Signature and concordance of virtual knots

Bodén, Hans; Chrisman, Micah; Gaudreau, Robin

doi:10.1512/iumj.2020.69.8215

Cited by 8 publications

(21 citation statements)

References 5 publications

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“…The C -parity enjoys a topological definition in terms of covering spaces that we now describe (for a similar construction in the case of the Gaussian parity for virtual knots, see [1,Section 5]). Given a link diagram D on Σ, suppose that γ ⊂ Σ is a simple closed curve with even intersection number with every component of D. Let π : Σ × I → Σ × I be a double cover of Σ × I formed by cutting two copies of Σ × I along γ × I, and identifying the resulting boundaries (see Figure 2).…”

Section: Topological Definitionmentioning

confidence: 99%

Minimal crossing number implies minimal supporting genus

Bodén

Rushworth

2021

Bull. London Math. Soc.

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A virtual link may be defined as an equivalence class of diagrams, or alternatively as a stable equivalence class of links in thickened surfaces. We prove that a minimal crossing virtual link diagram has minimal genus across representatives of the stable equivalence class. This is achieved by constructing a new parity theory for virtual links. As corollaries, we prove that the crossing, bridge, and ascending numbers of a classical link do not decrease when it is regarded as a virtual link. This extends corresponding results in the case of virtual knots due to Manturov and Chernov.

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Section: Topological Definitionmentioning

confidence: 99%

Minimal crossing number implies minimal supporting genus

Bodén

Rushworth

2021

Bull. London Math. Soc.

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“…The 1-handles may be arranged so that in the canonical projection p : Σ × I → Σ, the double points of the composition F ֒→ Σ × I → Σ appear only as band crossings (Figure 3, left). Calculations with such surfaces can be simplified using a 2-dimensional analogue of virtual knot diagrams called virtual Seifert surfaces [3,11]. Figure 3.…”

Section: 2mentioning

confidence: 99%

“…This can be most easily visualized by thickening the graph of standard saddle z = x 2 − y 2 in R 3 . After an isotopy, we obtain the frame at t = 3 8 . The cross-section of M then remains unchanged up to t = 5 8 .…”

Section: It Must Be Shown Thatmentioning

confidence: 99%

“…The main result in Section 3 is that virtual ∂-equivalence is generated by Š-equivalence. This is a generalization of S-equivalence of Seifert surfaces in S 3 . The Seifert surfaces in Figure 1 pairwise Š-equivalent but not pairwise S-equivalent.…”

mentioning

confidence: 93%

“…The effect of Š-equivalence on Seifert matrices is determined in Section 3.2. More precisely, for a Seifert surface F ⊂ Σ × I of a knot K we show how its pair θ ± K,F : H 1 (F ) × H(F ) → Z of directed Seifert forms [3] changes under Š-equivalence.…”

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confidence: 99%

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Algebraic concordance and almost classical knots

Chrisman¹,

Mukherjee²

2021

Preprint

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A virtual knot is called almost classical if it can be a represented by a homologically trivial knot K in a thickened surface Σ × [0, 1], where Σ is closed and oriented. Then K bounds a Seifert surface F ⊂ Σ × [0, 1]. In this paper, we study such Seifert surfaces up to two equivalence relations: virtual ∂-equivalence and virtual concordance. We will say that F0 ⊂ Σ0 × [0, 1] and F1 ⊂ Σ1 ×[0, 1] are virtually ∂-equivalent if K0 = ∂F0 and K1 = ∂F1 are equivalent as virtual knots. We give a complete set of operations relating any two virtually ∂-equivalent Seifert surfaces. Virtual concordance of Seifert surfaces, defined herein, is closely connected to virtual knot concordance. We study this relation using two generalizations of Levine's algebraic concordance group G F , called the uncoupled algebraic concordance group VG F and the coupled algebraic concordance group (VG, VG) F . Here F is a field of characteristic χ(F) = 2. The two groups are related by a pair of surjections π ± : (VG, VG) F → VG F such that G F embeds into the equalizer of π + and π − . Using the theory of isometric structures, we prove that VG F ∼ = I(F)⊕G F , where I(F) is the fundamental ideal of the Witt ring W(F). Complete invariants of VG F are obtained for F a global field with χ(F) = 0. Examples are calculated in the case that F = Q. For F = Z/2Z, we give a generalization of the Arf invariant.

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The Gordon–Litherland pairing for links in thickened surfaces

2022

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We extend the Gordon–Litherland pairing to links in thickened surfaces, and use it to define signature, determinant and nullity invariants for links that bound (unoriented) spanning surfaces. The invariants are seen to depend only on the [Formula: see text]-equivalence class of the spanning surface. We prove a duality result relating the invariants from one [Formula: see text]-equivalence class of spanning surfaces to the restricted invariants of the other. Using Kuperberg’s theorem, these invariants give rise to well-defined invariants of checkerboard colorable virtual links. The determinants can be applied to determine the minimal support genus of a checkerboard colorable virtual link. The duality result leads to a simple algorithm for computing the invariants from the Tait graph associated to a checkerboard coloring. We show these invariants simultaneously generalize the combinatorial invariants defined by Im, Lee and Lee, and those defined by Boden, Chrisman and Gaudreau for almost classical links. We examine the behavior of the invariants under orientation reversal, mirror symmetry and crossing change. We give a 4-dimensional interpretation of the Gordon–Litherland pairing by relating it to the intersection form on the relative homology of certain double branched covers. This correspondence is made explicit through the use of virtual linking matrices associated to (virtual) spanning surfaces and their associated (virtual) Kirby diagrams.

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Signature and concordance of virtual knots

Cited by 8 publications

References 5 publications

Minimal crossing number implies minimal supporting genus

Minimal crossing number implies minimal supporting genus

Algebraic concordance and almost classical knots

The Gordon–Litherland pairing for links in thickened surfaces

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