Quasi-triviality of quandles for link-homotopy

Quasi-triviality of quandles for link-homotopy

Ayumu Inoue Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro–ku, Tokyo, 152–8552 Japan ayumu.inoue@math.titech.ac.jp
Abstract.

We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.

Key words and phrases:
link, link-homotopy, quandle, quandle cocycle invariant
2010 Mathematics Subject Classification:
Primary 57M27; Secondary 57M99
The author is partially supported by Grant-in-Aid for Research Activity Start-up, No. 23840014, Japan Society for the Promotion of Science.

1. Introduction

Link-homotopy, introduced by Milnor [7], gives rise to an equivalence relation on oriented and ordered links in a 3-sphere. More precisely, two links are said to be link-homotopic if they are related to each other by a finite sequence of ambient isotopies and self-crossing changes, keeping the orientation and ordering. Here, a self-crossing change is a homotopy for a single component of a link, supported in a small ball whose intersection with the component consists of two segments, depicted in Figure 1. The classification problem of links up to link-homotopy is already solved by Habegger and Lin [2] completely. They gave an algorithm which determines whether given links are link-homotopic or not. On the other hand, a table consisting of representatives of all link-homotopy classes is still not known other than partial ones given by Milnor [7, 8] for links with 3 or fewer components and by Levine [5] for links with 4 components. We note that both of them utilized numerical invariants to obtain the tables.

Figure 1.

A quandle, given by Joyce [4], is an algebraic system consisting of a set together with a binary operation whose definition is strongly motivated in knot theory. Carter et al. [1] introduced quandle homology and gave invariants of links up to ambient isotopy using 2-cocycles of quandles. These invariants, called quandle cocycle invariants, are not invariant under link-homotopy in general. However, in this paper, we introduce the notion of quasi-triviality of quandles and show that quandle cocycle invariants associated with 2-cocycles of quasi-trivial quandles are invariant under link-homotopy (Theorem 3.5), modifying the definition of quandle homology slightly. We thus have a lot of numerical link-homotopy invariants, which might enable us to obtain a table consisting of representatives of all link-homotopy classes. At the end of this paper, as an application of Theorem 3.5, we show a famous fact that the Borromean rings, which is a 3-component link, is not trivial up to link-homotopy.

Throughout this paper, every link is assumed to be oriented, ordered and in a 3-sphere. We also assume that every link diagram is oriented and ordered.

2. Preliminaries

We devote this section to reviewing a quandle cocycle invariant. Recall that two links are ambient isotopic if and only if their diagrams are related to each other by a finite sequence of Reidemeister moves.

We first review the definition of a quandle. A quandle is a non-empty set equipped with a binary operation satisfying the following axioms:

  • For each , .

  • For each , a map () is bijective.

  • For each , .

The notion of homomorphisms of quandles is appropriately defined. The axioms (Q1), (Q2) and (Q3) of a quandle are closely related to the Reidemeister moves RI, RII and RIII respectively as follows.

An arc coloring of a link diagram by a quandle is a map satisfying the condition depicted in Figure 2 at each crossing. We call an element of a quandle assigned to an arc by an arc coloring a color of the arc. Suppose is a link diagram and a diagram obtained from by a Reidemeister move. Then, for each arc coloring of , we have a unique arc coloring of whose restriction to arcs unrelated to the deformation coincides with the restriction of . Indeed, the axioms (Q1), (Q2) and (Q3) of a quandle guarantee that we can perform RI, RII and RIII moves fixing colors of ends respectively (See Figure 3). Therefore, for a fixed quandle, the number of all arc colorings is invariant under ambient isotopy.

Figure 2.
Figure 3.

We next review quandle homology. Let be a quandle. Consider the free abelian group generated by all -tuples for each . We let . Define a map by

for , and . Then . Thus is a chain complex. Let be a subgroup of generated by -tuples with for some if , and let otherwise. It is routine to check that . Therefore, putting , we have a chain complex . Let be an abelian group. The -th quandle homology group with coefficients in is the -th homology group of the chain complex . The -th quandle cohomology group with coefficients in is the -th cohomology group of the cochain complex . We note that is a 2-cocycle if and only if satisfies the following conditions:

  • For each , .

  • For each , ,
    that is, .

Associated with a 2-cocycle of a quandle, we can define an weight of an arc coloring as follows. Suppose is a quandle and a 2-cocycle with coefficients in an abelian group . The -th weight of an arc coloring of a link diagram by associated with is a value

where the sum runs over the crossings of , each of which consists of under arcs belonging to the -th component and an over arc, is or depending on whether the crossing is positive or negative respectively, and denote colors of arcs around the crossing as depicted in Figure 2. We have the following theorem.

