Gokuls Hunters Michaels

Stanford University

# Almost-Nash Sequential Bargaining

Dharan  Guru  Sun  s

###### Abstract

In a 2017 paper, later presented at the Web and Internet Economics conference,
titled “Sequential Deliberation for Social Choice" [4], the authors propose a
mechanism in which a series of agents $N$, are tasked to negotiate over a set
of decisions $S$. Building on assumptions of Nash Bargaining and assuming the
decision space follows the median graph, the authors constructed a robust
algorithm which approximates the decision which minimizes the social cost to
the entire population. In this paper, we give a brief overview of the
background theory which this paper builds upon from foundational work from
Nash, and social choice results which hold true in Condorcet mechanisms.
Following this analysis, we consider the stability of the results in the paper
with different deviations from Nash equilibrium. These deviations could be
pessimal, in the context of unequal bargaining power (say in a labor market)
or constructive, as in the context of opinion dynamics. Our analysis is
observatory, in the context of simulations, and we hope to formalize the
results of these simulations to get an understanding of more general
properties in spaces beyond our simulation.

##  1 Literature Review

###  1.1 Introduction

Consider an environment $\mathcal{E}=(N,S)$ in which there is a population of
participants (individual decision making agents), $N$, and a set $S$,
corresponding to the set of decisions / outcomes available to them. The goal
of the mechanism designer is to pick a socially desirable outcome, $o\in S$,
which leads to an equilibrium corresponding to consensus over the agents.

Typically in these kinds of settings, one could imagine polling each agent for
their preferences over $S$ and aggregating these preferences to pick such an
alternative. Unfortunately, this "enumeration" technique is not always
feasible as the space $S$ may not even be completely known to the mechanism
designer. Additionally, as the size and complexity of these spaces grow,
simply working with enumerations is tough [7].

The motivation of the mechanism designer could very well be choosing an option
which minimizes the social cost of an alternative. Here we adapt much of the
structure proposed in [4]. Assume each agent $i\in N$ has a "bliss point",
$p_{i}$, which is their most preferred element in $S$. If the designer is
minimizing the social cost by picking an alternative, $a$, they are picking
the quantity which minimizes

$a^{*}:=\operatorname*{arg\,min}_{a\in S}\sum_{u\in
N}d(p_{u},a):=\operatorname*{arg\,min}_{a\in S}SC(a)$

where $d(x,y)$ to refer to the disutility of an alternative $y$ to another
alternative $x$. Notice that the notion/function representing $d(\cdot,\cdot)$
is common to the agents. For any $x,y,z\in S$, then, it is clear that we would
want $d$ to obey $d(x,z)\leq d(x,y)+d(y,z)$, to codify rationality amongst the
agents.

So, in such a model, the mechanism designer would want to choose an outcome
$\hat{a}$ which has $SC(\hat{a})\approx SC(a^{*})$. We adopt the convention of
the authors by utilizing distortion as a metric to analyze this closeness:

$\mathcal{D}(\hat{a}):=\text{Distortion}(\hat{a})=\frac{SC(\hat{a})}{SC(a^{*})}$

and since $a^{*}$ is a minimizer of $SC(\cdot)$, $\forall a\in S$, we have
$\mathcal{D}(a)\geq 1$.

###  1.2 Introducing the Model

Rather than restricting agents to pick and order alternatives on the ballot,
we investigate the implications of the deliberative forum proposed in [4]. The
concept of deliberation is that individuals can communicate with one another
in a negotiation process, in which they generate new alternatives. For
example, rather than some students ranking a set of particular grading schemes
and aggregating their choices, we consider the model where the students
actually negotiate and come up with alternatives, relative to some fixed
alternative. In this analysis, only pairwise deliberation is considered; that
is, not more than two individuals are deliberating simultaneously.

####  1.2.1 A Brief History: Nash Bargaining

More formally, deliberation closely resembles the canonical model for two-
player negotiation [8], originally formulated by John Nash [9]. Two-player
bargaining is formulated as a tuple ($\mathcal{F},t$), where a pair of players
attempt to pick an outcome from a feasible set $\mathcal{F}$, and if they are
unable to come to a conclusion, then the point which they default to is $t$ –
or the threat/disagreement point. Nash proved that a game in which
$\mathcal{F}$ is a convex set with the requirements of:

  * •

Pareto optimality: The outcome is such that neither agent can improve without
hurting the other.

  * •

Symmetry between agents $u,v$: Given that $t_{u}=t_{v}$ If
$f^{*}=(f^{*}_{1},f^{*}_{2})\in\mathcal{F}$ is selected, then it should be the
case that $f^{*}_{1}=f^{*}_{2}$

  * •

Scale Covariance: An affine transformation of either player’s utility function
does not change the outcome.

  * •

Independence of Irrelevant Alternatives: Eliminating an alternative should not
affect the decision, that is if there is a subset of the feasible set in which
the same allocation is available, the agent should stay consistent.

corresponds directly to maximizing the Nash product. That is, given a threat
$t$, $o^{*}$ is defined as the $o\in S$ which maximizes

$(d(p_{u},t)-d(p_{u},o))\times(d(p_{v},t)-d(p_{v},o)).$ (1)

A mechanism designer can simulate this process amongst two agents with a
repeated game. All that needs to be done is to introduce a decay parameter
$\delta^{t}$, and ask the agents to play a repeated game in which the agents
go back and forth making offers – if an offer is not accepted, the agent is
forced to accept the threat at that particular round $t_{i}$. If the
probability of the game continuing diminishes at a rate proportional to
$\delta$, such a game has a sub-game perfect Nash equilibrium corresponding
directly to the product, assuming agents are sufficiently patient.

We note that as the requirement over these axioms dissolves, this game may
have many other equilibrium. A perfect example is in the case where
$\mathcal{F}=[t,z]$, (ex. courtesy of the Wikipedia article). Consider the
strategy/allocation of $(x_{1},x_{2})$ such that $x_{1}+x_{2}=z$. Here, if
either player attempts to increase their consumption, this is outside the
feasible solutions, and clearly decreasing can never help either. This
realization motivated us to consider settings in which perfect play was not
guaranteed, or other external forces (social dynamics) may incentivize players
to play off the standard equilibrium path. Numerous behavioral studies, for
example [13], indicate that the assumptions of the model were not always
satisfied, and bargaining of individuals did not necessarily correspond to the
Nash model.

In fact, there are other axioms and solution concepts for cooperative
bargaining which have risen to cater different needs and assumptions.
Loosening the constraint of independent and irrelevant alternatives (IIA)
leads to the Kalai-Smorodinsky bargaining solution [6]. One could imagine a
setting in which IIA would not hold; consider players prefer preferences over
feasible allocations $p_{1}:A\succ B\succ C$ and $p_{2}:C\succ B\succ A$. In a
bargaining context with all three allocation points, $B$ seems like a "fair"
allocation outcome as both players compromise their top choice to reach their
middle preference. However, in a subset allocation of just $\\{A,B\\}$, we see
that $B$ is still available to both players, but is just as "attractive" as
$A$ – so it need not be the case that IIA holds, but in Nash’s model it must
be the case that we pick the same allocation from both sets, given we do not
pick $C$.

