N−Trivial Knots with Certain Configurations

Kohei Moro* and Ken’ichi Kuga

Department of Mathematics and Informatics,

Faculty of Science, Chiba University

It is a result of Ohyama [1] that the Conway polynomial of an n−trivial knot has vanishing terms in degrees less than n, except the zero-th term. This result is obtained by calculation tracing a relation tree encoding the process of cross changings and smoothings. Transformation of the knot along this process is observed using the configuration diagram encoding double point correspondence on the circle. Hence one way to push the above vanishing result a step further is to look at the configuration diagram.

We show that if there is a band-type member with zoro total crossing sign, defined below, in the triviality family of subsets of double points, the Conway polynomial has one more vanishing term. This band-type subset of double points is a simple and typical cause of the vanishing of lower terms in the Conway polynomial, and occurs in many examples, one of which will be illustrated.

Let K be an oriented knot in the 3−sphere S^{3}, and K̃ be a regular projection of K with the set of double points CK̃ = {c_{1}, c_{2}, …, c_{d}}. A planar diagram obtained by drawing line segments ℓ_{i} joining the two points on the circle S^{1} corresponding to each double point c_{i}, 1 ≤ i ≤ d, is called the configuration of K̃.(cf.[3])

Definition Let {A_{1}, A_{2}, …, A_{n}} be a family of pairwise disjoint subsets of CK̃. We say a member A_{k} of this family is of band-type, when there are two connected arcs α and β on the circle of the configuration diagram, such that no endpoints of the line segments ℓ_{i} sit in α ∪ β for c_{i} ∈ ∪_{j≠k}A_{j} and each line segment ℓ_{i} for c_{i} ∈ A_{k} has one endpoint in α and the other in β.

An obvious consequence of this definition is the following:

Lemma 1 Let A_{k} be a band-type member in {A_{1}, A_{2}, …, A_{n}} and fix one double point c_{i_{j}} ∈ A_{j} for each j ≠ k, then the number of the components of the link K{c_{i_{1}}, …, c_{i_{k}}, …, c_{i_{n}}} obtained by smoothing the fixed c_{i_{j}} ∈ A_{j}, j ≠ k, and a double point c_{i_{k}} ∈ A_{k} is constant for the choice of c_{i_{k}} ∈ A_{k}.

* This paper is a modified version of the first author’s master thesis written at Chiba University in March 1996.

Let {A_{1}, A_{2}, …, A_{n}} be a set of pairwise disjoint subsets of CK̃. The knot K is called n−trivial with respect to {A_{1}, A_{2}, …, A_{n}} when the knot obtained by changing crossings at every double point in A_{j_{1}} ∪ A_{j_{2}} ∪ … ∪ A_{j_{m}} is unknotted for any nonempty subset {A_{j_{1}}, A_{j_{2}}, …, A_{j_{m}}} of {A_{1}, A_{2}, …, A_{n}}.

This n−triviality is a property which guarantees the Conway polynomial of the knot to have vanishing terms in degrees less than n, except the zero-th term. In fact a calculation using relation trees and the n−triviality leads to the following result:

Proposition 1(Ohyama[1]) If a knot K is n−trivial with respect to {A_{1}, A_{2}, …, A_{n}} then the Conway polynomial ∇_{K} of K is written as

∇_{K}(z) = 1 + z^{n}∑_{1≤i_{j}≤r_{j},1≤j≤n}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{ni_{n}}∇_{K(i_{1},i_{2},...i_{n})}(z)

where we renumbered the double points in each A_{j} as A_{j} = {c_{j1}, c_{j2}, …, c_{jr_{j}}}, and ϵ_{ji} = ±1 denotes the sign of the crossing at c_{ji}. K(i_{1}, i_{2}, …, i_{n}) is the link obtained from K by changing crossings at c_{ji} for 1 ≤ i ≤ i_{j} −1, 1 ≤ j ≤ n, and smoothing at c_{ji_{j}}, for 1 ≤ j ≤ n.

To push this vanishing result a step further one might use the band type class of double points we defined above.

Theorem Let K be a 2m−trivial knot with respect to {A_{1}, A_{2}, …, A_{2m}}. Suppose that there is a band-type member A_{k} whose total sign of double points vanishes, i.e., ∑_{1≤i≤r_{k}}ϵ_{ki} = 0. Then the Conway polynomial ∇_{K} of K has vanishing terms in degrees less than 2m + 2, except the zero-th term.

Remark: For an n−trivial knot with n = 2m + 1, vanishing of the terms in degrees less than 2m + 2 (except the zero-th term) is generally true and is a direct consequence of Proposition 1.

It is not difficult to find examples of knots satisfying the assumption of the Theorem, One of those knots will be illustrated later.

Proof of Theorem is based on Lemma 1 and Lemma 2 below which then is based on Proposition 1.

Lemma 2 Let K be a 2m−trivial knot with respect to {A_{1}, A_{2}, …, A_{2m}}. Suppose

∑_{♯K(i_{1},i_{2},...i_{2m})=1}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}} = 0,

where ♯K(i_{1}, i_{2}, … i_{n}) denotes the number of components of the link K(i_{1}, i_{2}, … i_{n}).

Then the Conway polynomial ∇_{K} of K has vanishing terms in degrees less than 2m+2, except the zero-th term.

Proof of Lemma 2

For a μ−component link L, the Conway polinomial has the form

∇_{L}(z) = z^{μ−1}(a_{0} + a_{2}z^{2} + … + a_{2s}z^{2s}).

(cf. [4]). Hence if we denote by ˜∇_{L}(z) the polynomial obtained from ∇_{L}(z) by deleting its 0−th term,

∇_{K(i_{1},i_{2},...,i_{n})}(z) = δ_{1μ} + ˜∇_{K(i_{1},i_{2},...,i_{n})}(z),

where μ = ♯K(i_{1}, i_{2}, …, i_{n}).

Hence from Proposition 1 we have

∇_{K}(z) = 1 + z^{2m}∑_{♯K(i_{1},i_{2},...i_{n})=1}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}}
+ z^{2m}∑_{1≤i_{j}≤r_{j},1≤j≤2m}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}} ˜∇_{K(i_{1},i_{2},...i_{2m})}(z).

Hence from the assumption of the Lemma we have

∇_{K}(z) = 1 + z^{2m}∑_{1≤i_{j}≤r_{j},1≤j≤2m}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}} ˜∇_{K(i_{1},i_{2},...i_{2m})}(z).

Note that ♯K(i_{1}, i_{2}, … i_{n}) = an odd number from the construction. Hence the degree of each term in ˜∇_{K(i_{1},i_{2},...i_{2m})}(z) is at least 2. Hence the conclusion of Lemma follows.

Proof of Thorem

¿From Lemma 2 it suffices to show ∑_{♯K(i_{1},i_{2},...i_{2m})=1}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}} = 0. ¿From Lemma 1 the set

Γ = Γ(c_{ki_{k}}) = {(c_{1i_{1}}, …, ĉ_{ki_{k}}, …, c_{2m i_{2m}})|♯K(i_{1}, …, i_{k}, …, i_{n}) = 1} ⊂ ∏_{j≠k}A_{j}

is invariant for the choice of c_{ki_{k}} ∈ A_{k}. (ˆ() here indicates deletion of ().)

Hence from the assumption ∑_{1≤i≤r_{k}}ϵ_{ki} = 0. we have

∑_{♯K(i_{1},i_{2},...,i_{2m})=1}ϵ_{1i_{1}}ϵ_{2i_{2}}…ϵ_{2m i_{2m}} = ∑_{1≤i_{k}≤r_{k}}∑_{c∈Γ(c_{ki_{k}})}ϵ_{ki_{k}}∏_{c_{ji}∈c}ϵ_{ji}
= (∑_{1≤i_{k}≤r_{k}}ϵ_{ki_{k}})(∑_{c∈Γ}∏_{c_{ji}∈c}ϵ_{ji})
= 0,

which completes the proof of Theorem.

Example

The following knot diagram illustrates a 2−trivial knot K with respect to {A_{1} = {c_{11}}, A_{2} = {c_{21}, c_{22}}}. A_{2} is of band type as seen in the configuration diagram with respect to {A_{1}, A_{2}}, and satisfies ϵ_{21} + ϵ_{22} = 1 + (−1) = 0. Actually the Conway polynomial is ∇_{K} = 1 − z^{6}.

References

1. Ohyama, Y.: A numerical invariant of knots induced from their regular diagrams, Topology and its Applications 37, 249-255 (1990)

2. Ohyama, Y. and Y. Ogushi : On the triviality index of knots, Tokyo J. Math. Vol. 13, No 1 179-190 (1990)

3. Birman, Joan S. and Xiao-Song Lin : Knot polynomials and Vassiliev’s invariants, Inventiones Math. 111 225-270 (1993)

4. Burde, G. and H. Zieschang : Knots, de Gruyter Studies in Math. 5, Walter de Gruyter, Berlin, New york 1985

5. Conway, J. H. : An enumeration of knots and links, and some of their algebraic properties, Computational Problems in bstract Algebra, Pergamon Press, New York, 329-358 (1970)