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1. • CommentRowNumber1.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 13th 2016
   • (edited Feb 13th 2016)
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   Author: Richard Williamson
   Format: MarkdownItexWith a view to making an nLab page on the following, I would like to explain here a proof that the unknot is prime (i.e. if the connected sum $K_1 # K_2$ of a pair of knots is the unknot $U$, then both $K_1$ and $K_2$ are the unknot). I will first give a proof making use of the fact that the fundamental quandle of a knot is a complete invariant of it. Let us denote the fundamental quandle of a knot $K$ by $\Pi(K)$. Passing from $K_{1}$ to $K_{1} # K_{2}$ involves replacing one arc of $K_{1}$ by something, leaving the rest of $K_{1}$ untouched. There is an evident morphism of quandles $f : \Pi(K_{1} # K_{2}) \rightarrow \Pi(K_{1})$ which is the identity on all arcs of $K_{1} # K_{2}$ which come from $K_{1}$ (i.e. all arcs of $K_{1}$ except the one which is removed); and which sends all the remaining arcs of $K_{1} # K_{2}$ to the single arc of $K_{1}$ which was removed when passing from $K_{1}$ to $K_{1} # K_{2}$. This morphism $f$ of quandles is evidently surjective. Now, if $K_{1} # K_{2}$ is the unknot, then $\Pi(K_{1} # K_{2})$ is the trivial quandle. Because of the surjectivity of $f$, it follows that $\Pi(K_{1})$ is trivial. Since the fundamental quandle is a complete invariant of knots, we conclude that $K_{1}$ is the unknot. Since the connected sum of any knot $K$ with the unknot is simply $K$, we deduce that $K_{2}$ is also the unknot, as required. This argument is unbelievably easy, but as far as I know it is new. It generalises directly the well-known proof that if a connected sum of a 3-colourable knot with any other knot cannot be trivial. To give such a proof is stated as an open problem in 1.5 of 'The knot book' by Colin Adams. The classical proof of the primeness of the unknot is much more involved, and also is not 'diagrammatic': it involves Seifert surfaces. However, although the argument that I have given is remarkably simple, it relies on the fact that the fundamental quandle is a complete invariant, which is a deep and difficult fact, relying on work of Waldhausen on 3-manifolds. Thus, it cannot really be regarded as purely diagrammatic either, or as simple a proof as Colin Adams probably had in mind. In a future post in this thread, I will explain how to give another proof, which avoids appealing to the fact that the fundamental quandle is a complete invariant. This will make use of some ideas that I explored when working on a new proof of the Poincaré conjecture, which we are discussing in another thread.

   With a view to making an nLab page on the following, I would like to explain here a proof that the unknot is prime (i.e. if the connected sum $K_1 # K_2$ of a pair of knots is the unknot $U$, then both $K_1$ and $K_2$ are the unknot).

   I will first give a proof making use of the fact that the fundamental quandle of a knot is a complete invariant of it. Let us denote the fundamental quandle of a knot $K$ by $\Pi(K)$. Passing from $K_{1}$ to $K_{1} # K_{2}$ involves replacing one arc of $K_{1}$ by something, leaving the rest of $K_{1}$ untouched. There is an evident morphism of quandles $f : \Pi(K_{1} # K_{2}) \rightarrow \Pi(K_{1})$ which is the identity on all arcs of $K_{1} # K_{2}$ which come from $K_{1}$ (i.e. all arcs of $K_{1}$ except the one which is removed); and which sends all the remaining arcs of $K_{1} # K_{2}$ to the single arc of $K_{1}$ which was removed when passing from $K_{1}$ to $K_{1} # K_{2}$.

   This morphism $f$ of quandles is evidently surjective.

   Now, if $K_{1} # K_{2}$ is the unknot, then $\Pi(K_{1} # K_{2})$ is the trivial quandle. Because of the surjectivity of $f$, it follows that $\Pi(K_{1})$ is trivial. Since the fundamental quandle is a complete invariant of knots, we conclude that $K_{1}$ is the unknot. Since the connected sum of any knot $K$ with the unknot is simply $K$, we deduce that $K_{2}$ is also the unknot, as required.

