Multi-Skein Invariants for Welded and Extended Welded Knots and Links

N. Backes; M. Kaiser; T. Leafblad; E. I. C. Peterson; D. N. Yetter

Home/Archive/arXiv

arXiv · 1809.05874

Multi-Skein Invariants for Welded and Extended Welded Knots and Links

N. Backes·M. Kaiser·T. Leafblad·E. I. C. Peterson·D. N. Yetter

View original source ↗Read source document ↗Export metadata ↓

Abstract

The theory of welded and extended welded knots is a generalization of classical knot theory. Welded (resp. extended welded) knot diagrams include virtual crossings (resp. virtual crossings and wen marks) and are equivalent under an extended set of Reidemeister-type moves. We present a new class of invariants for welded and extended welded knots and links using a multi-skein relation, following Z. Yang's approach for virtual knots. Using this skein-theoretic approach, we find sufficient conditions on the coefficients to obtain invariance under the extended Reidemeister moves appropriate to welded and extended welded links.

Explore related subjects

math.GT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

N. Backes, M. Kaiser, T. Leafblad, E. I. C. Peterson, D. N. Yetter. 2018-09-16. Multi-Skein Invariants for Welded and Extended Welded Knots and Links. https://arxiv.org/abs/1809.05874

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

KEEP EXPLORING

Related papers

arXiv2026

Hyperbolic links associated to Hamiltonian subgraphs in simple $3$-polytopes

We build a large family of hyperbolic links with an explicit decomposition of the complement into right-angled hyperbolic polytopes of finite volume. Namely, in a series of papers A.D.Mednykn and A.Yu.Vesnin introduced a construction that for a given right-angled polytope $P$ in geometry $\mathbb L^3$, $\mathbb R^3$, $\mathbb S^3$, $\mathbb L^2\times \mathbb R$, $\mathbb S^2\times \mathbb R$ and a Hamiltonian cycle, theta-subgraph or $K_4$-subgraph $Γ$ in the $1$-skeleton of $P$ builds a geometric $3$-manifold $N(P,Γ)$ with an involution $τ$ such that $N(P,Γ)/\langleτ\rangle\simeq S^3$. The brach set of the corresponding $2$-sheeted branched covering $N(P,Γ)\to S^3$ is a link $C_Γ\subset S^3$ consisting of trivially embedded circles. This construction reformulated in the language of toric topology works for such a subgraph $Γ$ in any simple $3$-polytope $P$ and gives a topological $3$-manifold $N(P,Γ)$. We give a criterion when $S^3\setminus C_Γ$ has a complete hyperbolic structure of finite volume and generalize this criterion to similar links in $3$-manifolds different from $S^3$. We prove that hyperbolic links $C_Γ$ are parametrized by nonselfcrossing Eulerian cycles, Eulerian theta-subgraphs and Eulerian $K_4$-subgraphs in hyperbolic right-angled $3$-polytopes of finite volume in $\mathbb L^3$ with $0$, $2$ or $4$ finite vertices. The complement $S^3\setminus C_Γ$ is glued of $4$, $8$ or $16$ copies of the corresponding right-angled polytope. We give a criterion when the link $C_Γ$ consists of mutually unlinked circles and prove that if such a link is nontrivial, then it contains the Borromean rings. The latter problem is motivated by the Efimov effect in quantum mechanics.

math.GT↗

arXiv2026

Knots and the Sierpinski Tetrahedron

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.