The Artin relations describe both elementary exchanges and every change of crossing word under homotopy. The second assertion is the completeness part of the presentation.
The other input is Chapter 3's configuration-space model. The technical general-position prerequisite is stated in the expandable section below; the braid argument after that prerequisite is proved here.
5.1. The comparison homomorphism
Fix \(n\geq3\). Let \(D=\{z\in\mathbb C:|z|\leq1\}\), let \(D^\circ\) be its interior, and put \(q_j=-1+2j/(n+1)\) and \(Q=\{q_1,\ldots,q_n\}\). Write
\[F_n(D^\circ)=\{(z_1,\ldots,z_n)\in(D^\circ)^n:z_j\ne z_k\text{ for }j\ne k\},\qquad C_n(D^\circ)=F_n(D^\circ)/S_n.\]Let \(A_n\) denote the abstract group
\[\left\langle s_1,\ldots,s_{n-1}\;\middle|\;s_is_j=s_js_i\ (|i-j|\geq2),\quad s_is_{i+1}s_i=s_{i+1}s_is_{i+1}\ (1\leq i\leq n-2)\right\rangle.\]Theorem 5.1. The Artin isomorphism
There is an isomorphism \(\Phi:A_n\to\pi_1(C_n(D^\circ),Q)\) determined by \(\Phi(s_i)=[\sigma_i]\). Here \(\sigma_i\) is the counterclockwise elementary exchange.
Alphabeta Math: Artin completeness · configuration comparison
Products of loops traverse the first loop then the second. For \(c_i=(q_i+q_{i+1})/2\) and \(r=1/(n+1)\), define a loop \(\sigma_i\) of unordered configurations by fixing all other points and moving these two:
\[z_i(t)=c_i-r e^{\pi\mathrm{i}t},\qquad z_{i+1}(t)=c_i+r e^{\pi\mathrm{i}t},\qquad 0\leq t\leq1.\]Here \(\mathrm{i}^2=-1\), while the subscript \(i\) indexes the pair. Each point rotates counterclockwise about \(c_i\). The point starting on the left passes below, and the point starting on the right passes above. The two points stay distance \(2r\) apart; their circle lies in \(D^\circ\), and no fixed point lies on it. At \(t=1\) their positions have exchanged, so the unordered configuration returns to \(Q\). The intended map is \(\Phi(s_i)=[\sigma_i]\). We must first check that it respects the defining relations.
The proof of Theorem 5.1 follows after the crossing-word and homotopy lemmas below.
5.2. Crossing words
Definition 5.2. A generic crossing word
Lift a loop to labeled paths \(z_j(t)=x_j(t)+iy_j(t)\), starting at \((q_1,\ldots,q_n)\). Draw \((x_j(t),t)\), with time increasing upward. At a projected crossing, declare the strand with larger \(y\) to pass over the other. A crossing is positive when the strand immediately to its left before the crossing has smaller \(y\); this makes the loop above positive. Record \(s_i\) or \(s_i^{-1}\), where \(i,i+1\) are the crossing strands' positions in the current left-to-right order, not their original labels.
A generic loop has finitely many crossing times, each with exactly one equality \(x_j=x_k\), with \(\frac d{dt}(x_j-x_k)\ne0\). The crossing strands are adjacent in real-coordinate order. Because \(z_j\ne z_k\), their imaginary coordinates differ there, so the sign is defined.
Worked example for Definition 5.2
Counterexample: A nongeneric motion need not have isolated transverse crossings
Lemma 5.3. Reconstruction from crossings
A generic loop is homotopic to the chronological product of elementary exchanges recorded by its crossing word.
Proof.
The open set \(U\subset C_n(D^\circ)\) of configurations with distinct real coordinates has a unique left-to-right ordering. In those coordinates it is convex: the disk conditions and \(x_1<\cdots<x_n\) are convex. Thus every path segment in \(U\) can be straightened, and every connector to \(Q\) is unique up to homotopy.
Near one crossing, choose labels for the two crossing points and require all other real-coordinate inequalities to remain strict, allowing only this adjacent pair to change order. Also impose the fixed strict inequality between their imaginary coordinates. The resulting set \(W\) is convex in the labeled configuration space. It contains no collision: unequal real coordinates separate every other pair, and the imaginary inequality separates this pair. Both side intersections with the corresponding chambers are convex. Hence any passage through \(W\), closed using chamber connectors, represents the same loop as the appropriate \(\sigma_i^{\pm1}\). Use a truncated passage around the semicircle’s crossing, so its endpoints also satisfy the strict imaginary inequality; the remaining arcs lie in the real-order chambers. One can move the crossing configuration to this passage while preserving the inequalities; convexity gives the required homotopies. Subdivide the loop into chamber segments and single-crossing passages. Its class is the product of the recorded generators.
□5.3. The Artin relations
Lemma 5.4. Commutation and the three-strand relation
The elementary-loop classes satisfy the distant commutation and adjacent Artin relations.
Proof.
For \(|i-j|\geq2\), carry out the two exchanges in disjoint small disks about their respective midpoint circles. Allow their progress parameters to vary independently over a square. Every point in that square is a configuration without collisions. The two boundary routes perform the exchanges in opposite orders, proving \([\sigma_i][\sigma_j]=[\sigma_j][\sigma_i]\).
For the adjacent relation, isolate three consecutive points in an interior disk containing no other point. Fix their mean real coordinate at its centre and keep their imaginary coordinates at three small distinct heights \(y_A<y_B<y_C\). In their real coordinates, the walls \(x_A=x_B\), \(x_A=x_C\), and \(x_B=x_C\) are three lines through the origin in the two-dimensional difference plane. Take a small disk around their common intersection in this difference plane. Choosing both the heights and the disk radius sufficiently small keeps all three points inside the interior disk, with the other points fixed and separated in real coordinate. Throughout this disk the three points remain distinct because their heights are distinct. The two semicircular routes from the chamber \(A,B,C\) to the opposite chamber \(C,B,A\) cross the walls in orders \(AB,AC,BC\) and \(BC,AC,AB\). All six crossings are positive by the stated height order. Close the routes using paths in \(U\). The filled disk and chamber contractions give
\[[\sigma_i][\sigma_{i+1}][\sigma_i]=[\sigma_{i+1}][\sigma_i][\sigma_{i+1}].\]The disk may be chosen inside a small disk in \(D^\circ\), with all other points fixed. This proves the relation by an actual configuration-space homotopy. Consequently \(\Phi:A_n\to\pi_1(C_n(D^\circ),Q)\) is well defined. The preceding crossing-word argument proves it is surjective.
□Corollary 5.5. A surjective comparison homomorphism
The assignment \(s_i\mapsto[\sigma_i]\) defines a surjective homomorphism \(\Phi:A_n\to\pi_1(C_n(D^\circ),Q)\).
Worked example for Corollary 5.5
Counterexample: A surjective comparison map alone does not prove an isomorphism
Proof.
Lemma 5.4 verifies the relators, so the presentation universal property in Chapter 2 defines \(\Phi\). Lemma 5.6 provides a generic representative of each loop; Lemma 5.3 expresses that representative as a product of generator images.
□Prerequisites and proof of general position
Lemma 5.6. Generic representatives and homotopies
Every configuration loop has a generic representative. An endpoint-fixed homotopy between generic representatives may be chosen so that its crossing word changes at finitely many distinct parameters, each by a pair cancellation, a disjoint pair interchange, or a triple crossing.
Alphabeta Math: generic representatives · transversality
Counterexample: A nongeneric motion need not have isolated transverse crossings
Proof.
The differential-topology input consists of smooth approximation and transversality. We import these theorems from Hirsch, Differential Topology, Chapters 2 and 3, in their relative forms. The finite jet-transversality consequence needed here is explained below; no multijet theorem is needed. The following definitions and theorem statement specify the input; the application to braids is then checked below.
Smooth approximation means replacing a continuous map into an open subset of Euclidean space by an arbitrarily uniformly close map with derivatives of every order. If the original map is already smooth near a specified closed region, the replacement can agree there. For our parameter square, we use boundary collars as that region. This replacement is homotopic to the original whenever the straight interpolation remains in the target.
A smooth map \(f:M\to N\) is transverse to a smooth submanifold \(S\subset N\) if, whenever \(f(p)\in S\),
\[Df_p(T_pM)+T_{f(p)}S=T_{f(p)}N.\]Here \(T_pM\) is the vector space of infinitesimal directions at \(p\), and \(Df_p\) is the derivative. Equivalently, variations of the input supply every direction normal to \(S\). If \(S\) is defined locally by \(c\) independent equations, transversality says their derivatives after composition with \(f\) have rank \(c\). Its inverse image is then a smooth set of dimension \(\dim M-c\); it is empty if \(c>\dim M\). The number \(c\) is the codimension of \(S\).
An \(r\)-jet records a map's value and all its derivatives through order \(r\) at one source point, together with that point. Thus a 2-jet of a real function on our square records \(f,f_s,f_t,f_{ss},f_{st},f_{tt}\).
Imported transversality theorem. For a smooth family of maps whose joint evaluation is transverse to a smooth submanifold, almost every parameter gives a transverse map (parametric transversality, a consequence of Sard's theorem). Smooth maps transverse to finitely many specified smooth submanifolds can therefore be obtained by arbitrarily small perturbations, fixed where the conditions already hold. We use this relative theorem and smooth approximation from Hirsch, Chapters 2–3.
Here is why this also applies to our finite list of jet conditions. On a small source disk, perturb each coordinate by a polynomial of degree at most two times a smooth bump equal to one on a smaller disk. Its coefficients independently vary the value, first derivatives, and second derivatives at every point of that smaller disk. Thus the joint map from coefficients and source points into 2-jets is a submersion, hence transverse to every jet stratum. Finitely many smaller disks cover the compact region needing perturbation; combine their coefficient families and apply parametric transversality to the finitely many strata. Bumps are supported away from the fixed collars. On the remaining region the required conditions already hold and persist under sufficiently small perturbations. This establishes the special relative jet consequence we use, rather than importing braid genericity as a theorem.
A homotopy \(H(s,t)\), \(0\leq s,t\leq1\), of based loops lifts to \(F_n(D^\circ)\) because the square is simply connected and \(F_n\to C_n\) is a covering. The lift has the fixed starting tuple on \(t=0\) and a fixed permuted tuple on \(t=1\); the permutation is locally constant in \(s\), hence constant. First reparameterize time by a smooth nondecreasing map constant near 0 and 1 and strictly increasing between those collars. This produces constant endpoint collars by an endpoint-fixed homotopy. Approximate each boundary loop smoothly relative to these collars, then perturb its real coordinates transverse to the equality strata. These close approximations are homotopic to the original loops by straight interpolation upstairs. Attach those homotopies to the given square. Reparameterizing the square's homotopy coordinate similarly makes it independent of \(s\) near \(s=0,1\), equal there to the chosen smooth generic loops. Reparameterize its time coordinate to retain the constant endpoint collars. The square is now smooth on all four boundary collars: the \(t\)-collars have distinct real coordinates, and on the \(s\)-collars every crossing has nonzero \(t\)-derivative. Thus the relative approximation and perturbation hypotheses really hold there.
Compactness supplies a positive margin
\[\delta=\min_{s,t}\min\left(\min_j(1-|z_j(s,t)|),\ \min_{j < k}|z_j(s,t)-z_k(s,t)|\right) > 0.\]A uniform perturbation of each coordinate by less than \(\delta/4\), and its straight interpolation with the original lift, stays in the disk and avoids collisions. Thus approximation and perturbation preserve the homotopy class. All subsequent transversality perturbations can be chosen this small and fixed on the prescribed boundary collars.
Stratify the real-coordinate space by partitions into sets of equal coordinates. A block of size \(m\) imposes \(m-1\) independent equations. Transversality of a one-dimensional path avoids strata of codimension at least two and makes every codimension-one encounter transverse. Transversality of the two-dimensional homotopy avoids strata of codimension at least three. Its isolated codimension-two encounters therefore have either two disjoint equal pairs, or one triple of equal real coordinates. The triple's three pairwise equations have rank two, not three.
On a single-pair wall, write \(f=x_j-x_k\). Transversality makes \(f=0\) a smooth curve in the parameter square. Apply jet transversality also to the two equations \(f=f_t=0\). At such a point the derivative matrix of \((f,f_t)\) has determinant \(f_s f_{tt}\), so its required rank two gives \(f_s\ne0\) and \(f_{tt}\ne0\). These are ordinary folds, locally \(s-s_0=\pm(t-t_0)^2\) after smooth coordinate changes preserving the slicing parameter. Indeed, the implicit-function theorem writes the wall as a graph \(s=g(t)\), with \(g'(t_0)=0\) and \(g''(t_0)\ne0\). Taylor's formula gives \(g(t)-g(t_0)=(t-t_0)^2 h(t)\) with smooth nonzero \(h\); absorbing \(\sqrt{|h(t)|}\) into the time coordinate proves the fold form.
At a codimension-two intersection, both independent wall equations have independent differentials. Require also that each pairwise difference has nonzero \(t\)-derivative. At a triple, this includes all three pairs. Failures add a condition and have codimension at least three. Consequently the wall arcs meet transversely and their crossing times exchange order as \(s\) passes the event.
There are finitely many events. Each codimension-two intersection is isolated and belongs to a closed compact zero set. Fold points are likewise isolated; they cannot accumulate at intersections because every pairwise \(t\)-derivative there is nonzero. A limit at a single-pair wall is another fold, where the rank-two condition again gives isolation. The fixed collars contain no events. Compactness therefore makes the entire event set finite.
Separate their \(s\)-values without a further transversality theorem. Choose disjoint tiny neighborhoods of these finitely many parameter points and bumps \(\beta_r\) equal to one near each point and zero outside its neighborhood. The change of source coordinates
\[(s,t)\longmapsto\left(s+\sum_r\epsilon_r\beta_r(s,t),t\right)\]is a diffeomorphism fixed on the boundary collars when the constants \(\epsilon_r\) are sufficiently small. Compose the homotopy with its inverse. Near event \(r\), this merely translates its \(s\)-coordinate by \(\epsilon_r\). On the transition regions every wall has noncritical projection to \(s\); compactness and a sufficiently small first-derivative change preserve that fact. Thus no new fold appears, intersections are simply carried along, and their \(s\)-values can be chosen distinct.
Each nonexceptional time slice has finitely many transverse crossings: their zero sets are compact and isolated, with none on the endpoint collars. Between event parameters the implicit-function theorem follows each crossing time continuously; crossings cannot appear, disappear, or exchange order there. Their imaginary differences cannot change sign on a wall without a forbidden actual collision. The crossing word is therefore constant between events. This proves the exhaustive list, using only the stated smooth-approximation and parametric-transversality input.
□5.4. Completeness of the relations
Lemma 5.7. Homotopy invariance of the Artin word
Homotopic generic loops have crossing words defining the same element of \(A_n\).
Proof.
Now let two generic based loops be homotopic. Apply the preceding prerequisite to an endpoint-fixed homotopy. Sweeping \(s\) from 0 to 1, there are exactly three possible kinds of change.
- A fold creates or removes two crossings of the same pair. No other strand is between the pair locally, so their index is the same. Their imaginary order is unchanged, while their crossing directions are opposite. The subword is \(s_i s_i^{-1}\) or \(s_i^{-1}s_i\), which cancels.
- Two disjoint pairs cross at the same time. Their real-coordinate locations are distinct; otherwise four points would coincide in real coordinate, a forbidden codimension-three event. The pairs occupy disjoint adjacent positions, so \(|i-j|\geq2\). Their crossing times exchange order, giving \(s_i^\epsilon s_j^\eta\leftrightarrow s_j^\eta s_i^\epsilon\). These identities, including negative exponents, follow from \(s_is_j=s_js_i\).
- Three strands have the same real coordinate. Their imaginary coordinates are pairwise distinct, since actual collisions are excluded. The nonzero pairwise time derivatives mean that the three real-coordinate velocities are distinct. Thus their real ordering reverses across the event. The following calculation verifies both the crossing orders and their signs for every orientation of the event.
There is no Reidemeister-I event. Every strand is a graph over time, so a strand never intersects itself in the \((x,t)\) projection and never develops the local time maximum or minimum required for a curl. “Projected coincidence” in this proof always means equal real coordinates of different points; it never means that the points collide in the disk.
The six possible triple signs
Choose a short time interval about the event and label the strands \(A,B,C\) in their real order at its beginning. At the event their distinct velocities satisfy \(v_A>v_B>v_C\): just before it, \(x_A-x_B\) and \(x_B-x_C\) are negative. Each pair crosses exactly once on nearby slices by the implicit-function theorem, and the order at the interval's end is \(C,B,A\). Every crossing is adjacent. The only possible three-step reversals are consequently \(AB,AC,BC\) and \(BC,AC,AB\). To see that the two sides give different orders, write the crossing-time graphs as \(t=\tau_{AB}(s)\) and \(t=\tau_{BC}(s)\). Independence of the two wall differentials gives \(\tau'_{AB}(s_0)\ne\tau'_{BC}(s_0)\), so their times exchange order at \(s_0\). The \(AC\) crossing lies between them: \(x_A-x_C=(x_A-x_B)+(x_B-x_C)\), with both summands increasing in time near the event. This verifies the three-crossing move for either direction of the \(s\)-sweep and any original labeling.
Put \(a=s_i\), \(b=s_{i+1}\), and
\[p=\operatorname{sgn}(y_B-y_A),\qquad q=\operatorname{sgn}(y_C-y_A),\qquad r=\operatorname{sgn}(y_C-y_B).\]The local move is
\[a^p b^q a^r\quad\longleftrightarrow\quad b^r a^q b^p.\]Since the imaginary heights have a total order, only six of the eight sign triples can occur:
| Heights, lowest first | \((p,q,r)\) | Identity |
|---|---|---|
| \(A,B,C\) | \((+,+,+)\) | \(aba=bab\) |
| \(A,C,B\) | \((+,+,-)\) | \(aba^{-1}=b^{-1}ab\) |
| \(B,A,C\) | \((-,+,+)\) | \(a^{-1}ba=bab^{-1}\) |
| \(B,C,A\) | \((-,-,+)\) | \(a^{-1}b^{-1}a=ba^{-1}b^{-1}\) |
| \(C,A,B\) | \((+,-,-)\) | \(ab^{-1}a^{-1}=b^{-1}a^{-1}b\) |
| \(C,B,A\) | \((-,-,-)\) | \(a^{-1}b^{-1}a^{-1}=b^{-1}a^{-1}b^{-1}\) |
These are all consequences of one relation. From \(aba=bab\), multiply on the left by \(b^{-1}\) and on the right by \(a^{-1}\) to get the second identity. Multiplying instead on the left by \(a^{-1}\) and on the right by \(b^{-1}\) gives the third. Their inverses give the fifth and fourth, respectively, and inverting the original relation gives the sixth. The excluded triples \((+,-,+)\) and \((-,+,-)\) would require a cyclic ordering of three real heights, which is impossible.
All these moves take place in genuine configurations: near a triple event the distinct imaginary heights remain separated; near a double event the two crossing pairs remain separate; near a fold the crossing pair has distinct heights. Choosing sufficiently small parameter neighborhoods preserves these facts and the separation from the other points.
□5.5. Proof of the Artin isomorphism
Proof of Theorem 5.1.
Denote the crossing-word map by \(\Omega\), reserving \(\Psi\) for chapter 6. Assign to a loop its generic crossing word, interpreted in \(A_n\). A different generic approximation is homotopic to it, and the exhaustive event list shows that homotopic loops give equal elements of \(A_n\). Thus this defines a map \(\Omega:\pi_1(C_n(D^\circ),Q)\to A_n\). Concatenation concatenates crossing words; smoothing the join occurs in \(U\), so \(\Omega\) is a homomorphism.
