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The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots

Megan Fairchild

Abstract

The non-orientable 4-genus of a knot K in the three sphere is defined to be the minimum first Betti number of a non-orientable surface F in the four-ball so that K bounds F. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We also will view obstructions to a knot bounding a Möbius band given by the double branched cover of the three sphere branched over K.

The Non-Orientable 4-Genus of 11 Crossing Non-Alternating Knots

Abstract

The non-orientable 4-genus of a knot K in the three sphere is defined to be the minimum first Betti number of a non-orientable surface F in the four-ball so that K bounds F. We will survey the tools used to compute the non-orientable 4-genus, and use various techniques to calculate this invariant for non-alternating 11 crossing knots. We also will view obstructions to a knot bounding a Möbius band given by the double branched cover of the three sphere branched over K.
Paper Structure (11 sections, 17 theorems, 37 equations, 18 figures)

This paper contains 11 sections, 17 theorems, 37 equations, 18 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. Double Branched Cover
  4. RESULTS AND TECHNIQUES
  5. Constraints on Invariants
  6. Non-Oriented Band Moves
  7. Linking Form Calculation
  8. Results
  9. Knot Floer Homology
  10. SPECIAL CASES
  11. Concordance

Key Result

Theorem 1.1

For the 185 non-alternating 11 crossing knots, The remaining 6 knots have $\gamma_{4}(K) = 1$ or $2$.

Figures (18)

  • Figure 3.1: Band Moves
  • Figure 3.2: Checkerboard coloring for $11n_{155}$
  • Figure 3.3: left: $\eta (C) = 1$, right: $\eta (C) = -1$
  • Figure 4.1: A non-oriented band move from $11n_{38}\stackrel{0}{\longrightarrow} 3_1$
  • Figure 4.2: Non-oriented band moves from the knots $11n_{17}, 11n_{40}, 11n_{159},$$11n_{166}, 11n_{177}, \text{ and } 11n_{178}$ to knots with non-orientable genus 1.
  • ...and 13 more figures

Theorems & Definitions (21)

  • Theorem 1.1
  • Proposition 2.1: Proposition 2.3 in MY
  • Corollary 2.2: Corollary 2.4 in MY
  • Proposition 2.3: Proposition 2.4 in N10
  • Corollary 2.4: Corollary 3 in GL
  • Theorem 2.5: Theorem 4 in GL
  • Theorem 2.6: Theorem 11 in GL
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 11 more