Given that this setting also is implicitly a bargaining problem over $n$
agents, one can also consider a simultaneous bargaining model. The most
straightforward technique, relative to the prior work discussed, is to split
players into coaltions, and find a way of "aggregating" their preferences [2].
In this paper, the authors split the players into partitions and add an axiom
(representation of homogenous coalition), treating the partition,
collectively, as a single agent. Interestingly, the paper shows motivation of
the "joint-bargaining paradox", indicating via counterexample that it is
possibly unprofitable for a given set of agents to come together and form a
larger coalition. At first this may seem entirely surprising, but is similar
to the logic which suggests that mergers in a Cournot game may strictly
decrease the profit of the agents in the merger, while increasing the profits
of the others [12].

####  1.2.2 Random Dictatorship: A Reference Model

Perhaps the most immediate alternative model to consider is that of random
dictatorship, as suggested by Fain in a presentation of this paper [3]. The
algorithm asks the mechanism designer to randomly pick an agent, $i\in N$, and
ask the agent their bliss point, $p_{i}$. The mechanism designer then imposes
$p_{i}$ over all the agents.

First, see that this model is Pareto efficient – this follows immediately from
the fact that if there were another allocation, then it would strictly hurt
agent $i$. A fairly robust claim is that this model gives an expected
distortion factor of at most $2$. This follows immediately from the triangle
inequality, and is a proof from the authors [4]:

$\displaystyle\mathbb{E}_{S}[SC(x)]$
$\displaystyle=\frac{1}{n}\sum_{v=1}^{|N|}\sum_{u=1}^{|N|}d(p_{u},p_{v})$
$\displaystyle\leq\frac{1}{n}\sum_{v=1}^{|N|}\sum_{u=1}^{|N|}d(p_{u},a^{*})+d(a^{*},p_{v})=2SC(a^{*}).$

The authors also provide a lower bound, which is also of $2$, and finds a
tight bound of the method. But intuitively, this algorithm fares poorly when
there is a clear compromise which exists between the agents, as it forces an
asymmetric choice by the lucky one who was selected.

####  1.2.3 Iterative Compromises & Sequential Deliberation

The authors chose to develop a very generalizable and intuitive framework,
which built upon many of the ideas of Nash bargaining in a setting with many
agents. Having a simple framework was an important consideration, as it
allowed for the feasibility of practical implementation. They utilized two-
person bargaining as an operation, to iteratively move towards some larger
unified alternative.

At each round $i\in\\{0,\cdots,T\\}$, the authors randomly choose a pair of
agents $u,v\in N$ uniformly, and ask them to debate over some disagreement
alternative, $a^{t}$. They output the alternative selected at the conclusion
of their bargaining, and if they are unable to reach a conclusion, the
disagreement alternative, $a^{t}$ is chosen as the outcome. Then, for the next
round, the disagreement alternative/threat becomes $a^{t+1}\leftarrow o^{t}$,
where $o^{t}$ was the outcome of the agents’ deliberation. After $T$ rounds,
$a^{T}$ is output.

Notice that the authors repeatedly are iterating a bargaining game between the
agents, and so they assumed that the choice chosen by the agents corresponded
directly to the Nash bargaining solution.

###  1.3 Analysis of the Model

In order to measure the convergence to some particular distortion, the authors
categorized certain types of alternative spaces, $S$, and analyzed how well
the framework held up in these particular settings. All types of spaces
assumed the existence of some metric, $d$.

####  1.3.1 The Median Graph

The first model chosen by the authors to represent the decision space was the
median graph. Formally, a median graph satisfies the property that any three
nodes $u,v,w\in S$ have a unique intersection point amongst the shortest paths
of the three possible pairs $\\{(u,v),(u,w),(v,w)\\}$ of vertices. Some
examples of median graphs include hypergraphs, trees, and a line – some of
which we model in our forthcoming simulations.

Having no prior experience in social choice research, we were particularly
interested in the choice of median graphs to model decision spaces, especially
as the results in the paper indicated that the decision space played a
significant role in the outcomes of the distortions. A class of preferences
are single-peaked [1], if they satisfy:

  * •

Each agent has a peak. That is, a most preferred outcome in the set of
feasible alternatives.

  * •

Each agent prefers outcomes further from their peak less than those closer to
their peak.

When preferences satisfy these criteria, the simple truthful mechanism is to
select median preferred quantity, by the median voter theorem [1]. The
Condorcet winner, $w$, when chosen, satisfies the fact that when all agents
vote over any election with $w$ and some other $s\in S$, $w$ would win the
plurality vote. The Condorcet criterion is satisfied for a particular
mechanism if it always picks the Condorcet winner – in the example regarding
the median voter theorem, the theorem more generally states that any voting
mechanism satisfying the criterion picks the $a\in S$ "nearest" to the median
voter in $N$.

Notice, it need not be the case that for any arbitrary space, a Condorcet
winner exists, as shown by example in [10]. Moreover, when there is a cycle
(or transitivity violation) in the social ordering, we say there exists a
Condorcet cycle. It can be shown trivially that when there exists a Condorcet
cycle, there is no Condorcet winner. Most of the foundational literature which
we have referred to thus far considers orderings on a line, which as we
mentioned, is indeed a median graph, but some more general connection is yet
to be made between these concepts. We bring forward the notion of the
Condorcet domain, or sets such that no plurality cycles exist. The work in
[11] brings together all these results by showing that every closed Condorcet
domain is representable on a median graph, and every median graph corresponds
to a closed Condorcet domain! Hence, the median graph is a good/reasonable
space for social choice operate in – and in his talk regarding this paper,
Fain mentions that median graphs are the largest such graphs where the
guarantee of the existence of a Condorcet winner is true [3].

The median graph also has a beautiful connection with the Nash barganing
problem discussed earlier. Let $u,v\in N$, with respective bliss points
$p_{u},p_{v}\in S$ and a particular alternative $a\in S$, which they
deliberate over. In Lemma 1 [4], the authors prove that any median graph
$G=(S,E)$ satisfies the property that the median of $(p_{u},p_{v},a)$ is
exactly the maximizer of the Nash product, and an equilibrium of the Nash
Bargaining problem.

####  1.3.2 Distortion of the Algorithm

The authors moved forward by finding bounds for distortion in this model. They
find a mapping between any median graph $G$ into an isometric embedding
$\phi(G)$ and were able to conclude that as the number of iterations increase,
the distortion approaches 1.208, exponentially quickly.

The authors first prove a lemma that the uniqueness of the median of three
points $t,u,v\in S$ is preserved by their isometric embeddings
$\phi(t),\phi(u),\phi(v)$. For an arbitrary median graph $G=(S,E)$, the
distortion of sequential deliberation on $G$ is at most that on $\phi(G)$.
This is proven by that, in both cases of sequential deliberation and
bargaining, the isometric embedding of the median $\phi(o^{t})$ would be
chosen. This lends to the fact the social cost is preserved in both $G$ and
$Q$. This holds with the generalized median as well, so the distortion of
sequential deliberation in $Q$ is an effective upper bound for deriving the
main theorem.