   This argument is unbelievably easy, but as far as I know it is new. It generalises directly the well-known proof that if a connected sum of a 3-colourable knot with any other knot cannot be trivial. To give such a proof is stated as an open problem in 1.5 of ’The knot book’ by Colin Adams.

   The classical proof of the primeness of the unknot is much more involved, and also is not ’diagrammatic’: it involves Seifert surfaces.

   However, although the argument that I have given is remarkably simple, it relies on the fact that the fundamental quandle is a complete invariant, which is a deep and difficult fact, relying on work of Waldhausen on 3-manifolds. Thus, it cannot really be regarded as purely diagrammatic either, or as simple a proof as Colin Adams probably had in mind.

   In a future post in this thread, I will explain how to give another proof, which avoids appealing to the fact that the fundamental quandle is a complete invariant. This will make use of some ideas that I explored when working on a new proof of the Poincaré conjecture, which we are discussing in another thread.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 13th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexNice! Looking at Wikipedia, I gather that a standard proof invokes additivity of knot genus. Since you seem to be well-informed about knots: what is a good reference for the well-definedness of connected sum for (oriented) knots?

   Nice!

   Looking at Wikipedia, I gather that a standard proof invokes additivity of knot genus.

   Since you seem to be well-informed about knots: what is a good reference for the well-definedness of connected sum for (oriented) knots?

3. • CommentRowNumber3.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 14th 2016
   • (edited Feb 14th 2016)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexYes, exactly. Regarding the connected sum being well-defined, the book of Colin Adams that I mentioned in #1 explains the idea nicely in 1.2 (pg.10-11). This idea is as follows. Suppose that we use two different arcs of $K_{1}$ to form $K_{1} # K_{2}$. Pick one of the two cases. Then we can 'shrink' $K_{2}$ until it is very small, and slide it around $K_{1}$ until we reach the arc used in the second case, whereupon we make $K_{2}$ larger again. The only thing that could go wrong is if, after sliding, the 'isthmus' joining $K_{1}$ and $K_{2}$ 'points the wrong way' (i.e. into $K_{1}$ rather than out from it). Asking that orientations be preserved exactly prevents this situation from occurring. If you do not have access to Adams' book, you can see some pictures at [this page](http://homepages.warwick.ac.uk/~maaac/lecture12.html) (from a beautiful course on knot theory by Brian Sanderson, that I took as an undergraduate at Warwick!).

   Yes, exactly.

   Regarding the connected sum being well-defined, the book of Colin Adams that I mentioned in #1 explains the idea nicely in 1.2 (pg.10-11). This idea is as follows. Suppose that we use two different arcs of $K_{1}$ to form $K_{1} # K_{2}$. Pick one of the two cases. Then we can ’shrink’ $K_{2}$ until it is very small, and slide it around $K_{1}$ until we reach the arc used in the second case, whereupon we make $K_{2}$ larger again. The only thing that could go wrong is if, after sliding, the ’isthmus’ joining $K_{1}$ and $K_{2}$ ’points the wrong way’ (i.e. into $K_{1}$ rather than out from it). Asking that orientations be preserved exactly prevents this situation from occurring.

   If you do not have access to Adams’ book, you can see some pictures at this page (from a beautiful course on knot theory by Brian Sanderson, that I took as an undergraduate at Warwick!).

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 14th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexYeah, in fact I thought of that idea myself soon after posting. I'd be interested in seeing something more formal.

   Yeah, in fact I thought of that idea myself soon after posting. I’d be interested in seeing something more formal.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 14th 2016
   • (edited Feb 14th 2016)
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   Author: Todd_Trimble
   Format: MarkdownItexActually, Richard, your description of "what goes wrong" doesn't quite make sense to me, or at least I don't have a clear picture of what you are trying to say. I see the problem more in the ambiguity of how you perform the forming of the isthmus in the first place, not in a "clash" that may happen during a slide. That is: for unoriented knots, there is no clear instruction, after you excise small arcs from each of the knots, which way one is to join up the loose ends (orientations take care of that problem).

   Actually, Richard, your description of “what goes wrong” doesn’t quite make sense to me, or at least I don’t have a clear picture of what you are trying to say. I see the problem more in the ambiguity of how you perform the forming of the isthmus in the first place, not in a “clash” that may happen during a slide. That is: for unoriented knots, there is no clear instruction, after you excise small arcs from each of the knots, which way one is to join up the loose ends (orientations take care of that problem).

