Knots in S_g × S¹, their essential diagrams and virtual knots

2025/12/06 by Kim, Seongjeong
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · doi:10.48550/arxiv.2512.06300

openalex publication_date 2025/12/06 · openalex created_date 2025/12/10 · openalex updated_date 2026/07/28

Abstract

In \citeKim it is shown that knots in S_g × S¹ can be presented by virtual diagrams with a decoration, so called, \em double lines. In this paper, we study the essential diagram for each knot in S_g × S¹, which has the minimal number of double lines. We prove that virtual knot theory is embedded in the theory of knots in S_g× S¹. In the same time, one can obtain knots in S²× S¹ from 2-component links L = K\sqcup T where T is a trivial knot. By using knots in S² × S¹, we study the minimality and separability of such classical links.

Citations

• Parity functors
• Fibered Knots and Virtual Knots
• Virtual Knot Invariants Arising From Parities
• KBSM of the product of a disk with two holes and S¹
• Stable Equivalence of Knots on Surfaces and Virtual Knot Cobordisms

Related

• On links in S_g × S¹ and its invariants
• Knots in S_g × S¹ and winding parities
• Braids for Knots in S_g × S¹ and the affine Hecke algebra
• Liftings of knots in S_g × S¹ and covering of virtual knots
• Virtual Knots and Links