Theorem 2.1 (Carter et al. [1]).

Let be a quandle and a 2-cocycle with coefficients in an abelian group . Then an weight of an arc coloring by associated with is invariant under Reidemeister moves.

Proof.

Let be an arc coloring of a link diagram by . Consider to perform a Reidemeister move for this colored diagram. An RI move for a segment of the -th component only adds or subtracts to or from with some (See the upper left of Figure 3). It does not change because satisfies the condition (C1). An RII move, by which a segment of the -th component passes under a some arc, only adds or subtracts to or from with some (See the upper right of Figure 3). An RIII move adds to with some if the innermost arcs related to the deformation belong to the -th component (See the bottom of Figure 3). Since satisfies the condition (C2), it does not change . ∎

Suppose again that is a quandle and a 2-cocycle with coefficients in an abelian group . Theorem 2.1 says that, for each link and index , the multiset consisting of -th weights of all arc coloring of a diagram of by associated with does not depend on the choice of the diagram. Thus is invariant under ambient isotopy. The -th quandle cocycle invariant of associated with is this multiset .

3. Quasi-trivial quandle and link-homotopy

Although the number of all arc colorings is invariant under ambient isotopy, it is not invariant under self-crossing changes in general. Indeed, the trefoil knot is deformed into the unknot by a self-crossing change, but they are distinguished by the numbers of all arc colorings. Thus a quandle cocycle invariant is not invariant under link-homotopy in general. On the other hand, in this section, we show that a self-crossing change on a diagram also relates arc colorings of the original and deformed diagrams uniquely if we use a certain quandle, named a quasi-trivial quandle, for arc colorings. Since two links are link-homotopic if and only if their diagrams are related to each other by a finite sequence of Reidemeister moves and self-crossing changes on diagrams, the number of all arc colorings by a quasi-trivial quandle is invariant under link-homotopy. Furthermore, modifying the definition of quandle homology slightly, we will have a quandle cocycle invariant which is invariant under link-homotopy.

To define a quasi-trivial quandle, we first review the automorphism group and inner automorphism group of a quandle. The automorphism group of a quandle is a group consisting of all automorphisms of together with a product given by the composition of maps. The axioms (Q2) and (Q3) of a quandle guarantee that, for each , the bijection is an automorphism of . The inner automorphism group is the subgroup of generated by all automorphisms . We call an element of an inner automorphism of .

Definition 3.1.

A quandle is said to be quasi-trivial if for each and .

As mentioned above, we have the following proposition.

Proposition 3.2.

Let be a quasi-trivial quandle. Suppose is a link diagram and a diagram obtained from by a self-crossing change. Then, for each arc coloring of by , we have a unique arc coloring of whose restriction to arcs unrelated to the deformation coincides with the restriction of .

Proof.

Assume that the self-crossing change is performed at a crossing . Let denote colors of arcs around by as depicted in Figure 2. Since under arcs and an over arc around belong to the same component, we have

with some , and (See the left-hand side of Figure 4). Let denote the inner automorphism . By the assumption that is quasi-trivial, . Thus, remarking that , we have a unique arc coloring of whose restriction to arcs unrelated to the deformation coincides with the restriction of (See the right-hand side of Figure 4).

Figure 4.

Remark 3.3.

For a link , we have the knot quandle , defined by Joyce [4], which is invariant under ambient isotopy (The same notion is given by Matveev [6]). An arc coloring of a diagram of by a quandle is just a diagrammatic presentation of a homomorphism .

We further have the reduced knot quandle of , given by Hughes [3], which is the quasi-trivialization of . Suppose is a quasi-trivial quandle. Then each homomorphism factors through a homomorphism . Conversely, each homomorphism uniquely lifts to a homomorphism . Thus an arc coloring of a diagram of by is just a diagrammatic presentation of a homomorphism . Hughes showed that is invariant under link-homotopy. It gives an alternative proof for Proposition 3.2.

Proposition 3.2 says that, for a fixed quasi-trivial quandle, the number of all arc colorings is invariant under link-homotopy. However, weights of arc colorings are not invariant under self-crossing changes in general, even if we use quasi-trivial quandles for arc colorings. We thus have to modify the definition of quandle homology slightly, as follows, so that weights are invariant under self-crossing changes.

Let be a quasi-trivial quandle. Define the free abelian groups and boundary maps as in Section 2. Let be a subgroup of generated by -tuples with for some and -tuples for some if . Let if . It is easy to see that . Thus, putting , we have a chain complex .

