Tatsuya Tsukamoto

ON HABIRO'S Cn–MOVES AND VASSILIEV INVARIANTS OF ORDER n

Recently it has been proved by Habiro that two knots K1 and K2 have the same Vassiliev invariants of order less than or equal to n if and only if K1 and K2 can be transformed into each other by a finite sequence of Cn+1–moves. In this paper, we show that the difference of the Vassiliev invariants of order n between two knots that can be transformed into each other by a Cn–move is equal to the value of the Vassiliev invariant for a one-branch tree diagram of order n.

The fourth Skein Module and the Montesinos-Nakanishi Conjecture for 3-algebraic links

We study the fourth skein module of 3-manifolds, based on the skein relation b0L0 + b1L1 + b2L2 + b3L3 = 0 and a framing relation L(1) = aL (a, b0, b3 invertible). We give necessary conditions for trivial links to be linearly independent in the module. We show how elements of the skein module behave under the n-move and we compute the values for (2, n)-torus links and twist knots as elements of the skein module. Using mutants and rotors, we find different links which represent the same element in the skein module. We also show that algebraic links (in the sense of Conway) and closed 3-braids are linear combinations of trivial links. We introduce the concept of n-algebraic tangles (and links) and analyze the skein module for 3-algebraic links. As a by product we prove the Montesinos-Nakanishi 3-moves conjecture for 3-algebraic links (including 3-bridge links). For links in S3, the structure of our skein module suggests the existence of three new polynomial invariants of unoriented framed (or unframed) links. One of them would generalize the Kauffman polynomial of links and another one could be used to analyze amphicheirality of links (and may work better than the Kauffman polynomial). In the conclusion, we speculate that our new knot invariants are related to a deformation of the symplectic quotient of braid groups.

The almost alternating diagrams of the trivial knot

Bankwitz characterized the alternating diagrams of the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize the almost alternating diagrams of the trivial knot. As a corollary, we determine the unknotting number one alternating knots with the property that the unknotting operation can be done on its alternating diagram.

A criterion for almost alternating links to be non-splittable

We give a sufficient condition for an almost alternating link diagram to represent a non-splittable link. The main theorem gives us a way to see if a given almost alternating link diagram represents a splittable link without increasing numbers of crossings of diagrams in the process. Moreover, we show that almost alternating links with more than two components are non-trivial.

Knot-inevitable projections of planar graphs

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

Simple-ribbon fusions on non-split links

In [Simple ribbon fusions and genera for links, J. Math. Soc. Japan 68 (2016) 1033–1045], we introduced special types of fusions, so-called simple-ribbon fusions on links, and showed that they never decrease the genera of links. In this paper, we also gave a geometric condition for a simple-ribbon fusion to preserve a link type, and it turns out that a “complicated” simple-ribbon fusion may preserve a link type if the link is split. In this paper, we show that only “clearly trivial” simple-ribbon fusion preserves a link type if the link is non-split. This enables us to give a lower bound for the difference between genera of knots related by a simple-ribbon fusion.

FREE SELF DELTA-TRIVIALITY OF DELTA-SPLIT LINKS

A free self delta-trivial link is a link which can be transformed into a trivial link by trivializing a component only with self delta-moves on the component, one by one in any order. We show that if a link spans mutually disjoint disks whose singularities are arcs of the non-clasp-type and loops, then the link is free self delta-trivial. We also study free self sharp-triviality and free self pass-triviality of a link which spans mutually disjoint disks whose singularities are mutually disjoint simple arcs of the clasp-type.

A FACTORIZATION OF THE CONWAY POLYNOMIAL AND COVERING LINKAGE INVARIANTS

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the -invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t-1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S3. This generalizes a result of Hoste.

Any Knot Is Inevitable in a Regular Projection of a Planar Graph

For any knotted planar graphHinR3, there exist a planar graphGand its regular projectionG?R2such that every diagram obtained fromGcontains a subdiagram that representsH.

ON NON-SELF LOCAL MOVES

Gusarov and Habiro introduced a Cm move, that is strongly related to Vassiliev invariants. In this note, we study a special kind of Cm move, called a non-self Cm move. We show that two links can be transformed into each other by a finite sequence of non-self Cm moves if and only if (1) the two links can be transformed into each other by a finite sequence of Cm moves, and (2) the knot types of corresponding components coincide.