The intersection polynomials of a long virtual knot I: Definitions and properties

2025/12/05 by Nakamura, Takuji, Nakanishi, Yasutaka, Satoh, Shin +1 · 1 citation
#57K16 #FOS: Mathematics #Geometric Topology (math.GT) #Primary 57K12 #Secondary 57K14

paper · doi:10.48550/arxiv.2512.05366

Abstract

We introduce twelve polynomial invariants for long virtual knots, called intersection polynomials, extending and refining the three intersection polynomials for virtual knots. They are defined via intersection numbers of cycles on a closed surface, considering the order of over- and under-crossings. We study their fundamental properties including behavior under symmetries, crossing changes, and concatenation products. All are finite-type invariants of degree two under crossing changes, but not under virtualizations, and we examine their relation to the closure and the values at t=1 of their derivatives.

Citations

• The intersection polynomials of a virtual knot I: Definitions and calculations
• An Affine Index Polynomial Invariant of Virtual Knots
• On a Generalization of Alexander Polynomial for Long Virtual Knots
• New Invariants of Long Virtual Knots
• Finite Type Invariants of Classical and Virtual Knots

Cited by

• The intersection polynomials of a long virtual knot II: Two supporting genera and characterizations

Related

• The intersection polynomials of a virtual knot I: Definitions and calculations
• The intersection polynomials of a long virtual knot II: Two supporting genera and characterizations
• Intersection graph and writhe polynomial
• Intersection formulas for parities on virtual knots
• On a Generalization of Alexander Polynomial for Long Virtual Knots