The explicit loop \(\sigma_i\) has exactly one positive crossing, hence \(\Omega\Phi(s_i)=s_i\). The chamber decomposition proved that every loop is homotopic to the product of the generator loops in its word, hence \(\Phi\Omega\) is the identity too. We have proved
\[\boxed{A_n\ \cong\ \pi_1(C_n(D^\circ),Q).}\] □Corollary 5.8. Null-homotopic words
A word in the \(s_i^{\pm1}\) defines a null-homotopic configuration loop if and only if free insertions, cancellations, and Artin relation moves transform it into the empty word.
Worked example for Corollary 5.8
Counterexample: A surjective comparison map alone does not prove an isomorphism
Proof.
Injectivity of \(\Phi\) identifies null-homotopic words with the identity of \(A_n\). The finite relation-move criterion of Chapter 2 then gives the assertion.
□Sources and further reading
- Joan S. Birman, Braids, Links, and Mapping Class Groups, Annals of Mathematics Studies 82, Princeton University Press, 1974, Chapter 1. A standard reference for geometric braids and their presentation.
- Morris W. Hirsch, Differential Topology, Graduate Texts in Mathematics 33, Springer, 1976, Chapters 2–3. Source for the relative smooth-approximation and parametric-transversality theorems used here; the required finite jet consequence is derived in the prerequisite section.
- Alphabeta Math, Artin Presentation Completeness and Braid Combing, and Braids as Fundamental Groups of Configuration Spaces. Directly related companion expositions; the proof above makes its own general-position input and sign conventions explicit.
Worked examples
These calculations and proofs use this series’ chronological loop products, inverse-endpoint mapping classes, and counterclockwise positive point exchanges. Related statements link to their examples.
Worked example 1. Read a generic crossing using positions rather than labels
Applies to: Theorem 5.1, Definition 5.2, Lemma 5.3, Corollary 5.5.
Proof and calculation.
For \(n=3\), the single \(\sigma_1\) has real-coordinate difference \(x_1-x_2=-\tfrac12\cos\pi t\), with its only zero at \(t=1/2\) and derivative \(\pi/2\ne0\). At that time \(y_1=-1/4<y_2=1/4\), so the recorded crossing is positive \(s_1\). Tracing \(\sigma_1\) followed by \(\sigma_2\) gives the word \(s_1s_2\): just before the second crossing the left-to-right labels are \((2,1,3)\), so the exchanging original labels are 1 and 3, but their current positions are 2 and 3. The subscript is therefore 2, not an original-label index. Corners at the time join lie in the chamber of distinct real coordinates; reparametrizing each factor to be constant near its endpoints smooths the join without changing either transverse crossing. Thus the crossing reconstruction sends this actual loop to its chronological word, and the generator map sends that word back to this loop.
Worked example 2. A disjoint-exchange homotopy with no collisions
Applies to: Lemma 5.4, Lemma 5.7.
Proof and calculation.
For \(n=4\), the first and third exchanges have centers \(-2/5,2/5\) and radius \(1/5\). Choose disjoint supporting disks of radius \(1/4\); their centers are \(4/5\) apart. Performing the two rotations with independent parameters \((u,v)\in[0,1]^2\) keeps the two points in each disk separated by \(2/5\), and keeps the two pairs in disjoint disks. Thus it defines a continuous square of configurations. The two routes \((0,0)\to(1,0)\to(1,1)\) and \((0,0)\to(0,1)\to(1,1)\) have the same endpoints and are homotopic through this square. Their crossing words are \(s_1s_3\) and \(s_3s_1\), explicitly realizing distant commutation and its invariance under a disjoint crossing-time interchange.
Worked example 3. An adjacent relation and a null word with an explicit certificate
Applies to: Theorem 5.1, Lemma 5.4, Lemma 5.7, Corollary 5.8.
Proof and calculation.
