Narrow Results

Recently it has been proved by K. Taniyama, S. Yamada and the author that for any natural number n and any knot K, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to n coincide with those of K. In this paper we give another proof of this result by using web diagrams.

It is shown by Y. Ohyama, K. Taniyama and S. Yamada that for any natural number n and any knot K, there are infinitely many unknotting number one knots, whose Vassiliev invariants of order less than or equal to n coincide with those of K. We analyze it for delta unknotting number, and obtain the following. For any natural number n and any oriented knots K and M with a₂(K)≠a₂(M) there are infinitely many knots J_m such that the delta distance between J_m and M coincide with |a₂(K)-a₂(M)| and whose Vassiliev invariants of order less than or equal to n coincide with those of K. Here a₂(K) is the second coefficient of the Conway polynomial of K.

By the same way for knots, it can be proved that if a link L is transformed into L′ by a C_n-move, the difference of the Vassiliev invariants of order n between L and L′ is equal to the value of the one-branch tree diagram corresponding to the C_n-move. In this paper we show that a C_n-move for a link whose one-branch tree diagram has a cut edge cannot change the Vassiliev invariant of order n. And from this, we can give some examples of links that have the same Vassiliev invariants of order less than n and cannot be transformed into each other by C_n-moves.

In this paper, we show that when any nonnegative integer n and any knot K are given, there exist infinitely many unknotting number one knots J_m (m = 1,2, ⋯) such that their Vassiliev invariants of order less than or equal to n coincide with those of K and each J_m satisfies the following: The delta unknotting number of J_m is determined by the order two Vassiliev invariant of K and for almost cases, the clasp-pass distance between J_m and the twist knot with the same order two invariant is determined by the order three invariant of K.

We show that for each positive integer n>1 a simple C_n-move on knots, introduced by Habiro, is equivalent to the (non-oriented) insertion in a knot via a tangle map the geometric pure braid n-commutator of the form p_n=[p_n-1,n,[p_n-2,n-1,…, [p_1,2,p_0,1]…]∈P_n+1, where P_n+1 is the subgroup of pure braids of the braid group B_n+1 on n+1 strands. We relate the insertions of the pure braid commutators in knots to the operations of inverting and mirroring the knots.

We consider how C_n-move has an influence on the Vassiliev invariant of order m. In other words, we investigate the variance of the value of concrete Vassiliev invariant of order m by a C_n-move. We choose the mth coefficient of the Conway polynomial, a_m, as a concrete Vassiliev invariant of order m, and study the variance of that by a C_n-move for n≤4. And we get that when we give a natural number k, there is a pair of knots satisfying that they are transformed into each other by a C_n-move and the difference of a_m of them is equal to k in almost case under the condition that n<m and n≤4.

It is well-known that the coefficient of z^m of the Conway polynomial is a Vassiiev invariant of order m. In this paper, we show that for any given pair of a natural number n and a knot K, there exist infinitely many knots whose Vassiliev invariants of order less than or equal to n and Conway polynomials coincide with those of K.

A local move is a pair of tangles with same end points. Habiro defined a system of local moves, C_n-moves, and showed that two knots have the same Vassiliev invariants of order ≤ n - 1 if and only if they are transformed into each other by C_n-moves. We define a local move, β_n-move, which is obtained from a C_n-move by duplicating a single pair of arcs with same end points. Then we immediately have that a C_n+1-move is realized by a β_n-move and that a β_n-move is realized by twice C_n-moves. In this note we study the relation between C_n-move and β_n-move, and in particular, give answers to the following questions: (1) Is a β_n-move realized by a finite sequence of C_n+1-moves? (2) Is C_n-move realized by a finite sequence of β_n-moves?

The HOMFLY polynomial, which is an invariant of an oriented knot or link L, is the two-variable polynomial P(L; t, z) ∈ ℤ[t^±1, z^±1]. If we arrange the HOMFLY polynomial by the variable z, then the coefficient of z^k is the polynomial of t, which is called the P_k polynomial. Let be the nth derivative of the P_k polynomial of a link L evaluated at 1. It is a Vassiliev invariant of order less than or equal to max {n + k, 0}. Let K and K′ be oriented knots such that K′ is obtained from K by a C_n-move (n ≥ 3). We show that the value is 0 or ±n! · 2ⁿ by using Jacobi diagrams and moreover it is determined by the planarity of the Jacobi diagram corresponding to the C_n-move. By the result, we can decide the above value only by the knot diagrams of K and K′.

It is shown that two knots can be transformed into each other by C_n-moves if and only if they have the same Vassiliev invariants of order less than n. Consequently, a C_n-move cannot change the Vassiliev invariants of order less than n and may change those of order more than or equal to n. In this paper, we consider the coefficient of the Conway polynomial as a Vassiliev invariant and show that a C_n-move changes the nth coefficient of the Conway polynomial by ±2, or 0. And for the 2mth coefficient (2m > n), it can change by p or p + 1 for any given integer p.

A local move called a C_n-move is closely related to Vassiliev invariants. A C_n-distance between two knots K and L, denoted by d_Cn(K, L), is the minimum number of times of C_n-moves needed to transform K into L. Let p and q be natural numbers with p > q ≥ 1. In this paper, we show that for any pair of knots K₁ and K₂ with d_Cn(K₁, K₂) = p and for any given natural number m, there exist infinitely many knots J_j(j = 1, 2, …) such that d_Cn(K₁, J_j) = q and d_Cn(J_j, K₂) = p - q, and they have the same Vassiliev invariants of order less than or equal to m. In the case of n = 1 or 2, the knots J_j(j = 1, 2, …) satisfy more conditions.

A local move called a C_n-move is closely related to Vassiliev invariants. The C_n-distance between two knots K and L, denoted by d_Cn(K, L), is the minimal number of C_n-moves needed to transform K into L. In the case of n ≥ 3, we show that for any pair of knots K₁ and K₂ with d_Cn(K₁, K₂) = 1 and for any given natural number m, there exist infinitely many knots J_j (j = 1, 2, …) such that d_Cn(K₁, J_j) = d_Cn(J_j, K₂) = 1 and they have the same Vassiliev invariants of order less than or equal to m.

The Jones polynomial $V_L(t)$ for an oriented link $L$ is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer $n\ge 3$, we show that: (1) the difference of Jones polynomials for two oriented links which are $C_n$-equivalent is divisible by ${(t-1)}^n(t^2+t+1)(t^2+1)$, and (2) there exists a pair of two oriented knots which are $C_n$-equivalent such that the difference of the Jones polynomials for them equals ${(t-1)}^n(t^2+t+1)(t^2+1)$.

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 C_n+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 C_n–move is equal to the value of the Vassiliev invariant for a one-branch tree diagram of order n.