This paper classifies link diagrams on nonorientable surfaces under region crossing changes. We prove that for a link diagram L on the nonorientable surface <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$N_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation>, the rank of its incidence matrix <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M_L$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>L</mi> </msub> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {rank}(M_L)=r-n-1+\operatorname {rank}(N_L)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>rank</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>L</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>r</mi> <mo>-</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>+</mo> <mo>rank</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>N</mi> <mi>L</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where r is the number of regions, n is the number of link components, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$N_L$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>N</mi> <mi>L</mi> </msub> </math></EquationSource> </InlineEquation> is the homology matrix. This formula determines the number of equivalence classes of link diagrams under region crossing changes as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^{c -\operatorname {rank}(M_L)}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>c</mi> <mo>-</mo> <mo>rank</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mi>L</mi> </msub> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>. We also characterize the region crossing change admissible crossing sets via bi-colorings and homology classes, extending results from orientable to nonorientable settings.

Region crossing change on nonorientable surfaces

期刊论文
发表日期：

2026年6月26日

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Zhiyun Cheng,
Qian Liao,
Jingze Song,
Xinyuan Zeng

摘要

This paper classifies link diagrams on nonorientable surfaces under region crossing changes. We prove that for a link diagram L on the nonorientable surface $N_g$ $N_{g}$ , the rank of its incidence matrix $M_L$ $M_{L}$ satisfies $\operatorname {rank}(M_L)=r-n-1+\operatorname {rank}(N_L)$ $rank (M_{L}) = r - n - 1 + rank (N_{L})$ , where r is the number of regions, n is the number of link components, and $N_L$ $N_{L}$ is the homology matrix. This formula determines the number of equivalence classes of link diagrams under region crossing changes as $2^{c -\operatorname {rank}(M_L)}$ $2^{c - rank (M_{L})}$ . We also characterize the region crossing change admissible crossing sets via bi-colorings and homology classes, extending results from orientable to nonorientable settings.