Suppose that is an abelian group. The -th quasi-trivial quandle homology group with coefficients in is the -th homology group of the chain complex . The -th quasi-trivial quandle cohomology group with coefficients in is the -th cohomology group of the cochain complex . We note that is a 2-cocycle if and only if satisfies the conditions (C1), (C2) in Section 2 and the following condition:

  • For each and , .

Proposition 3.4.

Let be a quasi-trivial quandle and a 2-cocycle with coefficients in an abelian group . Then an weight of an arc coloring by associated with is invariant under self-crossing changes on diagrams.

Proof.

Let be an arc coloring of a link diagram by . A self-crossing change at a crossing of the diagram subtracts from with some and if the crossing consists of arcs belonging to the -th component (See the right-hand side of Figure 4). Since satisfies the condition (C3), it does not change . ∎

By Theorem 2.1 and Proposition 3.4, we have the following theorem immediately.

Theorem 3.5.

Let be a quasi-trivial quandle and be a 2-cocycle with coefficients in an abelian group . Then, for each link and index , the -th quandle cocycle invariant is invariant under link-homotopy.

Remark 3.6.

A trivial quandle is a non-empty finite set equipped with a binary operation satisfying for each . Carter et al. [1] showed that, for each 2-cocycle with coefficients in an abelian group and index , the -th quandle cocycle invariant is completely determined by the linking numbers of . Since the linking numbers are invariant under link-homotopy, is also invariant under link-homotopy. It is easy to see that each trivial quandle is quasi-trivial and its 2-cocycle satisfies not only the conditions (C1) and (C2) but the condition (C3).

4. Example

In this final section, as an application of Theorem 3.5, we show a famous fact that the Borromean rings is not trivial up to link-homotopy.

Let be a set consisting of twelve elements , and (). Define a binary operation on by Table 1 whose -entry denotes with being the -entry and the -entry. Then with is in fact a quasi-trivial quandle. Furthermore, define an element by Table 2 whose -entry denotes with being the -entry and the -entry. It is routine to check that is a 2-cocycle.


Table 1.

Table 2.

Suppose and are diagrams of the trivial 3-component link and the Borromean rings depicted in Figure 5 respectively. Since has no crossing, for each arc coloring of by and index , is always equal to 0. On the other hand, we have an arc coloring of by , depicted in Figure 5, satisfying for each index . It says that for each index . Thus, by Theorem 3.5, the Borromean rings is not link-homotopic to the trivial 3-component link .

Figure 5.
Remark 4.1.

Since the number of all arc colorings of by is equal to the number for , we essentially have to compute the quandle cocycle invariants to distinguish and up to link-homotopy.

Appendix. Decomposition of quandle cocycle invariant

In this appendix, we see that each quandle cocycle invariant can be decomposed into multisets which are still invariant under ambient isotopy or link-homotopy. It seems that these multisets are more useful than a quandle cocycle invariant itself to classify links.

We first note that the inner automorphism group of a quandle naturally acts on the quandle. Therefore each quandle is decomposed into the orbits of the inner automorphism group.

Let be a quandle and the orbit decomposition of . Remark that each arc coloring of a link diagram maps arcs belonging to the same component in the same orbit. We thus call an arc coloring of a diagram of an -component link by to be in if it maps arcs belonging to the -th component in . Obviously, the number of all arc colorings in is invariant under ambient isotopy for each -tuple .

Let be a 2-cocycle with coefficients in an abelian group . For an -component link and -tuple , consider the multiset consisting of -th weights of all arc colorings of a diagram of in . Then obviously

and each is invariant under ambient isotopy.

Of course, the above arguments hold even if is a quasi-trivial quandle and a 2-cocycle. In this case, the number of all arc colorings in and the multiset are invariant under link-homotopy.

References

  • [1] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito, Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3947–3989.
  • [2] N. Habegger and X. S. Lin, The classification of links up to link-homotopy, J. Amer. Math. Soc. 3 (1990), no. 2, 389–419.
  • [3] J. R. Hughes, Link homotopy invariant quandles, J. Knot Theory Ramifications 20 (2011), no. 5, 763–773.
  • [4] D. Joyce, A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23 (1982), no. 1, 37–65.
  • [5] J. P. Levine, An approach to homotopy classification of links, Trans. Amer. Math. Soc. 306 (1988), no. 1, 361–387.
  • [6] S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119 (161) (1982), 78–88 (English translation: Math. USSR-Sb. 47 (1984), 73–83).
  • [7] J. Milnor, Link groups, Ann. of Math. 59 (1954), 177–195.
  • [8] J. Milnor, Isotopy of links, Algebraic geometry and topology (A symposium in honor of S. Lefschetz), Princeton Univ. Press, Princeton, N. J., 1957, 280–306.
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