An Index-Graded Virtual Intersection Polynomial for Oriented Virtual Knots
- 1School of Mathematics, Liaoning Normal University, Dalian 116081, China
- 2School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Abstract
Polynomial invariants are important tools for studying virtual knots, and different constructions encode different aspects of the combinatorial and geometric information of a virtual knot diagram. Motivated by the interaction between crossing indices and smoothing operations, we construct a bivariate polynomial invariant PD(x,t) for oriented virtual knots. The construction combines the affine index, the absolute virtual intersection index, and the change of an auxiliary polynomial under a prescribed smoothing that produces one component. We prove that PD(x,t) is invariant under all oriented generalized Reidemeister moves and determine its behavior under orientation reversal and mirror image. We construct an infinite family for which the affine index polynomial, the virtual intersection polynomial PD(t), and the auxiliary polynomial QD(t) all vanish, whereas PD(x,t) distinguishes all members of the family. A second example shows that the variable x retains information that is lost after the specialization x=1. Moreover, using positive and negative resolutions of singular crossings, we prove that PD(x,t) has Vassiliev degree exactly one. These results provide a way to organize smoothing information according to the virtual intersection indices of the original crossings.
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BibTeX
@article{liang2026indexgraded,
title = {{An Index-Graded Virtual Intersection Polynomial for Oriented Virtual Knots}},
author = {Liang Liang and Bin Jiang and Fangyu Jin and Liyuan Ma},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14193601},
url = {https://www.mdpi.com/2227-7390/14/19/3601},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - An Index-Graded Virtual Intersection Polynomial for Oriented Virtual Knots
AU - Liang Liang
AU - Bin Jiang
AU - Fangyu Jin
AU - Liyuan Ma
PY - 2026
DA - 2026/10/04
JO - Mathematics
DO - 10.3390/math14193601
UR - https://www.mdpi.com/2227-7390/14/19/3601
AB - Polynomial invariants are important tools for studying virtual knots, and different constructions encode different aspects of the combinatorial and geometric information of a virtual knot diagram. Motivated by the interaction between crossing indices and smoothing operations, we construct a bivariate polynomial invariant PD(x,t) for oriented virtual knots. The construction combines the affine index, the absolute virtual intersection index, and the change of an auxiliary polynomial under a prescribed smoothing that produces one component. We prove that PD(x,t) is invariant under all oriented generalized Reidemeister moves and determine its behavior under orientation reversal and mirror image. We construct an infinite family for which the affine index polynomial, the virtual intersection polynomial PD(t), and the auxiliary polynomial QD(t) all vanish, whereas PD(x,t) distinguishes all members of the family. A second example shows that the variable x retains information that is lost after the specialization x=1. Moreover, using positive and negative resolutions of singular crossings, we prove that PD(x,t) has Vassiliev degree exactly one. These results provide a way to organize smoothing information according to the virtual intersection indices of the original crossings.
SN - 2227-7390
ER -