[Paper Review] On the concordance orders of knots

Julia Collins|arXiv (Cornell University)|Jun 4, 2012

Geometric and Algebraic Topology6 references3 citations

TL;DR

This thesis develops computational techniques using twisted Alexander polynomials and second-order invariants to determine the concordance orders of knots in the topological concordance group 𝒞. It establishes criteria for detecting infinite order and computes the orders of all but two of the 325 prime knots with up to 12 crossings, providing a complete classification of 9-crossing prime knots and proving that n-twisted doubles of the unknot are not slice for n ≠ 0,2.

ABSTRACT

This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small number of knots, or are designed to only deal with a specific knot. The thesis builds on the results of Herald, Kirk and Livingston [HKL10] and Tamulis [Tam02] to give a series of criteria, using twisted Alexander polynomials, for determining whether a knot is of infinite order in C. There are two immediate applications of these theorems. The first is to give the structure of the subgroups of the concordance group C and the algebraic concordance group G generated by the prime knots of 9 or fewer crossings. This should be of practical value to the knot-theoretic community, but more importantly it provides interesting examples of phenomena both in the algebraic and geometric concordance groups. The second application is to find the concordance orders of all prime knots with up to 12 crossings. At the time of writing of this thesis, there are 325 such knots listed as having unknown concordance order. The thesis includes the computation of the orders of all except two of these. In addition to using twisted Alexander polynomials to determine the concordance order of a knot, a theorem of Cochran, Orr and Teichner [COT03] is applied to prove that the n-twisted doubles of the unknot are not slice for n not 0 or 2. This technique involves analysing the `second-order' invariants of a knot; that is, slice invariants (in this case, signatures) of a set of metabolising curves on a Seifert surface for the knot. The thesis extends the result to provide a set of criteria for the n-twisted double of a general knot K to be slice; that is, of order 0 in C.

Motivation & Objective

• To develop general, computable criteria for determining the concordance order of knots in the topological concordance group 𝒞.
• To classify the algebraic and geometric concordance classes of all prime knots with 9 or fewer crossings.
• To determine the concordance order of all prime knots with up to 12 crossings, resolving the majority of previously unknown cases.
• To extend second-order sliceness obstructions to n-twisted doubles of arbitrary knots, proving non-sliceness for n ≠ 0,2 in the unknot case.
• To provide a practical computational framework applicable to a broad class of knots, beyond ad hoc methods.

Proposed method

• Utilizes twisted Alexander polynomials associated to metabelian representations to detect infinite order elements in the concordance group.
• Applies Casson-Gordon invariants and second-order signatures from metabolizing curves on Seifert surfaces to obstruct sliceness.
• Employs Witt group invariants and local invariants over ℝ and ℚₚ to analyze algebraic concordance classes.
• Combines computational algebra with topological constructions, including slice movies, to verify order 2 for specific knots.
• Uses the structure of the first homology of the double branched cover H₁(Σ₂; ℤ) and linking forms to compute invariants.
• Applies theorems from Cochran, Orr, and Teichner on second-order invariants to analyze the n-twisted double of a knot.

Experimental results

Research questions

• RQ1 Which knots in the topological concordance group have infinite order, and how can this be detected algorithmically?
• RQ2 What is the complete concordance classification of all prime knots with 9 or fewer crossings?
• RQ3 For which values of n is the n-twisted double of the unknot slice, and what invariants detect non-sliceness?
• RQ4 Can second-order invariants be used to determine the concordance order of n-twisted doubles of general knots?
• RQ5 What is the concordance order of each of the 325 prime knots with up to 12 crossings, and how many remain unresolved?

Key findings

• The thesis computes the concordance order for all but two of the 325 prime knots with up to 12 crossings, resolving the vast majority of previously unknown cases.
• It provides a complete classification of the subgroups of the concordance group 𝒞 and the algebraic concordance group 𝒢 generated by prime knots with 9 or fewer crossings.
• It proves that the n-twisted double of the unknot is not slice for any n ≠ 0,2, using second-order signature obstructions.
• It extends the second-order sliceness obstruction to the n-twisted double of any knot K, giving a criterion for when such a knot is of order 0 in 𝒞.
• It constructs explicit slice movies for knots like 11a₅ and 4₁, confirming they are of order 2 via saddle moves yielding unlinked unknots and order 2 knots.
• It identifies specific knots such as 9₈, 9₂₃, and 9₃₂ with non-slice twisted Alexander polynomials and non-square linking forms, confirming infinite order.