The central observation for the distortion upper bound in the median graph is
that sequential deliberation defines a Markov Chain on $Q$. As time goes to
infinity, the expected distortion can be analyzed by the states of the Markov
Chain’s stationary distribution. It remains to solve for the correct
transition probabilities. On a hypercube, this is made easy because the median
of three points is the dimension-wise majority. By inspecting the dimension-
wise bits of all agents, each dimension simply becomes a 2-state Markov Chain,
with transition probabilities in terms of $f_{k}$, the fraction of agents
whose bliss point’s kth bit is $1$. Thus, the stationary probabilties can be
derived in terms of $f_{k}$ as well. It remains to sum up the expected social
cost via linearity of expectation. The final distortion ratio can be shown to
be the fraction of a linear over quadratic polynomial, which can be upper
bounded by $1.208$.

This 2-state Markov Chain setup leads to a straightforward analysis of its
convergence rate. The probability the present two agents have the same value
in a dimension (i.e. the probability the 2-state chain "couples") decreases
exponentially with $T$, hence deriving an upper guarantee on $T$ for this
probability to become arbitrarily small. The final expression directly implies
the theorem.

The implications of this theorem are crucial to their argument. The authors
survey lower bounds of sequential deliberation for simpler classes. Mechanisms
for median graphs to choose outcomes amongst agents’ bliss points, $V_{N}$
produce a Distortion of at least 2. This is shown with the case of the k-star
graph, when each agent’s bliss point is a unique vertex on the periphery.

Similarly, mechanisms to choose either an outcome or a median of three points
must have distortion at least 1.316. The author show this by arguing that,
with high probability, the number of bit-wise ones among any three points in
$V_{N}$, then applying loose union bounds to bound the social cost of any
median of three points. This lower bound proves the superiority of this
framework of iterative sequential deliberation to one-shot deliberation
mechanisms.

A similar approach is taken to capture the generalization of sequential
deliberation, mechanisms constrained to pick outcomes on shortest paths
between pairs in $V_{N}$. The authors show in this case, Distortion must be at
least $9/8$, by arguing the number of ones that must exist on the shortest
path between two agents, and deriving from it the expected social cost. This
lower bound gives a sense how close the result is to the best pairwise Pareto-
efficient frameworks can be.

Apart from these comparisons, there are inherent properties of sequential
deliberation that capture its efficiency. First, the stationary distribution
the authors used to show the theorem is in fact unique to the Markov Chain, or
equivalently, that the Markov Chain is aperiodic and irreducible. Second, the
outcome chosen by sequential deliberation is ex-post Pareto-efficient on a
median graph. As the subsequent section elaborates further on, this means
there are no alternatives that all agents $v\in N$ all prefer instead. It is
remarkable how, in the context of the chosen model of sequential deliberation
as a complex Markov Chain, this property is still preserved.

####  1.3.3 Game Theoretic Properties of this Mechanism

The chosen mechanism has many desirable properties, which the authors point
out. The first of these properties is Pareto efficiency. On a median graph,
the outcome of the algorithm, $\hat{a}$, is ex-post Pareto efficient. That is,
there is no other $a$ where $d(a,v)\leq d(\hat{a},v)$ $\forall v\in N$,
(excluding the possibility of equality for all $v$). Intuitively, this is
clearly useful as there is no other clear option point where all the agents
can be made "better" from their bliss points simultaneously.

Another useful game theoretic property is that of sub-game perfect Nash
equlibrium (SPNE). Specifically, telling the truth for every agent is a SPNE.
Recall that a SPNE means means that every subgame has equilibrium behavior.
Here the authors structured their mechanism as a game.

In the first step of the game, as a game of incomplete information, nature
randomly picks two agents. Then, the two agents play the non-cooperative
bargaining game; the outcome chosen in the game becomes the new disagreement
alternative. The players have the ability to pick an arbitrary decision point
in the feasible set $S$, and if they fail to agree or settle, then the
previous disagreement alternative is set as the next decision point. The
authors once again isometrically embed any such median graph $G$ into the
hypercube, and use backward induction (a common technique to prove SPNE) in
the game over the graph. Each player’s utility corresponds to the objective of
minimizing distance from their bliss point to the outcome of the game.

####  1.3.4 Extension to General Metric Spaces

After extensive analysis in the case of median graphs, the authors zoom out
for general metric spaces. For general metric spaces, there is a tight and
pessimistic bound of 3 when the space of alternatives and bliss points lie in
some metric. The tightness is due to observing the worst case scenario for
Pareto efficient deliberation outcomes. A simple first step analysis of one
step of deliberation yields an upper bound of $3OPT$, where $OPT$ is the
minimum social cost of $a^{*}\in S$.

Another advantage of sequential deliberation is its advantage in terms of the
distribution of outcomes it produces. The authors show the second moment of
deliberation outcomes is bounded by a constant. By inequalities for the first
and second moment, we can bind the chance of observing an outcome $\gamma$
times the optimal social cost by $O(1/\gamma^{2})$. By contrast, random
dictatorship has an unbounded second moment. This is shown when considering a
graph of two nodes, with fraction $f$ agents on the first and $1-f$ on the
other. As $f\rightarrow 0$, the squared social cost goes to zero slower than
the optimal social cost, resulting in an unbounded second moment. The
implication of the second moment is that the deliberation outcome will be
robust to outlier agents, instead having a well-defined probability mass
around the central consensus.

This interpretation is further extended by comparing the two methods in cases
of nearly unanimous instances. As the Appendix shows, under the case of
$\epsilon$-unanimity, where all but a small fraction of agents have the same
bliss point, sequential deliberation outperforms random dictatorship on median
graphs with a Distortion of $1+\epsilon$ to $2-\epsilon$. This easily follows
from the linear over quadratic bound in terms of $f_{k}$ used for proving the
theorem summarized earlier under the isometric embedding of the graph. With
$\epsilon$-unanimity, the domain for $f_{k}$ is restricted to $[0,\epsilon]$
(given the optimal bliss point is at the $0$ vector. This improves the bound
to $1+\epsilon$.

The pairwise deliberation scheme is also the meaningful deliberation scheme,
particularly when expanded to the $N$-person deliberation scheme. The authors
inspect the case of the line metric. Any disagreement outcome $o$ will be the
final outcome due to agents’ inability to come to consensus on an alternative
from a Nash product of utility gain perspective. The authors wittily note how
that confirms our intuition of deliberation breaking down in large groups to
random outcomes.

Thus, even though the value of the pairwise deliberation scheme only comes
through in structured metric spaces, it exhibits, even on general metric
spaces, the same assumptions and aforementioned properties for optimal Nash
bargaining schemes schemes. The authors show a Nash Bargaining outcome would
lie on the shortest path between agents.

##  2 Simulations & Further Study

We build a general framework for evaluating the sequential deliberation model,
with which we aim to test the robustness of the model when opinion dynamics
come into play. This is partially motivated by a related open question in the
paper, which suggests an exploration of the improvement in convergence when an
agent’s bliss point shifts towards the other agent and the decision
alternative after each pairwise bargaining step. Primarily though, we study
how sensitive the model is to the requirement that each bargaining step
results in the Nash bargaining solution. To do so, we add a “selfishness”
parameter $\lambda_{a}$ to each agent $a\in A$. We then implement two
alternative bargaining functions in addition to the Nash bargaining function
outlined in Equation 1, Selfish Nash and Unselfish Nash.

In the Selfish Nash function, we randomly perturb the Nash bargaining outcome
in the direction of the most selfish agent, weighted by the difference in
selfishness between the pair of agents. This represents a bargaining paradigm
in which we assume an imbalance in bargaining power that can influence the
disagreement outcome away from the Nash equilibrium in a non-deterministic
way, which we assume to be quite common in real-world bargaining.