6. • CommentRowNumber6.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 14th 2016
   • (edited Feb 14th 2016)
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   Author: Richard Williamson
   Format: MarkdownItexRegarding #5: my understanding is that there is no problem with making an unambiguous _construction_ in the unoriented case. Given two knot diagrams, we can form a link diagram in which we have two disjoint components, namely these two knot diagrams. We then just replace a local piece as follows <pre> | | | | | | | | </pre> in which one of the vertical arcs belongs to one of the arcs and one belongs to the other, with a local piece as follows. <pre> | | ---- ---- | | </pre> One is of course allowed to rotate these figures, and apply planar isotopies which fix the endpoints. The problem is that the choice of arcs does not necessarily determine the resulting knot uniquely up to isotopy: up to isotopy, there are exactly two possibilities (which sometimes coincide). I like to think of this in the way that I tried to explain in #3. Namely, suppose that we begin with something that looks as follows, where $K$ is the remainder of a knot. <pre> | | --- K --- | | </pre> If we slide this figure around a knot, it may be that at some point it looks as follows. <pre> | | --- K --- | | </pre> Now, we cannot just 'flip' this picture: if we do so, we end up with a connected sum with the mirror image of $K$, and this could lead to a completely different knot. Thus we have to find a way to rule out this possibility. Bringing orientations into the picture, so that we are replacing <pre> ↑ | | | | | | ↓ </pre> with the following <pre> ↑ | --- --- | | | ↓ </pre> takes care of this: having chosen an orientation of each of our two knots, one of the two figures involving $K$ above occurs when the orientations match, and the other occurs when the orientations do not match, so if we restrict to requiring that the orientations match, the second figure cannot occur. With regard to formality, I think, if you will forgive me for saying so, that this is something of a red herring. I like to work in a piecewise-linear setting of one flavour or another. In such a setting, making use of subdivision, I think it is not too difficult to see that one could write down a formal procedure for 'sliding' a local part of a knot in the required manner. But such a proof would not add any insight, at least to me; so I would actually regard an argument as in #3 as entirely rigorous. I would say that there is always an understanding in knot theory, and other parts of geometric topology, that, if forced, one knows that one could write down a purely formal argument, but one does not do so, because it does not really help with arriving at understanding. It would be marvellous if there were some kind of formal system (perhaps of a pictorial nature) in which knot-theoretic arguments such as those of #3 could be expressed easily, and in a way which is not so far removed from the essence of the ideas. I quite often think about this kind of thing, but do not have any such system to propose at present. This is, I suppose, somewhat related to Grothendieck's remarks about the foundations of topology in Esquisse d'un programme, and elsewhere.

   Regarding #5: my understanding is that there is no problem with making an unambiguous construction in the unoriented case. Given two knot diagrams, we can form a link diagram in which we have two disjoint components, namely these two knot diagrams. We then just replace a local piece as follows

   
   |    |
   |    |
   |    |
   |    |
   
   

   in which one of the vertical arcs belongs to one of the arcs and one belongs to the other, with a local piece as follows.

   
   |    |
    ----
   
   
    ----
   |    |
   
   

   One is of course allowed to rotate these figures, and apply planar isotopies which fix the endpoints.

   The problem is that the choice of arcs does not necessarily determine the resulting knot uniquely up to isotopy: up to isotopy, there are exactly two possibilities (which sometimes coincide). I like to think of this in the way that I tried to explain in #3. Namely, suppose that we begin with something that looks as follows, where $K$ is the remainder of a knot.

   
        |
        |
     ---
    K
     ---
        |
        |
   
   

   If we slide this figure around a knot, it may be that at some point it looks as follows.

        |
        |
         ---
            K
         ---
        |
        |
   

   Now, we cannot just ’flip’ this picture: if we do so, we end up with a connected sum with the mirror image of $K$, and this could lead to a completely different knot.

   Thus we have to find a way to rule out this possibility. Bringing orientations into the picture, so that we are replacing

   
   ↑   |
   |   |
   |   |
   |   ↓ 
   
   

   with the following

   ↑   |
    ---
   
    ---
   |   |
   |   ↓ 
   

   takes care of this: having chosen an orientation of each of our two knots, one of the two figures involving $K$ above occurs when the orientations match, and the other occurs when the orientations do not match, so if we restrict to requiring that the orientations match, the second figure cannot occur.

   With regard to formality, I think, if you will forgive me for saying so, that this is something of a red herring. I like to work in a piecewise-linear setting of one flavour or another. In such a setting, making use of subdivision, I think it is not too difficult to see that one could write down a formal procedure for ’sliding’ a local part of a knot in the required manner. But such a proof would not add any insight, at least to me; so I would actually regard an argument as in #3 as entirely rigorous. I would say that there is always an understanding in knot theory, and other parts of geometric topology, that, if forced, one knows that one could write down a purely formal argument, but one does not do so, because it does not really help with arriving at understanding.

   It would be marvellous if there were some kind of formal system (perhaps of a pictorial nature) in which knot-theoretic arguments such as those of #3 could be expressed easily, and in a way which is not so far removed from the essence of the ideas. I quite often think about this kind of thing, but do not have any such system to propose at present. This is, I suppose, somewhat related to Grothendieck’s remarks about the foundations of topology in Esquisse d’un programme, and elsewhere.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeFeb 14th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexRichard, thanks very much for your time and your additional explanations. In #5 I was thinking of connected sum as an operation performed directly on *knots*, whereas your description seems to be in terms of an operation on *knot diagram* presentations. So when I referred back in #5 to an ambiguity in the forming of an isthmus in the unoriented case, I meant that the isthmus could look like your local picture in figure 2, or it could (in place of those two horizontal line segments) look like a crossing X shape. But in the oriented case, one of those two possible joinings would be disallowed, depending on the orientations of the vertical line segments you've drawn. For example, if the orientations both pointed down, the joining by horizontal line segments would obviously be disallowed. > With regard to formality, I think, if you will forgive me for saying so, that this is something of a red herring. (...) (I'll be brief, as musing philosophically about this is probably getting us off track.) Yes, I understand and also to some degree sympathize with that point of view. But I think there is some variation in attitudes towards this essential tension between intuitive explanation and formal proof. Surely most low-dimensional topologists adopt the point of view you've just expressed. But others from different cultures -- one example I have in mind is Joyal and Street when they wrote Geometry of Tensor Calculus I -- may agonize a little more over obtaining more formal proofs. Goresky and MacPherson in their Stratified Morse Theory have an interesting introduction which explains their efforts to establish a happy medium between geometric intuition and rigor -- involving, as they do, quite different brain modules -- and how these efforts led to new conceptual frameworks.

   Richard, thanks very much for your time and your additional explanations. In #5 I was thinking of connected sum as an operation performed directly on knots, whereas your description seems to be in terms of an operation on knot diagram presentations. So when I referred back in #5 to an ambiguity in the forming of an isthmus in the unoriented case, I meant that the isthmus could look like your local picture in figure 2, or it could (in place of those two horizontal line segments) look like a crossing X shape. But in the oriented case, one of those two possible joinings would be disallowed, depending on the orientations of the vertical line segments you’ve drawn. For example, if the orientations both pointed down, the joining by horizontal line segments would obviously be disallowed.

   With regard to formality, I think, if you will forgive me for saying so, that this is something of a red herring. (…)

   (I’ll be brief, as musing philosophically about this is probably getting us off track.)

   Yes, I understand and also to some degree sympathize with that point of view. But I think there is some variation in attitudes towards this essential tension between intuitive explanation and formal proof. Surely most low-dimensional topologists adopt the point of view you’ve just expressed. But others from different cultures – one example I have in mind is Joyal and Street when they wrote Geometry of Tensor Calculus I – may agonize a little more over obtaining more formal proofs. Goresky and MacPherson in their Stratified Morse Theory have an interesting introduction which explains their efforts to establish a happy medium between geometric intuition and rigor – involving, as they do, quite different brain modules – and how these efforts led to new conceptual frameworks.

