T in ˜ C, and show that the Jones polynomial of L can be defined as an intersection pairing between ˜ S and β ˜ T. Our construction is similar to one given by Lawrence, but more concrete. AMS Classification 57M25; 57M27, 20F36

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a4ad5f4fd4ff22c51c427ad811ae3950f6202123

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In Discrete Mathematics and Combinatorics

1. Tomotada Ohtsuki
2. G. Massuyeau
3. Simon Willerton
4. J. Scott Carter
5. Nathan M. Dunfield
6. A. Kricker
7. L. Rozansky
8. Brian Sanderson

In Geometry and Topology

1. Jacob Rasmussen
2. L. Rozansky
3. G. Massuyeau
4. Stavros Garoufalidis
5. Tomotada Ohtsuki
6. Toshie Takata
7. S. K. Hansen
8. Simon Willerton

In Mathematical Physics

1. Jacob Rasmussen
2. L. Rozansky
3. Stavros Garoufalidis
4. Tomotada Ohtsuki
5. Toshie Takata
6. S. K. Hansen
7. Simon Willerton
8. G. Massuyeau

Citations in the corpus

• Nilpotent slices, Hilbert schemes, and the Jones polynomial (2004, unknown): “At this point, we note that this picture is very similar to the one given by Bigelow [1] in his definition of the Jones polynomial.”
• Problems on invariants of knots and 3-manifolds (2004, background): “In the case of the Jones polynomial, no similar direct construction is known, but the approach of Bigelow [55] is tantalising.”