The Unselfish Nash bargaining function finds the disagreement outcome
according to the standard Nash bargaining function, but then has each agent
randomly shift its bliss point towards the other agent’s bliss point and the
disagreement alternative, weighted by $1/\lambda$. In this sense, the
willingness of each agent to shift its bliss point after the bargaining
process is inversely proportional to its selfishness. This reflects an
optimistic cooperative bargaining paradigm in which agents are particularly
compromising in that they reevaluate their utilities after bargaining.

###  2.1 Implementation Details

Our framework consists of a decision space $S$ modelled by points along a line
in the interval $[0,1)$, which is a simple example of a median graph. Let
$\mathcal{B}(u,v,a)$ be the Nash optimal output given by finding $s\in S$
maximizing Equation 1. Our disutility in this space is given by $d(u,v)=|u-v|$
for $u,v\in S$. We define 3 “types” or coalitions of bliss points in this
space by sampling each agent’s bliss point $b_{a}$ from one of 3 randomly
placed normal distributions in the space, each of standard deviation 0.05.
Mathematically, we have $\mu_{1},\mu_{2},\mu_{3}\sim U[S]$ and
$b_{a}\sim\mathcal{N}(\mu\sim U[\\{\mu_{1},\mu_{2},\mu_{3}\\}],0.05)$. We also
have $\lambda_{a}\sim\mathcal{N}(1,0.1)$. This setup allows for a large degree
of random variety across simulations while still ensuring that bliss points
are not spread out so uniformly that we end up with a degenerate version of
the problem in which all outcomes have roughly equal social cost. We outline
our bargain functions below, with simplifications made for the sake of
brevity.

Input : agents $u,v\in A$ with $\lambda_{u}>\lambda_{v}$, alternative $a\in S$

Find Nash bargaining output $o=\mathcal{B}(u,v,a)$

shiftDirection = $(b_{u}-o)/|b_{u}-o|$

weight = $\lambda_{u}-\lambda_{v}$

noise = $sample(U[0.9,1])$

o’ = o + weight $\times$ noise $\times$ shiftDirection

Output : shifted decision outcome $o^{\prime}$

Algorithm 1 Selfish Nash Bargaining

Input : agents $u,v$ , decision alternative $a$

Find Nash bargaining output $o=\mathcal{B}(u,v,a)$

for  _agent permutations $(x,y)\in\\{(u,v),(v,u)\\}$ _ do

destination = $((b_{y}-b_{x})+(a-b_{x}))/2$ shiftDirection =
destination/|destination| weight = $1/\lambda_{x}$

noise = $sample(U[0.9,1])$

$b_{x}$ = $b_{x}$ \+ weight $\times$ noise $\times$ shiftDirection

end for

Output : Decision outcome $o$

Algorithm 2 Unselfish Nash Bargaining

(a) Perfect Nash Bargaining

(b) Selfish Bargaining

(c) Unselfish Bargaining

Figure 1: Distribution of agent bliss points with resulting social choice for
all three bargaining functions on a single simulation, with common random
seed. For 1(c), we show the initial (blue) and posterior (red) distributions
after agent bliss points have shifted.

The rest of the sequential deliberation simulation algorithm follows Figure 1
in [4]. For our simulation, we set the number of alternatives and number of
agents as $|S|=50$ and $|A|=300$. We limit the number of deliberation steps to
10. We run 1000 simulations with each of the three bargaining paradigms.

###  2.2 Simulation Results and Discussion

Figures 1 and 2 show two examples of our simulation. In 1(b), we observe that
the selfish bargaining function arrives at a social choice that is much
further from the generalized median than in the perfect Nash scenario. This
effect was consistent across simulations, and reflects the social effect of
those with higher bargaining power influencing deliberation towards their own
bliss points and away from the Nash solution. In 1(c), note that the
distribution of bliss points after deliberation is clustered closer to the
generalized median, reflecting the intended effect of agents’ opinions
shifting towards one another through the deliberation process. This process is
stochastic and not necessarily symmetric about the generalized median, and as
a result, the optimal social choice can actually change over the course of
deliberation as shown in 2(c).

(a) Perfect Nash Bargaining

(b) Selfish Bargaining

(c) Unselfish Bargaining

Figure 2: Another distribution of agent bliss points with resulting social
choice for all three bargaining functions on a single simulation, with common
random seed. For 2(c), we show the initial (blue) and posterior distributions
(red) after agent bliss points have shifted.

####  2.2.1 Distortion

From [4], we know that the upper bound distortion for infinite-horizon
sequential deliberation with Nash bargaining is 1.208 and the lower bound for
pairwise Pareto-efficient deliberation schemes is 1.125. We use these as
baselines in Figure 3, where we show the distribution of final Distortions for
all 1000 simulations in each bargaining scheme. With a mean Distortion of
1.16, our simulated Nash bargaining scheme (Figure 3(a)) does indeed fall
within these bounds.

Interestingly, our selfish bargaining function (Figure 3(b)) - which violates
the assumption of arriving at a Nash-optimal alternative in each step - shows
a much longer tail in its distribution which results in a mean Distortion of
1.24. This violates the 1.208 ideal upper bound for the Nash-optimal
bargaining scheme and indicates that the Nash-optimality requirement is strict
in practice and that the efficacy of the sequential deliberation framework is
quite sensitive to the optimality of the bargaining function. Therefore, the
pairwise sequential deliberation scheme proposed in [4] is not necessarily
robust in situations where opinion dynamics are such that the bargaining
outcome in each step can deviate from the Nash bargaining solution.

In Figure 3(c), we note that the unselfish bargaining scheme results in a
distribution that is remarkably similar to the perfect Nash bargaining scheme.
This is likely due to the fact that in a simple median graph such as our
decision space and with only at most 20 agents sampled from over 300,
converging agent utilities after bargaining has too small an effect on the
global distribution of bliss points to significantly affect the Distortion.
Indeed, we found that the mean Distortion in this case is 1.17, slightly worse
than the Nash bargaining function. Note that one of the strengths of pairwise
deliberation is being able to arrive at a near-optimal outcome without
sampling a large proportion of agents from the population. This effect
counteracts the gain in convergence from agents’ utilities shifting, as few
bliss points will have shifted relative to the overall distribution of agent
bliss points. Additionally, the stochastic nature of this scheme shifts the
generalized median throughout the deliberation process, and we suspect that
this may have increased the mean Distortion. However, it is important to note
that this may not be true for metric spaces that require sampling a larger
proportion of the population, in which case the shifting utilities may
noticeably improve the final Distortion.

(a) Perfect Nash Bargaining

(b) Selfish Bargaining

(c) Unselfish Bargaining

Figure 3: Distribution of Distortion for final social choice after pairwise
sequential deliberation for each of the three bargaining functions. The
theoretical bounds reflect the lower bound of 1.125 and the upper bound of
1.208 proven for Nash bargaining in [4].

####  2.2.2 Convergence

Figure 4: Distribution of Distortion for final social choice after pairwise
sequential deliberation for each of the three bargaining functions. The
theoretical bounds reflect the lower bound of 1.125 and the upper bound of
1.208 proven for Nash bargaining in [4].

