Arrow diagrams on spherical curves and computations

We give a definition of an integer-valued function [Formula: see text] derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free [Formula: see text]-module generated by the arrow diagrams with at most [Formula: see text] arrows, called relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), and introduce another function [Formula: see text] to obtain [Formula: see text]. One of the main results shows that if [Formula: see text] vanishes on finitely many relators of Type ([Formula: see text]) (([Formula: see text]), ([Formula: see text]), ([Formula: see text]) or ([Formula: see text]), respectively), then [Formula: see text] is invariant under the deformation of type RI (strong RI[Formula: see text]I, weak RI[Formula: see text]I, strong RI[Formula: see text]I[Formula: see text]I or weak RI[Formula: see text]I[Formula: see text]I, respectively). The other main result is that we obtain new functions of arrow diagrams with up to six arrows explicitly. This computation is done with the aid of computers.

ANALISIS KESALAHAN SISWA DALAM MENYELESAIKAN SOAL RELASI DAN FUNGSI DI KELAS VII SMP NEGERI 4 BALAI JAYA

The concept of relations and functions in the school mathematics curriculum was first accepted by students in grade VIII. In evaluating students' understanding of the concept of relations and function material, there are mistakes made by students. Errors that occur can be in the form of misconceptions, principles, and operating errors. This study aims to describe the types of errors in solving the material relations and functions of the VIII grade students of SMP Negeri 4 Balai Jaya in the odd semester of the 2020/2021 school year. This type of research is descriptive-qualitative research, with the test as a data collection instrument. The subjects in this study were 27 students of class VIIIA. The results showed that on average 60% of students made misconceptions, 51% of students made mistakes in principle and 63% of students made operational errors. Students have not been able to present relations correctly using arrow diagrams, Cartesian diagrams and consecutive sets of pairs. Students also still have difficulty distinguishing relations and functions. In the matter of functions, the use of steps in manipulating algebraic calculations carried out by students is not quite right.

On Some Moves on Links and the Hopf Crossing Number

We consider arrow diagrams of links in $$S^3$$ S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$ k ∈ N . We study the equivalence classes of links in $$S^3$$ S 3 up to k-moves. For $$k=2$$ k = 2 , we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by $$-1$$ - 1 , when k is even. It follows that, for any $$k\ge 5$$ k ≥ 5 , there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces $$L_{p,1}$$ L p , 1 .

KONTSEVICH INTEGRAL FOR KNOTS AND VASSILIEV INVARIANTS

We review quantum field theory approach to the knot theory. Using holomorphic gauge, we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.

GAUSS DIAGRAM INVARIANTS OF LINKS IN M2 × R1

We construct an infinite series of invariants of Fiedler type (i.e. composed of oriented arrow diagrams arranged by elements of H1(M3)) for multicomponent links in M3 = M2 × R1, M2 orientable with π1(M2) ≠ {1}.

A diagrammatic approach to link invariants of finite degree

In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a self-contained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.