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This study proves that links with one negative crossing are strongly quasipositive, answering a key question in knot theory. It also characterizes quasipositive link diagrams and confirms a conjecture relating strong quasipositivity to the Bennequin inequality for knots.

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Area of Science:

  • Knot theory
  • Topology
  • Low-dimensional topology

Background:

  • Links with specific crossing properties are fundamental objects in knot theory.
  • Understanding quasipositivity is crucial for classifying and analyzing topological structures.
  • Stoimenow posed a question regarding links with a single negative crossing.

Purpose of the Study:

  • To prove that any link with a single negative crossing is strongly quasipositive.
  • To provide a complete characterization of link diagrams with a quasipositive canonical surface.
  • To apply these findings to classify strongly quasipositive knots and verify a conjecture.

Main Methods:

  • Diagrammatic methods in knot theory.
  • Analysis of Seifert's algorithm and the resulting canonical surface.
  • Investigating the Bennequin inequality and its relation to knot properties.

Main Results:

  • Established that links with a single negative crossing are strongly quasipositive.
  • Characterized link diagrams that yield a quasipositive canonical surface.
  • Identified strongly quasipositive prime knots up to 13 crossings.

Conclusions:

  • The study confirms that links with a single negative crossing possess strong quasipositivity.
  • A complete characterization of quasipositive canonical surfaces is now available.
  • The conjecture linking strong quasipositivity to the equality of the Bennequin inequality for knots with canonical surfaces realizing their genus is confirmed.