A Combinatorial Description of the Knot Concordance Invariant Epsilon

TL;DR

The paper delivers a fully combinatorial description of the knot concordance invariant $\varepsilon$ via grid homology, establishing its status as a concordance invariant and providing an explicit grid-based procedure to compute it. It obtains concrete values for $\varepsilon$ on negative torus knots and shows how $\varepsilon$ behaves under $(p,q)$-cabling, including preservation when the original invariant is nonzero and reduction to torus-knot values when it vanishes. It further proves that $\varepsilon=1$ for grid diagrams of positive braids, connecting to fibered and strongly quasipositive knot theory, and discusses the implications for connected sums and mirror symmetry. The work advances computability in knot concordance and clarifies how grid-homology invariants interact with cabling, connected sums, and braid closures, with future extensions to lens spaces anticipated.

Abstract

In this paper, we give a combinatorial description of the concordance invariant $\varepsilon$ defined by Hom in \cite{hom2011knot}, prove some properties of this invariant using grid homology techniques. We also compute $\varepsilon$ of $(p,q)$ torus knots and prove that $\varepsilon(\mathbb{G}_+)=1$ if $\mathbb{G}_+$ is a grid diagram for a positive braid. Furthermore, we show how $\varepsilon$ behaves under $(p,q)$-cabling of negative torus knots.

Figures (19)

• Figure 1: How to obtain a grid diagram for the Figure-8 knot. Notice that this diagram is $13\times 13$ and we can make it smaller by local isotopies, commutations and destabilizations
• Figure 2: Grid diagram for the left-handed trefoil (LHT), two grid states $\mathbf{x}$ and $\mathbf{y}$ are given by empty and full circles respectively. Darker shaded rectangle and lighter shaded rectangle are two rectangles from $\mathbf{x}$ to $\mathbf{y}$. Notice that darker shaded rectangle is counted by the differential whereas lighter shaded one is not since it is not an empty rectangle
• Figure 3: From left to right, we start with the standard grid diagram for the unknot $\mathbb{G}_{\mathcal{O}}$, reflect it with respect to a horizontal axis and then change $\mathbb{X}$ and $\mathbb{O}$ markings to obtain $\shortminus \mathbb{G}_{\mathcal{O}}$
• Figure 4: Grid diagram $\mathbb{G}_{-2,3}$ for the left-handed trefoil, grid index is 5 and $\sigma_{\mathbb{O}}=(1, 2, 3, 4, 5)$ and $\sigma_{\mathbb{X}}=(3, 4, 5, 1, 2)$
• Figure 5: The first picture on the top is a grid diagram for $T_{-2,3}$, while the first picture in the second row shows a part of reduced$CFK^{\infty}(T_{-2,3})$. The yellow dots in the grid picture indicate the grid state which is the generator of the $\mathbb{F}[U]$ part of $GH^{-}(T_{-2,3})$. The second and third pictures on top row show the effect of mirroring and interchanging the $\mathbb{X}$- and $\mathbb{O}$-markings on the diagram, respectively. In the second row, the yellow dot represents the generator of the free part of $HFK^{-}(T_{-2,3})$ and the next two pictures show the effect of mirroring and orientation reversal of $T_{-2,3}$ on that generator. In the final picture of both rows, the element represented by red dots kills the distinguished homology class.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2: ozsvath2015grid, Proposition 7.4.3
• Proposition 2.3: ozsvath2015grid, Proposition 7.1.1
• Lemma 2.4: ozsvath2015grid, Lemma 4.6.9
• Definition 2.5
• Definition 2.6: Alternative Definition