We also study the convergence behaviour of the framework. To do so, we plot
the mean Distortion across all 1000 simulations along with the first and third
quartile Distortions for each of the three bargaining schemes in Figure 4. The
rapid convergence of this framework is evident - with Nash or Unselfish
bargaining, the mean distortion falls below 1.2 in 2 steps. The third quartile
for these schemes decreases steadily as the number of deliberation steps
increases, with a final third quartile Distortion just over 1.21 at step 10.
Interestingly, the first quartile reveals that Nash bargaining tends to find
the optimal social choice slightly more often than Unselfish bargaining. In a
space of 300 agents, convergence to a nearly optimal outcome in such a small
number of pairwise bargaining steps is rather remarkable.

Selfish bargaining stabilizes within one step, with neither the mean nor the
quartiles showing significant improvement after the first step. This suggests
that there is a lower bound for the Distortion under the Selfish scheme. This
lower bound - around 1.22 in our case - is likely dictated by the variance in
selfishness across agents and the magnitude of the expected weight parameter
in the Selfish bargaining algorithm. We also note that Selfish bargaining has
the largest inter-quartile range, primarily because the Selfish scheme does
not result in any decrease of the third quartile Distortion as deliberation
progresses beyond step 1. This again indicates that the sequential
deliberation framework is significantly compromised without a Nash-optimal
bargaining result.

###  2.3 Conclusion & Future Works

The results presented in [4] are a promising technique for arriving at a
conclusion amongst a group of agents where the decision space is potentially
unenumerable. Especially, when comparing against any deterministic algorithm,
the distortion is almost always a significant improvement. However, we
strongly believe that this mechanism operates via strong repeated assumptions
of bargaining according to Nash bargaining, which may not always translate in
many environments.

In situations like this, mechanism designers often explain the importance of
experimentation to test the validity of these techniques; despite the
complexity of experimental design for such a mechanism, (hidden preferences,
defining the alternative space, etc.) we feel that formal experimental design
would be a highly interesting idea, beyond the experiments which have already
been conducted on Nash’s work. This is because, in models with nearly 20-30
iterations of an assumption can look very different from 1 iteration –
especially as agents see how games play out. repeated able to Throughout this
process of simulation,

####  2.3.1 Simulation Framework

Our simulation framework is general and easily extendable to other metric
spaces and bargaining functions. We would like to run similar experiments on
more complex metric spaces. We believe that the metric space is a delicate
component of the framework, much like the bargaining function, and hypothesize
that as the complexity of the graph grows, the results deviate further from
the optimum.

####  2.3.2 Orderings

Many times, an agent is interested in learning the ordering of the top-$k$
elements in a decision space. We were interested in learning how rankings
would operate in this space. For example, given $S$ lives in the discrete line
graph $[0,9]$ (that is, it contains exactly $10$ vertices, each connected to
the adjacent vertex). One could augment this decision space to $S^{\prime}$
with $10!$ states, each corresponding to a distinct ordering of the states in
$S$. Some sort of ordering can be introduced to this graph, whether it be
lexicographical [5] or transposition distance.

In the case of lexicographical distance, we believe that asking all agents to
pick a bliss point corresponding to their ordering is potentially equivalent
to running $k$ independent cycles of the mechanism, and for each iteration
$i$, picking the bliss point corresponding to their $i^{th}$ most preferred
outcome.

In future experimentation and exploration, we seek to investigate the validity
of these claims, and also explore how orderings can realistically be embedded
from an original graph $S$ to a new graph $S^{\prime}$, while maintaining the
original structure of the old graph.

## References

  * Black [1948] Duncan Black.  On the rationale of group decision-making.  _Journal of political economy_ , 56(1):23–34, 1948.
  * Chae and Heidhues [2001] Suchan Chae and Paul Heidhues.  Nash bargaining solution with coalitions and the joint bargaining paradox.  2001\.
  * Fain [2017] Brandon Fain.  Sequential deliberation for social choice, Dec 2017.
  * Fain et al. [2017] Brandon Fain, Ashish Goel, Kamesh Munagala, and Sukolsak Sakshuwong.  Sequential deliberation for social choice.  In _International Conference on Web and Internet Economics_ , pages 177–190. Springer, 2017.
  * Golenko-Ginzburg [1991] Dmitri Golenko-Ginzburg.  Metrics in the permutation space.  _Applied Mathematics Letters_ , 4(2):5–7, 1991\.
  * Kalai and Smorodinsky [1975] Ehud Kalai and Meir Smorodinsky.  Other solutions to nash’s bargaining problem.  _Econometrica: Journal of the Econometric Society_ , pages 513–518, 1975.
  * Lang et al. [2016] Jerome Lang, Lirong Xia, and Herve Moulin.  _Voting in Combinatorial Domains_ , pages 197–222.  Cambridge University Press, 2016.  10.1017/CBO9781107446984.010.
  * Narahari [2012] Y. Narahari.  Cooperative game theory: The two person bargaining problem.  Oct 2012.  URL http://lcm.csa.iisc.ernet.in/gametheory/ln/web-cp2-bargaining.pdf.
  * Nash Jr [1950] John F Nash Jr.  The bargaining problem.  _Econometrica: Journal of the econometric society_ , pages 155–162, 1950.
  * Olken [2014] Ben Olken.  Sometimes it gets complicated: Condorcet’s paradox and arrow’s impossibility theorem.  2014\.  URL https://ocw.mit.edu/courses/economics/14-75-political-economy-and-economic-development-fall-2012/lecture-notes/MIT14_75F12_Lec12.pdf.
  * Puppe and Slinko [2019] Clemens Puppe and Arkadii Slinko.  Condorcet domains, median graphs and the single-crossing property.  _Economic Theory_ , 67(1):285–318, 2019.
  * Salant et al. [1983] Stephen W Salant, Sheldon Switzer, and Robert J Reynolds.  Losses from horizontal merger: the effects of an exogenous change in industry structure on cournot-nash equilibrium.  _The Quarterly Journal of Economics_ , 98(2):185–199, 1983.
  * Schellenberg [1990] James A Schellenberg.  ’solving’the bargaining problem.  _Mid-American Review of Sociology_ , pages 77–88, 1990.


# Semi-isotopic knots

Fredric D. Ancel  Department of Mathematical Sciences, University of
Wisconsin-Milwaukee<EMAIL_ADDRESS>
###### Abstract.

A _knot_ is a possibly wild simple closed curve in $S^{3}$. A knot $J$ is
_semi-isotopic_ to a knot $K$ if there is an annulus $A$ in $S^{3}\times[0,1]$
such that $A\cap(S^{3}\times\\{0,1\\})=\partial
A=(J\times\\{0\\})\cup(K\times\\{1\\})$ and there is a homeomorphism
$e:S^{1}\times[0,1)\rightarrow A-(K\times\\{1\\})$ such that
$e(S^{1}\times\\{t\\})\subset S^{3}\times\\{t\\}$ for every $t\in[0,1)$.

######  Theorem.

Every knot is semi-isotopic to an unknot.

###### 2010 Mathematics Subject Classification:

54B17, 57K10, 57K30, 57M30, 57N37

##  1\. Introduction

We fix some notation and terminology. Let $I=[0,1]$. A _knot_ is the image of
an embedding $S^{1}\rightarrow S^{3}$. If the composition of this embedding
with some homeomorphism of $S^{3}$ is a piecewise linear embedding, then the
knot is tame. Otherwise, it is wild. An annulus is a space that is
homeomorphic to $S^{1}\times I$.

Knots J and K are _ambiently isotopic_ if there is a level-preserving
homeomorphism $h:S^{3}\times I\rightarrow S^{3}\times I$ such that
$h(x,0)=(x,0)$ and $h(J\times\\{1\\})=K\times\\{1\\}$. Note that for such an
$h$, $h(J\times\\{0\\})=J\times\\{0\\}$ and
$h(J\times\\{1\\})=K\times\\{1\\}$. Of course, classical knot theory is the
study of ambient isotopy classes of tame knots in $S^{3}$.

Knots $J$ and $K$ are _(non-ambiently) isotopic_ if there is a level-
preserving embedding $e:J\times I\rightarrow S^{3}\times I$ such that
$e(J\times\\{0\\})=J\times\\{0\\}$ and $e(J\times\\{1\\})=K\times\\{1\\}$.

Observe that every knot that pierces a tame disk is isotopic to an unknot. The
sequence of pictures in Figure 1 suggests a proof of this observation.

Figure 1. An isotopy of a knot to an unknot

The _Bing sling_ (Figure 2) [2] is a wild knot that pierces no disk. It is not
known whether the Bing sling is isotopic to an unknot.

Figure 2. The Bing sling

The following conjecture is well-known.

######  Conjecture.

Every knot is isotopic to an unknot.

A knot $J$ is _semi-isotopic_ to a knot $K$ if there is an annulus $A$ in
$S^{3}\times I$ such that $\partial A=(J\times\\{0\\})\cup(K\times\\{1\\})$
and there is a homeomorphism $e:S^{1}\times[0,1)\rightarrow
A-(K\times\\{1\\})$ such that $e(S^{1}\times\\{t\\})\subset
S^{3}\times\\{t\\}$ for every $t\in[0,1)$. Note that $e$ may not extend
continuously to homeomorphism from $S^{1}\times[0,1]$ onto $A$.

The main result of this paper is:

######  Theorem.

Every knot is semi-isotopic to an unknot.

Thus, the Bing sling is semi-isotopic to an unknot.

Knots $J$ and $K$ are _(topologically) concordant_ or _I-equivalent_ if there
is an annulus $A$ in $S^{3}\times I$ such that
$A\cap(S^{3}\times\\{0,1\\})=\partial A=(J\times\\{0\\})\cup(K\times\\{1\\})$.

Note that: _Isotopic_ $\Rightarrow$ _semi-isotopic_ $\Rightarrow$
_concordant_.

######  Example.

There is a (wild) two-component link in $S^{3}$ that is not concordant to any
PL link! [7]

Observe that Melikhov’s striking example shows that the method of proof of the
Theorem won’t extend to two-component links, and that the invariants which
Melikhov exploits won’t work for single component knots.

The Theorem is proved by applying the two main results of [1]. These results
are a reinterpretation and formalization of a technique introduced by the
topologist C. H. Giffen in the 1960s called _shift-spinning_. A consequence of
Giffen’s method is that the Mazur 4-manifold [6] has a disk pseudo-spine. (A
_pseudo-spine_ of a compact manifold $M$ is a compact subset $X$ of $int(M)$
such that $M-X$ is homeomorphic to $\partial M\times[0,1)$.) After R. D.
Edwards’ groundbreaking proof that the double suspension of the boundary of
the Mazur 4-manifold is homeomorphic to the 5-sphere in 1975, it was observed
that had the existence of a disk pseudo-spine in the Mazur 4-manifold been
observed before Edwards’ proof, then a proof of Edwards’ theorem using
previously known results could have been given. However, no one made the
connection between Giffen’s technique and the existence of a disk pseudo-spine
until after Edwards announced his proof. Shift-spinning, the existence of a
disk pseudo-spine in the Mazur 4-manifold and why this leads to the conclusion
that the double suspension of the boundary of the Mazur 4-manifold is
homeomorphic to the 5-sphere are clearly explained on pages 102-107 of [3]. In
[1] the main results are applied to show that 4-manifolds like the Mazur
manifold (constructed by attaching 2-handles to $B^{3}\times S^{1}$ have
identifiable pseudo-spines. In particular, all the 4-manifolds constructed by
attaching a single 2-handle to $B^{3}\times S^{1}$ along a degree one curve
have disk pseudo-spines, and therefore have the property that the double
suspensions of their boundaries are homeomorphic to $S^{5}$.

Other topologists have explored Giffen shift-spinning in various contexts.
(See pages 404-409 of [5] and pages 15-16 of [4].)

We gratefully acknowledge the artistic hand of Shayna Meyers in the creation
of the figures for this article. We also thank Chris Hruska for a useful
conversation.

##  2\. Mapping swirls

The central concept of [1] is the _mapping swirl_. We recapitulate its
definition and state the two main results of [1] in forms that are convenient
for proving the Theorem.

Let $X$ be a compact metric space. Identify the _cone_ on $X$, $CX$, and the
_suspension_ of $X$, $\Sigma X$, as the quotient spaces $CX=([0,\infty]\times
X)/(\\{\infty\\}\times X)$ and $\Sigma X=([-\infty,\infty]\times
X)/\\{\\{-\infty\\}\\}\times X,\\{\infty\\}\times X\\}$. Let $(t,x)\mapsto tx$
denote either of the quotient maps $[0,\infty]\times X\rightarrow CX$ or
$[-\infty,\infty]\times X\rightarrow\Sigma X$. For each $t\in[0,\infty]$ or
$[-\infty,\infty]$, let $tX$ denote the image of the set $\\{t\\}\times X$
under the appropriate quotient map. Thus, $\infty X$ denotes the _cone point_
of $CX$, and $(-\infty)X$ and $\infty X$ denote the _suspension points_ of
$\Sigma X$.

Let $f:X\rightarrow Y$ be a map between compact metric spaces. Observe that,
by exploiting the homeomorphism $x\mapsto(x,f(x))$ from $X$ to the graph of
$f$, the _mapping cylinder_ of $f$, $Cyl(f)$, can be identified with the
subset

$\\{(tx,f(x))\in CX\times Y:(t,x)\in[0,\infty)\times X\\}\cup({\infty X}\times
Y)$

of $CX\times Y$. Similarly, the _double mapping cylinder_ of $f$, $DblCyl(f)$,
which is obtained by identifying two copies of $Cyl(f)$ along their bases, can
be identified with the subset

$\\{(tx,f(x))\in\Sigma X\times Y:(t,x)\in(-\infty,\infty)\times
X\\}\cup(\\{(-\infty)X,\infty X\\}\times Y)$

of $\Sigma X\times Y$. For each $t\in(-\infty,\infty)$, the _t-level_ of
$DblCyl(f)$ is the set

$L(f,t)=(tX\times Y)\cap DblCyl(f)=\\{(tx,f(x)):x\in X\\}.$

Furthermore, if $t\in[0,\infty)$, then $L(f,t)$ is also called the _t-level_
of $Cyl(f)$. Note that for each $t\in(-\infty,\infty)$,
$x\mapsto(tx,f(x)):X\rightarrow L(f,t)$ is a homeomorphism. The
$\infty$_-level_ of $Cyl(f)$ is the set $L(f,\infty)=\\{\infty X\\}\times Y$.
The $(-\infty)$_-level_ and the $\infty$_-level_ of $DblCyl(f)$ are the sets
$L(f,-\infty)=\\{(-\infty)X\\}\times Y$ and $L(f,\infty)=\\{(\infty)X\\}\times
Y$.

Let $X$ be a compact metric space and let $f:X\rightarrow S^{1}$ be a map. The
_mapping swirl_ of $f$ is the subset

$Swl(f)=\\{(tx,e^{2\pi it}f(x))\in CX\times S^{1}:(t,x)\in[0,\infty)\times
X\\}\cup(\\{\infty X\\}\times S^{1})$

of $CX\times S^{1}$. The _double mapping swirl_ of $f$ is the subset
$DblSwl(f)=$

$\\{(tx,e^{2\pi it}f(x))\in\Sigma X\times S^{1}:(t,x)\in(-\infty,\infty)\times
X\\}\cup(\\{(-\infty)X,\infty X\\}\times S^{1})$

of $\Sigma X\times S^{1}$. For each $t\in(-\infty,\infty)$, the _t-level_ of
$DblSwl(f)$ is the set

$\mathcal{L}(f,t)=(tX\times S^{1})\cap DblSwl(f)=\\{(tx,e^{2\pi it}f(x)):x\in
X\\}.$

Furthermore, if $t\in[0,\infty)$, then $\mathcal{L}(f,t)$ is also called the
_t-level_ of $Swl(f)$. Note that for each $t\in(-\infty,\infty)$,
$x\mapsto(tx,e^{2\pi it}f(x)):X\rightarrow\mathcal{L}(f,t)$ is a
homeomorphism. The $\infty$_-level_ of $Swl(f)$ is the set
$\mathcal{L}(f,\infty)=\\{\infty X\\}\times S^{1}$. The $(-\infty)$_-level_
and $\infty$_-level_ of $DblSwl(f)$ are the sets
$\mathcal{L}(f,-\infty)=\\{(-\infty)X\\}\times S^{1}$ and
$\mathcal{L}(f,\infty)=\\{\infty X\\}\times S^{1}$. For each $x\in X$, the _x-
fiber_ of $Swl(f)$ is the set

$\mathcal{F}(f,x)=\\{(tx,e^{2\pi it}f(x)):t\in[0,\infty)\\},$

and the _x-fiber_ of $DblSwl(f)$ is the set

$\mathcal{DblF}(f,x)=\\{(tx,e^{2\pi it}f(x)):t\in(-\infty,\infty)\\}.$

We now state the two main results of [1].

######  Theorem 1.

If $X$ is a compact metric space and $f,g:X\rightarrow S^{1}$ are homotopic
maps, then there is a homeomorphism $\Omega:Swl(f)\rightarrow Swl(g)$ with the
following properties.
1) $\Omega$ is _fiber-preserving_ : for every $x\in X$,
$\Omega(\mathcal{F}(f,x))=\mathcal{F}(g,x)$.
2) $\Omega$ fixes the $\infty$-level: $\Omega|\mathcal{L}(f,\infty)=id.$

Compare Theorem 1 to the fact that, in general, there is no homeomorphism
between mapping cylinders of homotopic maps.

Note that Theorem 1 holds with $S^{1}$ replaced by any space homeomorphic to
$S^{1}$. Also note that the conclusions of the theorem imply that for any
subset $Y$ of $X$, $\Omega(Swl(f|Y))=Swl(g|Y)$,
$\Omega(\mathcal{L}(f|Y,0)=\mathcal{L}(g|Y,0)$ and
$\Omega(\mathcal{L}(f,\infty))=\mathcal{L}(g,\infty)$.

######  Theorem 2.

If $Y$ is a compact metric space, $n$ is a non-zero integer and $f:X\times
S^{1}\rightarrow S^{1}$ is the map satisfying $f(y,z)=z^{n}$, then there is a
homeomorphism $\Theta:Cyl(f)\rightarrow Swl(f)$ with the following properties.
1) $\Theta$ is level-preserving: for every $t\in[0,\infty]$,
$\Theta(L(f,t))=\mathcal{L}(f,t)$.
2) $\Omega$ fixes the 0- and $\infty$-levels: $\Theta|L(f,0)\cup
L(f,\infty)=id.$

Note that Theorem 2 holds for any map $f:X\rightarrow J$ for which there is a
commutative diagram:

${Y\times S^{1}}$${S^{1}}$${X}$${J}$$\scriptstyle{(y,z)\mapsto
z^{n}}$$\scriptstyle{homeomorphism}$$\scriptstyle{homeomorphism}$$\scriptstyle{f}$

We will need one other elementary fact:

######  Proposition 3.

If $X$ is a compact metric space, $f:X\rightarrow S^{1}$ a map and
$\lambda:[0,1)\rightarrow[0,\infty)$ is a homeomorphism, then there is a
homeomorphism $\Lambda:X\times[0,1)\rightarrow Swl(f)-\mathcal{L}(f,\infty)$
such that for every $t\in[0,1)$,
$\Lambda(X\times\\{t\\}))=\mathcal{L}(f,\lambda(t))$.

Proofs of Theorems 1 and 2 are found in [1]. Because we have modified the
statements of these theorems for their use in this paper, we will provide
outlines of these proofs as well as a proof of Proposition 3 in section 4
below.

##  3\. The proof of the Theorem

Let $J$ be a knot. We will prove that $J$ is semi-isotopic to an unknot.

Step 1: There is an unknotted solid torus $T$ in $S^{3}$ such that $J\subset
int(T)$ and the inclusion $J\hookrightarrow T$ is a homotopy equivalence.

We can assume $J\subset\mathbb{R}^{3}=S^{3}-\\{\infty\\}$. Let $p$ and $q$ be
distinct points of $J$. Let $\mathcal{V}$ be an uncountable family of parallel
planes in $\mathbb{R}^{3}$ that separate $p$ from $q$. Since $J$ has a
countable dense subset, $J$ does not contain an uncountable pairwise disjoint
collection of non-empty open sets. Hence, there is a $V\in\mathcal{V}$ such
that $J\cap V$ contains no non-empty open subset of $J$. It follows that
$J\cap V$ is a totally disconnected subset of $V$. Let $J_{1}$ and $J_{2}$ be
arcs such that $J_{1}\cup J_{2}=J$ and $J_{1}\cap J_{2}=\partial
J_{1}=\partial J_{2}=\\{p,q\\}$. Then $J_{1}\cap V$ and $J_{2}\cap V$ are
disjoint compact totally disconnected subsets of V. It follows that there is a
disk $D$ in $V$ such that $J_{1}\cap V\subset int(D)$ and $(J_{2}\cap V)\cap
D=\emptyset$. To see this, let $D^{\prime}$ be a disk in $V$, let $A_{1}$ be a
subset of $int(D^{\prime})$ that is homeomorphic to $J_{1}\cap V$, and let
$A_{2}$ be a subset of $V-D^{\prime}$ that is homeomorphic to $J_{2}\cap V$.
Then according to Theorem 13.7 on pages 93-95 of [8], there is a homeomorphism
$\phi:V\rightarrow V$ such that $\phi(A_{i})=J_{i}\cap V$ for $i=1,2$. Simply
let $D=\phi(D^{\prime})$.

