Peripheral structures of core groups

TL;DR

The paper introduces a peripheral refinement of the core group $AC(L)$ for virtual links by pairing meridians with orientation-sensitive longitudes into a peripheral structure ${\cal P}(D)$. It shows that the augmented invariant $(AC(L),{\cal M}(L))$ is equivalent to the $\,\pi$-orbifold group $O(L)$ with its peripheral data, unifying these perspectives for unoriented links and allowing detection of subtle phenomena such as noninvertibility and orientation-sensitive distinctions in the Borromean rings. The authors develop the involutory meridional automorphisms $\iota_{g_a}$, show that the corresponding automorphism group ${\cal I}(D)$ is determined by the meridians, and relate $AC(L)$ to $O(L)$ via a detailed comparison of peripheral structures, including a constructive link through $D^+$. They illustrate the strength of the peripheral data with examples: recasting Trotter’s noninvertibility within the core-group framework, distinguishing two 4-component links with identical $AC$ groups, and separating the two oriented Borromean rings. Overall, the work extends classical peripheral concepts to virtual links, provides a new classifying invariant for sufficiently complicated links, and demonstrates practical distinctions beyond the core group alone.

Abstract

The core group is an invariant of unoriented virtual links. We introduce a peripheral structure for the core group, in which the longitudes are sensitive to orientations. We show that the combination of the core group and its peripheral structure is equivalent, as a link invariant, to the combination of the $π$-orbifold group and its peripheral structure. Examples show that the peripheral structure of the core group can be used to verify noninvertibility of some knots and links.

Figures (5)

• Figure 1: The Hopf link, $H$.
• Figure 2: The crossing $c_{ij}$ has writhe $w_{ij}$.
• Figure 3: Diagram $D$ for K(7,-3,5)
• Figure 4: The links $L$ and $L'$.
• Figure 5: Two oriented versions of the Borromean rings.

Theorems & Definitions (46)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof