There are non homotopic framed homotopies of long knots
Let be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let be the closure of the union of all singular knots in with exactly one ordinary double point and such that the two resolutions represent the same (non singular) knot type. We call the inessential walls and we call the essential diagram space.
We construct a non trivial class in by an extension of the Kauffman bracket. This implies in particular that there are loops in which consist of regular isotopies of knots together with crossing changings and which are not contractible in (leading to the title of the paper).
We conjecture that our construction gives rise to a new knot polynomial for knots of unknotting number one.
1 Introduction and results
The study of knot spaces consists mainly of the study of spaces of non singular knots (compare , , , ). In this paper we change the point of view: we construct a 1-cocycle for some space which includes singular knots too. This 1-cocycle is identical zero on all loops which consist only of non-singular knots.
The space of all (possibly singular long) knots was introduced and studied by Vassiliev in the pioniering work . It is the space of all differentable maps which agree with outside of . This is a contractible space (compare ), let us call it . We identify then as usual a knot with its image . A knot is non-singular if is a smooth submanifold, otherwise it is called singular. Knots are oriented from the left to the right.
The singular knots form the discriminant of the space of all (long) knots. It has a natural stratification. The complement of the discriminant are the chambers. They correspond to the (non-singular) knot types. Vassiliev has introduced a filtration on the cohomology of the chambers. The 0-dimensional part are the well-known Vassiliev knot invariants (for all this compare , , ).
Instead of Vassiliev’s knot space we will study the space of long framed knots. We fixe a plane in 3-space which contains the and we fixe an orthogonal projection of the 3-space into this plane. is the space of all those (possibly singular) knots for which the restriction of on the knots is an immersion. The chambers of consist of knot diagrams up to isotopy and up to Reidemeister moves of type II and III (this is called usually regular isotopy of non singular knots). The points in project to immersed long planar curves. It is easy to see that the connected components of are in 1-1 correspondence with the regular homotopy classes of immersed long planar curves (standard at infinity). It is well known that the latter are in 1-1 correspondence with their Whitney index (i.e. the degree of the Gauss map). One easily sees that is a disjoint union of contractible spaces (numbered by the Whitney index).
The strata of codimension one of the discriminant are called the walls. They correspond to the knots which have exactly one ordinary double point as the only singularity. An ordinary double point of an oriented knot can be resolved in two different ways (compare Fig. 1).
A wall is called inessential if the two adjacent chambers coincide. The union of all inessential walls is denoted by . The space is called the essential diagram space and denoted by . A generic path in is called a framed homotopy.
We could associate to each connected component of a stratified space in the following way: the chambers of correspond to vertices. Each stratum in corresponds to an edge connecting the vertices corresponding to chambers adjacent to . Each stratum in gives rise to a 2-cell glued to all edges and vertices corresponding to those strata and those chambers which are adjacent to . And we can go on by gluing cells of dimension corresponding to strata of codimension .
The resulting space is not a CW-complex because it is not locally finite. (It is well known that there are e.g. infinitely many knot types of unknotting number one.) However, this space is evidently still simply connected.
Let us now consider the closure in a component of of the union of all inessential walls (i.e. we add all adjacent strata of higher codimension). This space is a rather mysterious object. For example, it seems not to be known wether or not the strata of correspond always to ”nugatory crossings” (for the definition see e.g. ). Moreover, I do not know wether or not the complement of is still connected in each component of .
The two knots shown in Fig.2 are not framed homotopic in .
But notice that the two knots shown in Fig.3 are framed homotopic in . The path is shown in the figure too. It uses the fact that the ”figure eight” knot can be unknotted both by a positive or a negative crossing change.
So, we do not know much about the components of the essential diagram space besides the fact that each component becomes simply connected (even contractible) if we add again the closure of the non-essential walls .
In this paper we describe a surprising phenomen: there are components of the essential diagram space which have non-trivial first homology groups. This result is far from beeing obvious and we do not know any other method to prove this. (In finite dimensions this would imply by Alexander duality that contains a cycle of codimension two in and this cycle is not homologically trivial in .)
Our prove uses a 1-cocycle which is constructed in the following way: Let be a singular knot and let be the extension of the Jones polynomial for singular links contained in  (abusing notation we denote a knot diagram for by too). It is defined as follows:
(In  we have chosen because the polynomial is homogenous in and .) The singular Kauffman bracket is defined at crossings as the usual Kauffman bracket (compare ) and at double points it is defined as shown in Fig. 4. Here, and are new independent variables. Notice that one of the smoothings induces only a piecewise orientation on the link diagram. is the writhe of the knot diagram (see e.g. ).
Let be a generic path in which connects a given non singular knot with the unknot. Here, the end points of the paths are allowed to move inside the chambers. A generic homotopy of such a path meets the codimension two part of the discriminant in a finite number of points which correspond to knots with exactly two ordinary double points or with exactly one ordinary cusp or with exactly one double point with equal tangent directions (see e.g.  and ).
We define a co-orientation on by saying that the positive normal direction corresponds to changing a negative crossing to a positive one. The path intersects transversally in a finite number of points. Let be such an intersection point. Abusing notation we denote the corresponding double point by too and we denote the corresponding singular knot by . Let be the intersection index of with at .
Let be an oriented path in which connects a non singular knot with a diagram of the unknot. The polynomial is defined by the following formula
It follows immediately from the definitions that the intersection index of is an invariant of paths in up to homotopy in , if the end points of the paths are fixed up to regular isotopy. Consequently, the writhe of the unknot at the end of the path is completely determined by the writhe of the knot and by the intersection index . The Whitney index is invariant under crossing changes. (As well known, non singular knots are regularly isotopic if and only if they are isotopic and they share the same writhe and the same Whitney index, compare e.g. .) In particular, the intersection index of each loop with is zero.
Let be an oriented loop in . Then the value of depends only on the homology class of . The induced cohomology class is non trivial.
We describe now the first loop in for which is non-trivial (we could not find a simpler example). Let us consider the knots and (from the Knot Atlas) with their standard diagrams. They are both amphicheiral. Each of them can be unknotted by either a positive or a negative crossing change. Let be the connected sum . Let be the homotopy which unknots by a positive crossing change (i.e. ) of and by a negative crossing change (i.e. ) of . Let be the homotopy which unknots by a negative crossing change of and by a positive crossing change of .
A calculation by hand gives the following result:
Here denotes the usual Kauffman bracket. Notice that the diagrams of and have vanishing writhe. Hence, the Kauffman bracket coincides with the Jones polynomial (see e.g. , ). We could change by Reidemeister moves of type I and consider the induced loop . The value of would change just by a standard factor. This shows that these components of are non-simply connected too.
The framed long knot (i.e. the diagram or like-wise the smooth submanifold ) in our example has unknotting number two. The value of looks like the product of some standard polynomial (which depends only on the unknotting number of ) with a linear combination of Jones polynomials of those knots of unknotting number one which are contained in the loop. It seems to be very difficult to prove this in general. But it leads us to the following conjecture.
Let be a framed long knot of unknotting number one. Let be a homotopy which unknots and such that (respectively ). Then is invariant under regular isotopy of .
Notice that apriori the intersection of the closure of the connected component of in with the closure of the component of the unknot could have an infinite number of connected components. Each of these components corresponds to an unknotting of and they could be all different!
We have verified this conjecture by hand for all diagrams with no more than 6 crossings (which is not a big deal).
Here is the writhe and is the Whitney index of the diagram which we unknot. is the right trefoil and is its mirror image, the left trefoil.
Our theorem shows that the situation is more complicated already for framed knots of unknotting number two. The corresponding conjecture would be that is now well defined up to combinations of Jones polynomials of framed knots with unknotting number one.
There is an extension of the HOMFLY-PT polynomial for singular links by Kauffman and Vogel . A natural idea would be to replace in the construction of the singular Kauffman bracket by their singular HOMFLY-PT polynomial. Surprisingly, this fails. We can not replace by our singular Alexander polynomial neither (see  and the next section).