Write \(a=s_1,b=s_2\) and take \(n=3\). Put \(\epsilon=1/100\), fix imaginary heights \((y_A,y_B,y_C)=(-\epsilon,0,\epsilon)\), and let the real-coordinate vector move in the plane \(x_A+x_B+x_C=0\) inside the disk of radius \(\epsilon\). Each point then has modulus at most \(\sqrt2\epsilon<1\), and distinct heights exclude collisions. Choose endpoints \((-\epsilon/2,0,\epsilon/2)\) and its negative, and take the two semicircles between them in that plane. Each crosses the three distinct equality lines once, with no endpoint crossing. Straight interpolation between the routes remains in the real-coordinate disk and keeps the same endpoints, so it is a genuine configuration homotopy. The two routes around the intersection of the real-coordinate walls reverse the order \(A,B,C\) by pair crossings \(AB,AC,BC\) and \(BC,AC,AB\). Every sign is positive because the strand initially to the left has smaller imaginary height. The routes are homotopic in the full little real-difference disk: collisions are impossible there because the three imaginary heights remain distinct. Connecting their ends through the convex distinct-real-coordinate chambers gives based loops with words \(aba\) and \(bab\).
Consequently the loop for \(w=aba b^{-1}a^{-1}b^{-1}\) is null: use this square homotopy to replace the first route by the second, then cancel each return passage with its reverse using \(H(t,s)=\gamma((1-s)r(t))\) from chapter 3. Algebraically the same certificate is \(aba b^{-1}a^{-1}b^{-1}\to bab b^{-1}a^{-1}b^{-1}\to\varepsilon\). This gives an explicit instance of both directions of the null-word criterion, with an actual homotopy and actual relation moves rather than an appeal to a word-solving algorithm.
Worked example 4. A mixed-sign triple event and the signs that cannot occur
Applies to: Lemma 5.6, Lemma 5.7.
Proof and calculation.
Take the height order \(y_A<y_C<y_B\), for example heights \(-2\eta,2\eta,0\) for labels \(A,B,C\), with \(\eta>0\). Then \((p,q,r)=(+,+,-)\). The two local routes therefore record \(aba^{-1}\) and \(b^{-1}ab\). Multiplying \(aba=bab\) on the left by \(b^{-1}\) and on the right by \(a^{-1}\) gives \(b^{-1}ab=aba^{-1}\), proving their equality without inventing a new relation. The imaginary heights ensure no point collision during the real-coordinate perturbation. In contrast \((p,q,r)=(+,-,+)\) would require \(y_B>y_A\), \(y_C<y_A\), and \(y_C>y_B\), a contradiction. Thus not every independent choice of crossing signs describes a possible triple event.
Worked example 5. Concrete transversality, jets, and an ordinary fold
Applies to: Lemma 5.6.
Proof and calculation.
For the real wall \(\{0\}\subset\mathbb R\), the family \(f_\epsilon(t)=t^2+\epsilon\) is transverse for every \(\epsilon\ne0\): if \(\epsilon>0\) it never reaches the wall, and if \(\epsilon<0\) its two roots \(\pm\sqrt{-\epsilon}\) have derivatives \(\pm2\sqrt{-\epsilon}\ne0\). At \(\epsilon=0\) the derivative vanishes at the root. This checks the definition and shows the exceptional parameter directly. On a local homotopy square with coordinates \((s,t)\), put \(f=t^2-s\). The wall \(f=0\) is the parabola \(s=t^2\); the derivatives of \((f,f_t)\) form \(\begin{pmatrix}-1&2t\\0&2\end{pmatrix}\), determinant \(-2\). For \(s<0\) there is no crossing and for \(s>0\) two crossings of opposite time-derivative signs. Keeping the two imaginary heights unequal makes this a collision-free local birth/cancellation of inverse crossings.
