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Non-orientable genus of a knot in punctured $\mathbb{C}P ^2$

Kouki Sato, Motoo Tange

Abstract

For any knot $K$ which bounds non-orientable and null-homologous surfaces $F$ in punctured $n\mathbb{C}P^2$, we construct a lower bound of the first Betti number of $F$ which consists of the signature of $K$ and the Heegaard Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$. By using this lower bound, we prove that for any integer $k$, a certain knot cannot bound any surface which satisfies the above conditions and whose first Betti number is less than $k$.

Non-orientable genus of a knot in punctured $\mathbb{C}P ^2$

Abstract

For any knot which bounds non-orientable and null-homologous surfaces in punctured , we construct a lower bound of the first Betti number of which consists of the signature of and the Heegaard Floer -invariant of the integer homology sphere obtained by -surgery along . By using this lower bound, we prove that for any integer , a certain knot cannot bound any surface which satisfies the above conditions and whose first Betti number is less than .

Paper Structure

This paper contains 13 sections, 11 theorems, 33 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Remark.
  3. Acknowledgements.
  4. Proof of Proposition \ref{['thm1']}
  5. Proof.
  6. Proof of Lemma \ref{['new lem']}
  7. Proof of Proposition \ref{['thm1']}.
  8. Proof of Theorem \ref{['thm2']}
  9. Proof of Theorem \ref{['thm2']}.
  10. Proof of Theorem \ref{['thm3']}
  11. Proof of Proposition \ref{['tangeprop']}.
  12. Proof of Lemma \ref{['claim1']}.
  13. Proof of Theorem \ref{['thm3']}.

Key Result

Theorem 1.1

Let $K \subset S^3$ be a knot. Then where $\sigma$ denotes the signature of $K$ and $d(S^3_1(K))$ the Heegaard-Floer $d$-invariant of the integer homology sphere obtained by $1$-surgery along $K$.

Figures (6)

  • Figure 1: the link $L$ (for $k \geq 0$)
  • Figure 2: The differential maps of the fundamental part $G_0$ of $CFK^\infty(9_{42})$ (Fig.14 in ozsvath-szabo2).
  • Figure 3: The chain complex of the fundamental part of $CFK^\infty(3_1)$.
  • Figure 4: The chain complex $G_0^{(2)}\{\max(i,j)\ge 0\}$ and the homological generator.
  • Figure 5: $9_{42}$ bounds Möbius band in $B^4$
  • ...and 1 more figures

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1.1: Batson,batson
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Theorem 1.5: Yasuhara, yasuhara
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1: Ozsváth and Szabó, ozsvath-szabo
  • Proposition 3.1
  • ...and 2 more