For any positive integers h and n, we show that a knot surgered elliptic surface <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$E(n)_{T(2,2h+1)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mi>h</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> for a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(2,2h+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mi>h</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-torus knot <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$T(2,2h+1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mi>h</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> admits a handle decomposition without 1- and 3-handles using a Kirby diagram derived from a Lefschetz fibration on it. As a corollary, an elliptic surface <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$E(1)_{2,2h+1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>E</mi> <msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mi>h</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> has such a handle decomposition.

Knot surgered elliptic surfaces without 1- and 3-handles for a $(2,2h+1)$-torus knot

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2026年4月4日

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Naoyuki Monden,
Reo Yabuguchi

摘要

For any positive integers h and n, we show that a knot surgered elliptic surface $E(n)_{T(2,2h+1)}$ $E {(n)}_{T (2, 2 h + 1)}$ for a $(2,2h+1)$ $(2, 2 h + 1)$ -torus knot $T(2,2h+1)$ $T (2, 2 h + 1)$ admits a handle decomposition without 1- and 3-handles using a Kirby diagram derived from a Lefschetz fibration on it. As a corollary, an elliptic surface $E(1)_{2,2h+1}$ $E {(1)}_{2, 2 h + 1}$ has such a handle decomposition.

Knot surgered elliptic surfaces without 1- and 3-handles for a \((2,2h+1)\)-torus knot

摘要