8. • CommentRowNumber8.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 15th 2016
   • (edited Feb 15th 2016)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexYou're welcome! I don't think that it matters all that much here whether one is working with knot diagrams or knots themselves, but I see your point now, thanks! I guess that I was starting with a construction on unoriented knots, and then viewing the introduction of orientations as a way (a priori there could be others, for instance if one happened to be working only with achiral knots, namely knots which are isotopic to their mirror image) to resolve the possible ambiguity coming from the choice of arcs; whereas you were starting with oriented knots, where one _has_ to make choices that respect orientation. Yes, certainly I agree that there are differences in perspective regarding formalisation of arguments in low-dimensional topology; and I also think that the foundational matter of finding a good formal system for expressing these arguments is very interesting, and wholeheartedly agree that the study of this can be very insightful (you may have noticed from other things I have written that I am very interested in foundational matters myself!). What I was getting at was more that I feel that if one takes 'writing down formally' to mean writing down something in one of the standard settings (piecewise-linear topology, for instance), then I think that it is, broadly speaking, understood what kind of techniques would be needed for that, so that this has more the flavour of a 'tedious exercise' than of something insightful. As it happens, I am not very keen on the kind of formalisation that is studied in the work of Joyal and Street. What they try to do is to write down some formal topological framework which makes 'argument by picture' rigorous. But to me it seems at least as difficult, if one really writes down the details, to translate an argument by picture into their framework as to figure out how to write it down using the structure of a braided monoidal category (or whatever). So I do not really see what is gained; that is to say, I would view this aspect of their work also as something of a 'red herring'. I realise that this point of view may be controversial! But certainly I would like to be able to write down arguments in low-dimensional topology in a nice formal system (some kind of 'synthetic knot theory'), and think that the study of such a formal system could be very interesting from a conceptual point of view. When one is only interested in local moves, as is often the case when constructing invariants, then I think that braid groups, or similar things (I have for instance a way to formalise link diagrams by means of a certain cubical 2-category), do a good job of providing a nice formal framework: one can work purely algebraically, and view the usual pictures just as notation. It is manipulations of link diagrams, such as the 'slides' we have been discussing, which are better thought of as taking place in one go, rather than being broken down into local moves, for which I think there does not exist a good formal system at present.

   You’re welcome! I don’t think that it matters all that much here whether one is working with knot diagrams or knots themselves, but I see your point now, thanks! I guess that I was starting with a construction on unoriented knots, and then viewing the introduction of orientations as a way (a priori there could be others, for instance if one happened to be working only with achiral knots, namely knots which are isotopic to their mirror image) to resolve the possible ambiguity coming from the choice of arcs; whereas you were starting with oriented knots, where one has to make choices that respect orientation.

   Yes, certainly I agree that there are differences in perspective regarding formalisation of arguments in low-dimensional topology; and I also think that the foundational matter of finding a good formal system for expressing these arguments is very interesting, and wholeheartedly agree that the study of this can be very insightful (you may have noticed from other things I have written that I am very interested in foundational matters myself!). What I was getting at was more that I feel that if one takes ’writing down formally’ to mean writing down something in one of the standard settings (piecewise-linear topology, for instance), then I think that it is, broadly speaking, understood what kind of techniques would be needed for that, so that this has more the flavour of a ’tedious exercise’ than of something insightful.

   As it happens, I am not very keen on the kind of formalisation that is studied in the work of Joyal and Street. What they try to do is to write down some formal topological framework which makes ’argument by picture’ rigorous. But to me it seems at least as difficult, if one really writes down the details, to translate an argument by picture into their framework as to figure out how to write it down using the structure of a braided monoidal category (or whatever). So I do not really see what is gained; that is to say, I would view this aspect of their work also as something of a ’red herring’. I realise that this point of view may be controversial!

   But certainly I would like to be able to write down arguments in low-dimensional topology in a nice formal system (some kind of ’synthetic knot theory’), and think that the study of such a formal system could be very interesting from a conceptual point of view. When one is only interested in local moves, as is often the case when constructing invariants, then I think that braid groups, or similar things (I have for instance a way to formalise link diagrams by means of a certain cubical 2-category), do a good job of providing a nice formal framework: one can work purely algebraically, and view the usual pictures just as notation. It is manipulations of link diagrams, such as the ’slides’ we have been discussing, which are better thought of as taking place in one go, rather than being broken down into local moves, for which I think there does not exist a good formal system at present.