Crossing an inessential wall does not change the knot type but it changes the framing. If we would replace framed isotopy by isotopy then our construction would just lead to a version of the Jones polynomial (as it happened in the first version of the present paper).
Let be a generic path in the essential diagram space (i.e. a framed homotopy).
First we observe that is invariant under all homotopies of which do not change the intersection with the strata of the discriminant . This comes from the fact that is an isotopy invariant of framed singular knots (i.e. only Reidemeister moves of type II and III and the additional moves for singular links and are allowed, compare  and also ).
Next we need some simple facts from singularity theory which can be proven with the same methods as e.g. in the Appendix of  or . Only the following accidents can occure in a generic homotopy of a path in and they can occure only a finite number of times.
I. becomes tangential (in an ordinary tangent point) to a stratum of .
II. passes transversally to a stratum of codimension 2, which consists of the transverse intersection of two strata of codimension 1. We denote these strata by .
III. passes transversally to a stratum of codimension 2, which consists of a (single) double point where the two branches are tangential. We denote these strata by .
IV. passes transversally through an ordinary triple point.
All other paths in the homotopy are just generic paths in .
The important point is that the following accident can occure in but not in :
V. passes transversally to a stratum of codimension 2, which consists of an ordinary cusp. We denote these strata by .
The strata are just (a generic part of) boundaries of inessential walls in , but all our paths are in .
It follows immediately from the definitions that all our polynomials are invariant under accidents of type I.
In order to prove Theorem 1 we have to show that for the meridional loops of and of .
We show the diagrams for the meridional loop of in Fig. 5. After smoothing the crossings and the double points we are left with exactly the four diagrams shown in Fig. 6. They are in general independent. The meridional loop of leads to the following equations:
Here, is the usual Kauffman bracket. in general if and only if each coefficient of is zero. Therefore we obtain the unique (non trivial) solution
Each singular link in the framed homotopy has exactly one double point. Therefore we do not loose information by setting , and we obtain exactly the definition of .
Notice that it could happen that exactly one of the four strata of in Fig. 5 is non essential. In this case we could not push our path in through the corresponding stratum of by a small homotopy. Our theorem implies that sometimes we can not even do it by a big homotopy.
We show the diagrams for the meridional loop of in Fig. 7. The value of is a non trivial multiple of a Kauffman bracket. Consequently, is in general non zero.
We show the diagrams for the meridional loop of in Fig. 8.
follows from the invariance of under the move for singular links (compare ). (We had to choose in Fig. 8 a type of a Reidemeister II move and orientations on the branches. Taking mirror images or changing orientations of branches leads to the same result.)
The stratum of an ordinary triple point is not smooth. But one easily sees that the meridians form theta-graphs. Using the invariance of under passing it suffices to prove the invariance for the loops in just one of the theta graphs (compare ). The three arcs in the theta-graph are shown in Fig. 10,11 and 12. It suffices hence to prove that .
We show the calculation of in Fig. 12. The first equality in Fig. 12 uses twice the fact that is invariant under passing . The calculation of is completely analogous and is left to the reader.
It is clear that we can replace homotopy by homology in all our constructions because the values are in a commutative ring. We have proven that depends only on the homology class of in . The example in the introduction shows that induces a non trivial cohomolgy class. The theorem is proven.
It remains to prove our assertion from Remark 1. We recall the definition of the singular HOMFLY-PT polynomial from  in Fig. 13. Here, are new independent variables. We want to replace by this polynomial. A calculation for the meridional loop of leads to the equation shown in Fig. 14, which has the unique solution . But then the singular HOMFLY-PT polynomial is no longer sensitive for crossing changes of links.
The failure of the singular Alexander polynomial from  is also caused by the stratum . It leads to the equation , which makes the invariant uninteresting. We leave the verification to the reader.
Acknowledgements— I am grateful to the referee for his useful comments.
Institut de Mathématiques de Toulouse
Université Paul Sabatier et CNRS (UMR 5219)
118, route de Narbonne
31062 Toulouse Cedex 09, France
- 2000 Mathematics Subject Classification: 57M25. Key words and phrases: classical and singular links, discriminant, knot polynomials, state models
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