[Paper Review] Milnor's triple linking numbers and derivatives of genus three knots

JungHwan Park|arXiv (Cornell University)|Mar 30, 2016

Geometric and Algebraic Topology15 references3 citations

TL;DR

This paper investigates Milnor's triple linking numbers of derivatives of genus three algebraically slice knots, proving that for such knots, the difference in triple linking numbers between any two derivatives associated with a fixed metabolizer forms a subgroup of the integers generated by a specific expression involving the Seifert matrix. A key result shows that for the unknot with a fixed Seifert surface and metabolizer, all integers can be realized as triple linking numbers of derivatives, and connected sums of three genus one algebraically slice knots admit at least one derivative with non-zero triple linking number.

ABSTRACT

A derivative of an algebraically slice knot $K$ is an oriented link disjointly embedded in a Seifert surface of $K$ such that its homology class forms a basis for a metabolizer $H$ of $K$. We show that for a genus three algebraically slice knot $K$, the set $\{ \barμ_{\{γ_1,γ_2,γ_3\}}(123) - \barμ_{\{γ'_1,γ'_2,γ'_3\}}(123)| \{γ_1,γ_2,γ_3\}$ and $\{γ'_1,γ'_2,γ'_3\}$ are derivatives of $K$ associated with a metabolizer $H\}$ contains $n\cdot \mathbb{Z}$ where $n$ is determined by a Seifert form of $K$ and a metabolizer $H$. As a corollary, we show that it is possible to realize any integer as a Milnor's triple linking number of a derivative of the unknot on a fixed Seifert surface with a fixed metabolizer. In addition, we show that a knot, which is a connected sum of three genus one algebraically slice knots, has at least one derivative which has non-zero Milnor's triple linking number.

Motivation & Objective

• To understand the behavior of Milnor’s triple linking numbers for derivatives of genus three algebraically slice knots.
• To determine the set of possible differences in triple linking numbers between different derivatives associated with a fixed metabolizer.
• To show that any integer can be realized as a triple linking number of a derivative of the unknot on a fixed Seifert surface with a fixed metabolizer.
• To prove that a connected sum of three genus one algebraically slice knots has at least one derivative with non-zero triple linking number.

Proposed method

• The paper uses a symplectic basis for the first homology of a genus three Seifert surface, with a metabolizer H spanned by three homology classes.
• It constructs a Seifert matrix M with respect to the basis {a₁,b₁,a₂,b₂,a₃,b₃}, where the entries encode linking data and the metabolizer condition β_F vanishes on H.
• The main result is derived from a formula involving the entries of the Seifert matrix: ((a−1)(b−1)(c−1) − abc + x₁x₂ + y₁y₂ + z₁z₂), which generates the set of differences in triple linking numbers.
• The proof relies on properties of Seifert forms, metabolizers, and the definition of Milnor’s triple linking number as an invariant of link concordance.
• Corollaries are derived by specializing the main theorem to the unknot and to connected sums of genus one algebraically slice knots.
• The argument uses number-theoretic techniques, including GCDs and Bézout identities, to construct new metabolizers and transform Seifert matrices.

Experimental results

Research questions

• RQ1 What is the structure of the set of differences in Milnor’s triple linking numbers between different derivatives of a genus three algebraically slice knot associated with a fixed metabolizer?
• RQ2 Can any integer be realized as a Milnor’s triple linking number of a derivative of the unknot on a fixed Seifert surface with a fixed metabolizer?
• RQ3 Does a connected sum of three genus one algebraically slice knots always admit at least one derivative with non-zero triple linking number?
• RQ4 How does the choice of Seifert matrix and metabolizer affect the range of possible triple linking numbers for derivatives?
• RQ5 Is the set S_{K,H} of differences in triple linking numbers exactly equal to the integer span of the expression derived from the Seifert matrix?

Key findings

• For any genus three algebraically slice knot K with a fixed metabolizer H, the set of differences in Milnor’s triple linking numbers between derivatives associated with H contains n·((a−1)(b−1)(c−1) − abc + x₁x₂ + y₁y₂ + z₁z₂) for all n ∈ ℤ.
• For the unknot with a fixed Seifert surface and metabolizer H, the set of all possible triple linking numbers of its derivatives is exactly ℤ, meaning every integer is realizable.
• The connected sum of three genus one algebraically slice knots has at least one derivative with non-zero Milnor’s triple linking number.
• The expression ((e₁−1)(e₂−1)(e₃−1) − e₁e₂e₃) is non-zero for certain choices of Seifert matrices of genus one knots, ensuring non-triviality in the connected sum case.
• The main theorem shows that the set S_{K,H} contains a non-trivial subgroup of ℤ, and the paper leaves open whether equality holds.
• The result demonstrates that even for the simplest knot (the unknot), the triple linking numbers of derivatives can be arbitrarily large in absolute value, showing complexity in the invariant's behavior.