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Link groups of Kishino knot stacks

Blake K Winter

TL;DR

The paper defines and analyzes the stack invariant for virtual links, constructing $S_{a_i}(L)$ by stacking copies of $L$ and its vertical reflection and treating the resulting link groups and quandles as invariants of the original link. Focusing on Kishino knots, it computes the stack groups for seven knots and shows that two-layer stacks distinguish five of them from the unknot, while for two knots the stack groups are free and thus not detectable by group/quandle invariants alone, though their Jones polynomials do detect nontriviality. A key general result is that if $S_{+-}(K)$ and $S_{++}(K)$ are free on $2m$ generators, any $n$-layer stack is free on $nm$ generators, implying intrinsic limitations of the invariants with deeper stacking for certain knots. The work highlights both the power and limits of stack-based algebraic invariants and connects these with explicit Jones polynomial computations, raising questions about the relationship between stack and original Jones polynomials.

Abstract

For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and the fundamental group of such a link may be used to detect whether the link is nontrivial and whether it is nonclassical in some cases. We show that this group is able to distinguish five Kishino knots from the unknot using a stacked pair. However, for two other Kishino knots, the group and quandle of any stack invariant will be free with a number of generators equal to the number of copies in the stack, although the Jones polynomial of the stacks is able to detect their nontriviality.

Link groups of Kishino knot stacks

TL;DR

The paper defines and analyzes the stack invariant for virtual links, constructing by stacking copies of and its vertical reflection and treating the resulting link groups and quandles as invariants of the original link. Focusing on Kishino knots, it computes the stack groups for seven knots and shows that two-layer stacks distinguish five of them from the unknot, while for two knots the stack groups are free and thus not detectable by group/quandle invariants alone, though their Jones polynomials do detect nontriviality. A key general result is that if and are free on generators, any -layer stack is free on generators, implying intrinsic limitations of the invariants with deeper stacking for certain knots. The work highlights both the power and limits of stack-based algebraic invariants and connects these with explicit Jones polynomial computations, raising questions about the relationship between stack and original Jones polynomials.

Abstract

For any virtual link, a class of new links can be defined called stacks, in which copies of the virtual link are placed on top of one another. The resulting virtual link depends only on the virtual isotopy class of the original link, and the fundamental group of such a link may be used to detect whether the link is nontrivial and whether it is nonclassical in some cases. We show that this group is able to distinguish five Kishino knots from the unknot using a stacked pair. However, for two other Kishino knots, the group and quandle of any stack invariant will be free with a number of generators equal to the number of copies in the stack, although the Jones polynomial of the stacks is able to detect their nontriviality.
Paper Structure (3 sections, 1 theorem, 24 equations, 8 figures)

This paper contains 3 sections, 1 theorem, 24 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Stack Invariant
  3. Kishino Knot Calculations

Key Result

Theorem 1

Let $K$ be a virtual link with $S_{+-}(K)$ and $S_{++}(K)$ having link groups that are free on $2m$ generators, where $m$ is the number of components of $K$. Then any stack with $n$ layers for $K$ will have a link group that is free on $nm$ generators.

Figures (8)

  • Figure 1: The knot $K$ with the vertical double on the right and the double on the left.
  • Figure 2: The knot $K_{switch}$ with the vertical double on the right and the double on the left.
  • Figure 3: The knot $K_{alt}$ with the vertical double on the right and the double on the left.
  • Figure 4: The knot $K_{v}$ with the vertical double on the right and the double on the left.
  • Figure 5: The knot $K_{5}$ with the vertical double on the right and the double on the left.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 2