Proof checks for the prose jet and approximation tools
A 2-jet at \((0,0)\) is realized by the polynomial
\[P=a+bs+ct+(d/2)s^2+est+(f/2)t^2.\]The six values \((P,P_s,P_t,P_{ss},P_{st},P_{tt})\) there are exactly \((a,b,c,d,e,f)\). Thus varying these coefficients has identity derivative onto jet coordinates, a concrete submersion. Two independent wall equations \((s,t)=0\) have rank 2 and one isolated solution on a square; three equations \((s,t,\epsilon)=0\) have none when \(\epsilon\ne0\), illustrating the codimension distinction. A smooth approximation to \(|t|\) is \(\sqrt{t^2+\epsilon^2}\), with error at most \(\epsilon\), since its difference from \(|t|\) is \(\epsilon^2/(\sqrt{t^2+\epsilon^2}+|t|)\le\epsilon\). Finally, if every pair of moving points and every boundary distance is at least \(\delta>0\), perturbing each coordinate by less than \(\delta/4\) leaves pair distances greater than \(\delta/2\) and boundary distances greater than \(3\delta/4\), also throughout straight interpolation. These explicit examples check how the tools are used; they do not replace the general relative approximation and transversality theorems imported in the original proof.
Worked example 6. A spatial circle is not a Reidemeister-I event
Applies to: Lemma 5.3, Lemma 5.7, and the prose exclusion of self-strand Reidemeister-I moves.
Proof and calculation.
For \(n=3\), move only the first point by \(z_1(t)=-1/2+\epsilon(e^{2\pi it}-1)\), with \(\epsilon=1/100\), and keep \(z_2=0,z_3=1/2\). The moving point stays within \(2\epsilon\) of \(-1/2\), so its modulus is at most \(13/25<1\), and its distances from the other marks are at least \(12/25\) and \(49/50\). Thus this is a genuine collision-free based loop. Its real coordinate is at most \(-1/2\), so there are no projected crossings with another strand and its word is empty. The homotopy \(z_1(t,s)=-1/2+(1-s)\epsilon(e^{2\pi it}-1)\) obeys the same bounds and fixes the endpoints, proving it null directly.
The space-time trace is a small helix, not a projected curl: the graph \((\operatorname{Re}z_1(t),t)\) cannot intersect itself at different times, since equality of its second coordinates forces the times equal. Its time coordinate is strictly increasing even where the real-coordinate derivative vanishes. Therefore a turn in the spatial motion does not supply the local time reversal or same-strand self-crossing needed for a Reidemeister-I move. This explicitly tests the graph-over-time restriction used in the homotopy-event proof.
Counterexamples and distinctions
Each example refutes the stated stronger claim while retaining the hypotheses explicitly checked in its proof.
Counterexample 1. A surjective comparison map alone does not prove an isomorphism
Related statements: Corollary 5.5, Corollary 5.8.
Claim refuted. Surjectivity on generators, or a matching permutation, proves that a comparison map is injective.
Proof.
The homomorphism \(A_3\to S_3\), \(a\mapsto(12),b\mapsto(23)\), is onto: the two transpositions generate \(S_3\), and both triple products equal \((13)\), so the relation holds. But \(a^2\) maps to 1 and is nontrivial because its well-defined exponent sum is 2. Thus the surjectivity statement has a genuinely separate injectivity requirement. For the Artin comparison, it is the analysis of all homotopy events and the inverse crossing-word map, not merely generation, that supplies it.
Counterexample 2. A nongeneric motion need not have isolated transverse crossings
Related statements: Definition 5.2, Lemma 5.6.
Claim refuted. Every continuous collision-free braid can be read immediately as a finite sequence of transverse projected crossings.
Proof.
Reparametrize \(\sigma_1\) by a continuous nondecreasing surjection \(\lambda:[0,1]\to[0,1]\) which equals \(1/2\) throughout \([1/3,2/3]\), linear on the other two intervals. The configuration remains collision free because it visits the same distinct configurations as \(\sigma_1\). Yet the moving points have equal real coordinates at every time of that whole interval, not at isolated crossing times. The homotopy \(\sigma_1((1-s)\lambda(t)+st)\) is endpoint fixed and removes the pause. Thus a generic representative exists here by an explicit deformation, but the initial continuous representative does not satisfy the hypotheses of Definition 5.2. Reading its continuum of projected equalities as infinitely many braid generators would be incorrect.
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