9. • CommentRowNumber9.
   • CommentAuthorRichard Williamson
   • CommentTimeFeb 15th 2016
   • (edited Feb 15th 2016)
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexI will now outline an argument for giving a proof of the primeness of the unknot which does not involve fundamental quandles. We begin by observing that there is a surjection from the set of arcs of $K_{1} # K_{2}$ to the set of arcs of $K_{1}$, defined in the same way as the morphism $f$ of #1, namely sending all arcs of $K_{2}$, along with the connecting arcs of the isthmus joining $K_{1}$ to $K_{2}$, to the arc which is removed from $K_{1}$, which I'll denote by $x$. Now, suppose that we apply a Reidemeister move to $K_{1} # K_{2}$. If this Reidemeister move involves only the arcs of $K_{1}$ (considering the two arcs of the 'isthmus' joining $K_{1}$ to $K_{2}$ to both be $x$, i.e. to both be an arc of $K_{1}$), then we apply the same Reidemeister move to $K_{1}$. Otherwise, we do nothing to $K_{1}$, but send any new arcs created by the move to $x$, and map the remaining arcs, even if they have been moved, split into two, or now cross other arcs (allowing crossing themselves, i.e. applying an $R_{1}$ move), to the same arc of $K_{1}$ as before. Either way, we still have a surjection from the arcs of $K_{1} # K_{2}$ (after having applied the local move) to the arcs of $K_{1}$ (after having applied the local move, if we are in this case). We can continue in this way for any number of manipulations of $K_{1} # K_{2}$, always having a corresponding $K_{1}$ for which there is a surjection from the arcs of $K_{1} # K_{2}$ to the arcs of $K_{1}$. Now, if $K_{1} # K_{2}$ is trivial, we can manipulate it using local moves to the unknot. We deduce that there is a surjection from the set of arcs of the unknot, which consists of only one arc, to the set of arcs of the corresponding $K_{1}$. Hence the latter must only have one arc, and thus it must be the unknot or a figure of eight, so that it is isotopic to the unknot, and hence the original $K_{1}$ is isotopic to the unknot, as required. I would be interested to hear any thoughts! Everything rests of course on the well-definedness of the procedure for modifying the surjection.

   I will now outline an argument for giving a proof of the primeness of the unknot which does not involve fundamental quandles.

   We begin by observing that there is a surjection from the set of arcs of $K_{1} # K_{2}$ to the set of arcs of $K_{1}$, defined in the same way as the morphism $f$ of #1, namely sending all arcs of $K_{2}$, along with the connecting arcs of the isthmus joining $K_{1}$ to $K_{2}$, to the arc which is removed from $K_{1}$, which I’ll denote by $x$.

   Now, suppose that we apply a Reidemeister move to $K_{1} # K_{2}$. If this Reidemeister move involves only the arcs of $K_{1}$ (considering the two arcs of the ’isthmus’ joining $K_{1}$ to $K_{2}$ to both be $x$, i.e. to both be an arc of $K_{1}$), then we apply the same Reidemeister move to $K_{1}$. Otherwise, we do nothing to $K_{1}$, but send any new arcs created by the move to $x$, and map the remaining arcs, even if they have been moved, split into two, or now cross other arcs (allowing crossing themselves, i.e. applying an $R_{1}$ move), to the same arc of $K_{1}$ as before. Either way, we still have a surjection from the arcs of $K_{1} # K_{2}$ (after having applied the local move) to the arcs of $K_{1}$ (after having applied the local move, if we are in this case).

   We can continue in this way for any number of manipulations of $K_{1} # K_{2}$, always having a corresponding $K_{1}$ for which there is a surjection from the arcs of $K_{1} # K_{2}$ to the arcs of $K_{1}$.

   Now, if $K_{1} # K_{2}$ is trivial, we can manipulate it using local moves to the unknot. We deduce that there is a surjection from the set of arcs of the unknot, which consists of only one arc, to the set of arcs of the corresponding $K_{1}$. Hence the latter must only have one arc, and thus it must be the unknot or a figure of eight, so that it is isotopic to the unknot, and hence the original $K_{1}$ is isotopic to the unknot, as required.

   I would be interested to hear any thoughts! Everything rests of course on the well-definedness of the procedure for modifying the surjection.