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Invariance and naturality of knot lattice homology and homotopy

Seppo Niemi-Colvin

Abstract

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice homology is an invariant of the smooth knot type of a generalized algebraic knot in a rational homology sphere. In that case, knot lattice homology can be realized as the cellular homology of a doubly-filtered homotopy type, which is itself invariant. Along the way, we show that the topological link type of a generalized algebraic link determines the topology of the minimal plumbing resolution for the nested singularity type used to create it. Knot lattice homotopy is a natural invariant in that diffeomorphisms of the knot that play suitably well with the minimal good resolution will provide a contractible space of morphisms between the doubly-filtered knot lattice spaces associated to any presentation.

Invariance and naturality of knot lattice homology and homotopy

Abstract

Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from potentially singular complex algebraic surfaces and complex curves inside them. We prove that knot lattice homology is an invariant of the smooth knot type of a generalized algebraic knot in a rational homology sphere. In that case, knot lattice homology can be realized as the cellular homology of a doubly-filtered homotopy type, which is itself invariant. Along the way, we show that the topological link type of a generalized algebraic link determines the topology of the minimal plumbing resolution for the nested singularity type used to create it. Knot lattice homotopy is a natural invariant in that diffeomorphisms of the knot that play suitably well with the minimal good resolution will provide a contractible space of morphisms between the doubly-filtered knot lattice spaces associated to any presentation.
Paper Structure (30 sections, 21 theorems, 88 equations, 10 figures)

This paper contains 30 sections, 21 theorems, 88 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Avenues for Future Work
  3. Links of Singularity and Generalized Algebraic Links
  4. Types of Links of Singularity and Their Resolutions
  5. Links of Normal Complex Surface Singularities
  6. Generalized Algebraic Links
  7. Checking Blow-Ups Suffice
  8. Motivations for the Equivalence Type
  9. A Cubical Construction of Lattice Homology
  10. The Lattice Chain Complex
  11. Generators of the Lattice Chain Complex
  12. The Maslov Grading on the Lattice Chain Complex
  13. Differential on the Lattice Chain Complex
  14. The Filtered Homotopy Type
  15. Other Properties and Structure
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $(Y,K)$ be a generalized algebraic knot in a rational homology sphere with a $\mathop{\mathrm{Spin}}\nolimits^c\,$ structure $\mathfrak{t}$. The chain homotopy equivalence type of depends only $(Y,K)$ and $\mathfrak{t}$ and not the particular graph with unweighted vertex $G_{v_0}$ used to represent it.

Figures (10)

  • Figure 1: Dimensionally reduced figures for an algebraic knot, a link of a normal complex surface singularity, and a generalized algebraic knot.
  • Figure 2: A resolution in the three manifold context and for a curve inside it. The extra sphere in Subfigure \ref{['subfig:knotsingres']} signifies that further blow-ups may be needed to resolve the curve. Furthermore, the curve pierces the sphere to emphasize that it should be transverse to the sphere, and in fact locally represented as a fiber of the plumbing.
  • Figure 3: An example of a graph depicting a generalized algebraic knot \ref{['subfig:tref']}, along with the various blow-up moves that can be done to said graph that occur away from the curve defining the knot (\ref{['subfig:genblowup']},\ref{['subfig:vertexblowup']},\ref{['subfig:edgeblowup']}) and those that affect the curve defining the knot (\ref{['subfig:vertexblowupspec']} and \ref{['subfig:edgeblowupspec']}).
  • Figure 4: An example of the lattice homology filtration. Subfigure \ref{['subfig:ExamplePlumbing']} gives the plumbing for this example. Subfigure \ref{['subfig:LatExample']} gives the ellipses when using the quadtratic form $\frac{K^2+2}{4}$ with the value of the height function labeled on the vertices, i.e. the characteristic cohomology classes. Subfigure \ref{['subfig:LatExampleCubed']} shows this height function discretized so that the height of a cube is the same height as its lowest vertex; this gives the filtration for lattice homology. In order to make each figure readable on their own, the colors of the ellipses in Figure \ref{['subfig:LatExample']} are lighter than those for the filtration in figure \ref{['subfig:LatExampleCubed']}. However, the relative shadings to agree, and for example the darkest ellipse in Figure \ref{['subfig:LatExample']} and the darkest cubes in Figure \ref{['subfig:LatExampleCubed']} are both at height 0.
  • Figure 5: A figure illustrating an example of the computation of knot lattice homology. Figure \ref{['subfig:ExamplePlumbingKnot']} gives the plumbing and knot used for this example. Figure \ref{['subfig:knotLatExample']} shows both the original ellipses used to create this example with $h_U$ in blue and the translated ellipses created using $h_V$ in magenta. These are done with transparency so that darker ellipses have higher values on the height functions and more magenta means $h_V$ is higher and more blue means $h_U$ is higher. Inside the vertices of Figure \ref{['subfig:knotLatExample']} are the values $(h_U,h_V)$. Figure \ref{['subfig:knotLatticeExampleCubed']} reproduces the first height function, while \ref{['subfig:knotLatticeExampleCubedRed']} provides the discritized second height function. Heights of less than -14 were rendered as white.
  • ...and 5 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 57 more