Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.4348

Segment number of knots

Makoto Ozawa
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a new numerical knot invariant, termed the \textit{segment number}, which is derived from partitioned knot diagrams subject to specific over/under-crossing constraints. We prove that a knot is non-trivial if and only if its segment number is at least 3. Furthermore, we investigate the structural properties of the directed graph associated with a minimal segment number presentation. Specifically, we show that for any minimal presentation, the underlying graph is connected and cannot be a path. Finally, we discuss the relationship between the segment number and the bridge number, providing bounds and conjectures for future study. We also conjecture that the bridge number $b(K)$ provides a lower bound for the segment number.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Makoto Ozawa. 2026-02-18. Segment number of knots. https://arxiv.org/abs/0705.4348

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Arborescent links and modular tails

We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a conjectural classification of the modularity of tails of the colored Jones polynomial for alternating knots.

math.GT↗
arXiv2026

Instanton and pillowcase homology of the $(-2,3,q)$ pretzel knots

Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot $T(3,5)$ by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots $T(3,n)$, we compute the Floer complex and its $\mathbb{Z}/4$ grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank $q+2$ for every odd $q\ge3$, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.

math.GT↗
arXiv2026

Bounded cohomology of transformation groups in all degrees

Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge2$. Let $\mathcal T_MG$ be one of the following transformation groups: $Homeo_0(M,μ)$, $Diff_0(M,μ)$, $Symp_0(M, ω)$ (in case $M$ is symplectic) and $Ham(M, ω)$ (in case $M$ is symplectic). Denote by $\overline{H}_b^d(\mathcal T_M)$ reduced bounded cohomology of $\mathcal T_M$ in degree $d$, and by $\overline{EH}^d(\mathcal T_M)$ reduced exact bounded cohomology of $\mathcal T_M$ in degree $d$. In this paper we prove that if $n=2$ and $M$ is a closed surface $Σ_g$ or a disc $\mathbb D$, then for every $d\ge 2$ $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, $\dim\overline{EH}^d(\mathcal T_M)=\infty$ if $\mathcal T_M$ is either $Diff_0(Σ_g,μ)$ or $Ham(Σ_g, ω)$, or $Ham(\mathbb{D}, ω)$; or $g>1$. In case $n>2$, under certain conditions on $π_1(M)$, we prove that for every degree $d\ge 2$ $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for $\mathcal T_M$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

math.GT↗