Let h(K), g1(K), gH(K) and t(K) be the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot K in the 3-sphere S3, respectively. It is known that gH(K) − 1 = t(K) ⩽ g1(K) ⩽ h(K) ⩽ gH(K). Then a natural question arises: under what conditions do the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot become equal? We provide the necessary and sufficient conditions for those equalities and use these to show that for each integer n ⩾ 1, there are infinitely many knots in each of the following three families <Equation ID="Equa"> <EquationSource Format="TEX">$A_{n}=\{K|h(K)=n<g_{H}(K)\},\\B_{n}=\{K|g_{1}(K)=n<h(K)\},\\C_{n}=\{K|t(K)=n<g_{1}(K)\}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>A</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>K</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo><</mo> <msub> <mi>g</mi> <mrow> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace linebreak="newline" /> <msub> <mi>B</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>K</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi>g</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo><</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace linebreak="newline" /> <msub> <mi>C</mi> <mrow> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>K</mi> <mrow> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo><</mo> <msub> <mi>g</mi> <mrow> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </math></EquationSource> </Equation>This resolves a conjecture by Morimoto (2005) that each of these families is nonempty.

Comparing h-genera, bridge-1 genera and Heegaard genera of knots

期刊论文
发表日期：

2026年5月12日

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Ruifeng Qiu,
Chao Wang,
Yanqing Zou

摘要

Let h(K), g1(K), gH(K) and t(K) be the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot K in the 3-sphere S³, respectively. It is known that gH(K) − 1 = t(K) ⩽ g1(K) ⩽ h(K) ⩽ gH(K). Then a natural question arises: under what conditions do the h-genus, bridge-1 genus, Heegaard genus and tunnel number of a knot become equal? We provide the necessary and sufficient conditions for those equalities and use these to show that for each integer n ⩾ 1, there are infinitely many knots in each of the following three families $A_{n}=\{K|h(K)=n<g_{H}(K)\},\\B_{n}=\{K|g_{1}(K)=n<h(K)\},\\C_{n}=\{K|t(K)=n<g_{1}(K)\}.$ $A_{n} = {K | h (K) = n < g_{H} (K)}, B_{n} = {K | g_{1} (K) = n < h (K)}, C_{n} = {K | t (K) = n < g_{1} (K)} .$

This resolves a conjecture by Morimoto (2005) that each of these families is nonempty.