In the paper ‘From Curves to Currents’, the authors gave a definition of essential crossing [Reference Martínez-Granado and ThurstonMGT21, Def. 2.6] that is too permissive. The main results remain true (with natural interpretation), and only one small part of the proof is affected.
For the issue with the definition, on the pair of pants P, consider the curve
$\gamma $
shown solid below, with marked crossing x.
If we take the natural generators
$a,b$
of
$\pi _1(P,*)$
shown dashed, then
$\gamma $
is in the free homotopy class of
$[ab]$
. To see that the x fits [Reference Martínez-Granado and ThurstonMGT21, Def. 2.6], note it divides
$\gamma $
into two subpaths, representing
$p = bab$
and
$q = B$
(adopting the convention that capital letters mean inverse, and connecting x to
$*$
by the short path). Since
$pq = ba$
and
$qp = ab$
, the corresponding lifts of
$\gamma $
to
$\widetilde { P}$
cross at infinity since their axes
$\overrightarrow {ab}$
and
$\overrightarrow {ba}$
cross. But x is not an essential crossing in any reasonable sense: it can clearly be isotoped away, and if we take lengths with respect to any Riemannian metric,
$\ell ([p Q]) = \ell ([ab^3])> \ell ([ab]) = \ell ([pq])$
, and not the other way around.
As a result, [Reference Martínez-Granado and ThurstonMGT21, Lemma 2.8] is wrong. Instead, we essentially take the conclusion of that lemma as a definition.
Definition 1. A representative
$\gamma $
of a multi-curve C is minimal if it has only transverse self-intersections and a minimum number of them. A crossing that appears in some minimal representative is said to be essential and induces a reduction relation
$C {\mathrel {\searrow }} C_1$
and
$C {\mathrel {\searrow }} C_2$
for the two different ways of smoothing the crossing.
‘Some’ above can be replaced by ‘every’:
Lemma 2. If there is a crossing x in some minimal representative
$\gamma $
of C, then for any other representative
${\gamma }'$
, there is a corresponding crossing
${x}'$
giving the same smoothings.
Proof. If
$\gamma $
is connected, by a result of Hass and Scott [Reference Hass and ScottHS94],
$\gamma '$
can be turned into the minimal form
$\gamma $
using Reidemeister I, II and III moves, with the Reidemeister I and II moves being used only in the forward (simplifying) direction. Since we know that
$\gamma $
has a crossing of the desired type, we can trace the crossings backwards through these moves: there is a bijection between the crossings before and after a Reidemeister III move that does not change the homotopy types of curves achievable by a single smoothing, and we can ignore the additional crossings created by backwards Reidemeister I and II moves.
For multi-curves, Hass and Scott show that we need to add one more move, swapping the position of two components that are homotopically powers of the same primitive curve
$\delta $
[Reference Hass and ScottHS94, pp. 31–32]. Again, this operation induces a bijection between crossings that preserves the curves that result from smoothing.
To give a more workable and algebraic criterion for essential crossings, we adopt a group-theoretic point of view. For
$a,b \in \pi _1(S,*)$
, write
$[a]$
for the free homotopy class of curves represented by a; algebraically, this is the union of the conjugacy classes of a and of A. Write
$[a][b]$
for the multi-curve
$[a] \cup [b]$
. Say that a factorization
$[ab]$
reduces if some (equiv. every) minimal representative of
$[ab]$
has a crossing so that one smoothing gives
$[a][b]$
and the other smoothing gives
$[aB]$
. Similarly say that
$[a][b]$
reduces if some/every minimal representative has a crossing whose smoothings give
$[ab]$
and
$[aB]$
. (This is an abuse of notation since the reduction is not invariant under individual conjugation of a and b.)
For
$a,b \in \pi _1(S,*)$
, let
$a * b$
be a 4-valent graph mapped into S obtained by taking the union of the paths
$a,b$
joined at the basepoint
$*$
, and let
$[a * b]$
be its free homotopy class. Free homotopy means in particular invariance under simultaneous conjugation:
$[a * b] = [caC * cbC]$
. A taut representative of
$[a * b]$
is one with a minimum number of transverse crossings.
Finally, if we fix a hyperbolic structure on S, a nontrivial element
$a \in \pi _1(S,*)$
determines a geodesic
$\overrightarrow {a}$
in
$\widetilde {S}$
, the unique geodesic that stays a bounded distance from
$a^k \cdot \widetilde {*}$
, with endpoints
$a^+$
and
$a^-$
. Depending on the cyclic order of the endpoints up to orientation reversal, we say that
$\overrightarrow {a}$
and
$\overrightarrow {b}$
are
-
• parallel if they appear in order
$a^-,a^+,b^+,b^-$
; -
• anti-parallel if they appear in order
$a^-,a^+,b^-,b^+$
; and -
• cross if they appear in order
$a^-,b^-,a^+,b^+$
.
We can now reformulate essential crossings in the (interesting) first case of [Reference Martínez-Granado and ThurstonMGT21, Def. 2.6], when a and b are not common powers.
Proposition 3. Let
$a,b \in \pi _1(S,*)$
be nontrivial elements that are not common powers. Then in the following three trialities, the same possibility holds in each case.
-
1. (Reduction) One of the following factorizations reduces to the other two:
(a)
$[ab]$
; (b)
$[aB]$
; or (c)
$[a][b]$
. -
2. (Geometric) For every taut embedding of
$a * b$
, around the 4-valent vertex we see in cyclic order, up to orientation reversal -
3. (Algebraic) The geodesics
$\overrightarrow {a}$
and
$\overrightarrow {b}$
are(a) parallel; (b) anti-parallel; or (c) cross.
Proof. Since a and b are not common powers, the endpoints of
$\overrightarrow {a}$
and
$\overrightarrow {b}$
are all distinct from each other, so the algebraic triality is mutually exclusive and exhaustive. For the geometric triality, for any given embedding of
$a * b$
, one of the three given possibilities holds, but it is not a priori clear that the same possibility holds for all taut embeddings. For the reduction triality, the possibilities are mutually exclusive (we cannot have circular reductions since at each reduction, length with respect to any Riemannian metric strictly decreases), but now it is not clear that any of the three possibilities holds.
We start with the geometric triality. Consider any taut representative
$\rho $
of
$[a * b]$
in one of the cases (a), (b) or (c). Consider the (concrete) multi-curve
$\gamma $
given by replacing the
$4$
-valent vertex with a crossing. If we pick orientations on
$\gamma $
correctly, this is an embedding of
$[ab]$
,
$[aB]$
or
$[a][b]$
, respectively, with a distinguished crossing x.
We claim that
$\gamma $
is minimal. Otherwise, there is some reducing isotopy using RIII moves and some RI/RII reduction [Reference Hass and ScottHS94]. Any RIII move on
$\gamma $
can be copied with an isotopy of
$\rho $
, as can an RI or RII reduction of
$\gamma $
that does not involve x. An RI reduction at x is only possible if a or b is the identity, and an RII reduction involving x also reduces
$\rho $
, reducing the number of intersections of
$\rho $
by one. In any case, this contradicts minimality of
$\rho $
or nontriviality of
$a,b$
.
We have thus shown that one of
$[ab]$
,
$[aB]$
or
$[a][b]$
has a minimal representative
$\gamma $
with a crossing x that can be smoothed to give the other two multi-curves. Since we cannot have a circular chain of essential smoothings (for instance. because we get strict inequalities of hyperbolic length), it follows that the cases are exclusive and thus that every taut representative of
$[a * b]$
has the same cyclic order around the 4-valent vertex. Combined, this proves equivalence of the first two trialities.
For the remaining equivalence, suppose first we are in case (c) of the geometric triality. For the fixed hyperbolic structure, take
$\gamma $
to be the geodesic representative of the free homotopy class
$[a][b]$
, intersecting at the distinguished point
$x \in S$
, and let
$\overrightarrow {a}$
and
$\overrightarrow {b}$
be the respective lifts passing through some lift
$\widetilde {x}$
of x. Then
$\overrightarrow {a}$
and
$\overrightarrow {b}$
intersect only once and thus cross at infinity, so we are also in case (c) of the algebraic triality.
If we are in case (a) of the geometric triality, again take the geodesic representative
$\gamma $
of
$ab$
, with distinguished crossing x. For concreteness, if necessary, reverse the orientation of S so the branches appear in the order shown below. If we perform the disconnected (oriented) smoothing at x, we get two components
$\gamma _a \in [a]$
and
$\gamma _b \in [b]$
; these concrete curves are typically not minimal. However,
$\gamma _a$
and
$\gamma _b$
are regular isotopic to a +minimal curve [Reference ThurstonThu14, Lemma 5.5] (i.e., there is no RI move required in making them minimal). This implies that their lifts
$\widetilde {\gamma _a}$
and
$\widetilde {\gamma _b}$
to
$\widetilde {S}$
passing through a neighborhood of
$\widetilde {x}$
are embedded in the disk (see, for example, [Reference ThurstonThu14, Lemma 2.21]). Now consider the endpoints
$a^+, a^-$
of
$\widetilde {\gamma _a}$
. As an oriented curve,
$\widetilde {\gamma _a}$
is geodesic outside of neighborhoods of lifts of x, and inside those neighborhoods turns to the left. It then follows that
$(ab)^+, a^+, a^-, (ba)^-$
occur in that cyclic order on
$\partial \widetilde {S}$
, and similarly,
$(ab)^-, b^-,b^+,(ba)^+$
occur in that cyclic order:
It therefore follows that
$\overrightarrow {a}$
and
$\overrightarrow {b}$
are parallel, so we are in case (a) of the algebraic triality.
Case (b) follows in the same way, interchanging
$b \leftrightarrow B$
.
The case when a and b are common powers is handled by the following result.
Proposition 4. If
$a = c^k$
and
$b = c^\ell $
are common powers of a primitive element c with
$k,\ell \ge 1$
, then the factorization
$[ab]$
reduces.
Proof. See Hass and Scott [Reference Hass and ScottHS85, Lemma 1.12].
As a result of Proposition 3, [Reference Martínez-Granado and ThurstonMGT21, Def. 2.6] is correct when looking at crossing between two different components of a multi-curve. There is only one place in the proof of our main results where we smoothed a crossing of one component with itself – namely, step (iii
$'$
) of the proof of [Reference Martínez-Granado and ThurstonMGT21, Lemma 9.1(b)]. In the terminology there, we must verify that
$\overrightarrow {\alpha _1}$
and
$\overrightarrow {\beta _1}$
are parallel, whereas the written proof uses [Reference Martínez-Granado and ThurstonMGT21, Lemma 12.3] to prove that
$\overrightarrow {\gamma _1} = \overrightarrow {\alpha _1 \beta _1}$
and
$\overrightarrow {\gamma _2} = \overrightarrow {\beta _1 \alpha _1}$
cross, which, as explained above, is not sufficient. However, by [Reference Martínez-Granado and ThurstonMGT21, Lemma 12.2],
$\overrightarrow {\alpha _1}$
and
$\overrightarrow {\beta _1}$
both cross
$\overrightarrow {\delta }$
in the same sense, so they cannot be anti-parallel. To see that
$\overrightarrow {\alpha _1}$
and
$\overrightarrow {\beta _1}$
cannot cross (and so must be parallel), use the lemma below and the established fact that
$\overrightarrow {\gamma _1}$
and
$\overrightarrow {\gamma _2}$
cross.
Lemma 5. If a and b are two hyperbolic elements of
${\mathit {PSL}}(2,{\mathbb {R}})$
with crossing axes, then
$ab$
and
$ba$
are also hyperbolic with parallel axes.
Proof. We will prove that the endpoints of the various geodesics must be laid out as follows, up to orientation reversal:
By the North-South dynamics of a and b on the circle,
$ab$
maps the interval
$(a^+,b^+)$
strictly inside itself, which implies that it is hyperbolic with attracting endpoint
$(ab)^+$
in that interval; similarly,
$ab$
maps
$(a^-,b^-)$
to a strict superset of itself, so
$(ab)^- \in (a^-,b^-)$
. The actions of a and b interchange
$(ab)^- \leftrightarrow (ba)^-$
and
$(ab)^+ \leftrightarrow (ba)^+$
, forcing the configuration shown. (The endpoints
$(ab)^\pm ,(ba)^\pm $
are necessarily on the portions of the circle where a and b move in opposite directions.)
Remark 6. A very similar argument shows that if
$\overrightarrow {a}$
and
$\overrightarrow {b}$
are parallel, then
$\overrightarrow {ab}$
and
$\overrightarrow {ba}$
cross. So in the self-crossing case, the condition in [Reference Martínez-Granado and ThurstonMGT21, Def. 2.6] is necessary but not sufficient.
Competing interest
The authors have no competing interest to declare.
Financial support
The first author was supported by the Luxembourg National research Fund AFR/Bilateral-ReSurface 22/17145118. The second author was supported by the National Science Foundation under Grant Number DMS-2110143.
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