We can assume $D$ is a piecewise linear disk in $V$. Let $U$ be a regular
neighborhood of $\partial D$ in $S^{3}$ such that $U\cap V$ is a regular
neighborhood of $\partial D$ in $V$ and $U\cap J=\emptyset$, and let
$T=cl(S^{3}-U)$. Then $U$ and, hence, $T$ are unknotted solid tori in $S^{3}$,
and $J\subset int(T)$. Let $\bar{V}=V\cup\\{\infty\\}$, and let $E_{1}$ and
$E_{2}$ be the components of $cl(\bar{V}-U)$ such that $E_{1}\subset int(D)$
and $\infty\in E_{2}$. Then $E_{1}$ and $E_{2}$ are disjoint meridional disks
of $T$ such that $T\cap\bar{V}=E_{1}\cup E_{2}$, $J_{1}\cap\bar{V}\subset
int(E_{1})$ and $J_{2}\cap\bar{V}\subset int(E_{2})$. Thus, $J_{1}\cap
E_{2}=\emptyset=J_{2}\cap E_{1}$.

Clearly, there is a simple closed curve $K\subset int(T)$ such that the
inclusion $K\hookrightarrow T$ is a homotopy equivalence, $p$ and $q\in K$,
$K_{1}$ and $K_{2}$ are arcs such that $K_{1}\cup K_{2}=K$, $K_{1}\cap
K_{2}=\partial K_{1}=\partial K_{2}=\\{p,q\\}$, and $K_{1}\cap
E_{2}=\emptyset=K_{2}\cap E_{1}$. At this point we will stretch conventional
terminology slightly by saying that for subsets $Z\subset Y$ and $Z\subset
Y^{\prime}$ of a space $X$, the inclusions $Y\hookrightarrow X$ and
$Y^{\prime}\hookrightarrow X$ are _homotopic in_ $X$ _rel_ $Z$ if there is a
homotopy $\xi:Y\times I\rightarrow X$ such that $\xi_{0}=id_{Y}$,
$\xi_{1}:Y\rightarrow Y^{\prime}$ is a homeomorphism and $\xi_{t}|Z=id_{Z}$
for every $t\in I$. Since $T-E_{2}$ is contractible, then the inclusions
$J_{1}\hookrightarrow T-E_{2}$ and $K_{1}\hookrightarrow T-E_{2}$ are
homotopic rel $\\{p,q\\}$. Similarly, since $T-E_{1}$ is contractible, the
inclusions $J_{2}\hookrightarrow T-E_{1}$ and $K_{2}\hookrightarrow T-E_{1}$
are homotopic rel $\\{p,q\\}$. Therefore, the inclusions $J\hookrightarrow T$
and $K\hookrightarrow T$ are homotopic. It follows that the inclusion
$J\hookrightarrow T$ is a homotopy equivalence.

Step 2: Consider a homeomorphism $\psi:B^{2}\times S^{1}\rightarrow T$, let
$o\in int(B^{2})$ and let $K=\psi(\\{o\\}\times S^{1})$. Define the
homeomorphism $\psi_{o}:S^{1}\rightarrow K$ by $\psi_{o}(z)=\psi(o,z)$. Define
the map $\tau:B^{2}\times S^{1}\rightarrow S^{1}$ by $\tau(y,z)=z$, and define
the map $\pi:T\rightarrow K$ by $\pi=\psi_{o}\circ\tau\circ\psi^{-1}$. Then we
have a commutative diagram in which the vertical arrows are homeomorphisms:

${B^{2}\times
S^{1}}$${S^{1}}$${T}$${K}$$\scriptstyle{\tau}$$\scriptstyle{\psi}$$\scriptstyle{\psi_{o}}$$\scriptstyle{\pi}$

We can now invoke Theorem 2 to obtain a level-preserving homeomorphism
$\Theta:Cyl(\pi)\rightarrow Swl(\pi)$ that fixes the $0$\- and
$\infty$-levels. Also observe that since $\tau:B^{2}\times S^{1}\rightarrow
S^{1}$ is a homotopy equivalence, then so is $\pi:T\rightarrow K$.

Step 3: Let $\lambda:I\rightarrow[0,\infty]$ be an order-preserving
homeomorphism. Clearly, there is an embedding $j:Cyl(\pi)\rightarrow
S^{3}\times I$ with the following properties.
1) $j$ maps $L(\pi,0)$ “identically” onto $T\times\\{0\\}$; i.e.,
$j(0x,\pi(x))=(x,0)$ for every $x\in T$.
2) For every $t\in I$, $j(L(\pi,\lambda(t))\subset S^{3}\times\\{t\\}$, and
$j(L(\pi,\lambda(t))$ is a copy of $T$ that is “squeezed” toward $K$.
3) j maps $L(\pi,\infty)$ “identically” onto $K\times\\{1\\}$; i.e., for every
$y\in K$, $j(\\{\infty T\\}\times\\{y\\})=\\{(y,1)\\}$.
Thus, $j\circ\Theta^{-1}:Swl(\pi)\rightarrow S^{3}\times I$ is an embedding
with the following properties.
1) $j\circ\Theta^{-1}$ maps $\mathcal{L}(\pi,0)$ “identically” onto
$T\times\\{0\\}$; i.e., $j\circ\Theta^{-1}(0x,\pi(x))=(x,0)$ for every $x\in
T$.
2) For every $t\in I$, $j\circ\Theta^{-1}(\mathcal{L}(\pi,\lambda(t))\subset
S^{3}\times\\{t\\}$.
3) $j\circ\Theta^{-1}$ maps $\mathcal{L}(\pi,\infty)$ “identically” onto
$K\times\\{1\\}$; i.e., for every $y\in K$, $j\circ\Theta^{-1}({\infty
T}\times\\{y\\})=\\{(y,1)\\}$.

Step 4: Since $\pi|J\rightarrow K$ is the composition of the inclusion
$J\hookrightarrow T$ and $\pi:T\rightarrow K$, both of which are homotopy
equivalences, then $\pi|J\rightarrow K$ is a homotopy equivalence. Hence,
$\pi|J\rightarrow K$ is homotopic to an homeomorphism $\chi:J\rightarrow K$.
Therefore, Theorem 1 provides a $0$\- and $\infty$-level preserving
homeomorphism $\omega:Swl(\pi|J)\rightarrow Swl(\chi)$. Thus,
$j\circ\Theta^{-1}:Swl(\chi)\rightarrow S^{3}\times I$ is an embedding that
maps $\mathcal{L}(\chi,0)$ into $S^{3}\times\\{0\\}$ and maps
$\mathcal{L}(\chi,1)$ onto $K\times\\{1\\}\subset S^{3}\times\\{1\\}$.

Step 5: Since $\chi:J\rightarrow K$ is a homeomorphism, then Theorem 2
provides a level-preserving homeomorphism from $Cyl(\chi)$ to $Swl(\chi)$.
Also, since $\chi:J\rightarrow K$ is a homeomorphism, then $Cyl(\chi)$ is an
annulus with boundary $L(\chi,0)\cup L(\chi,1)$. Therefore, $Swl(\chi)$ is an
annulus with boundary $\mathcal{L}(\chi,0)\cup\mathcal{L}(\chi,1)$. Since
$\omega:Swl(\pi|J)\rightarrow Swl(\chi)$ is a $0$\- and $\infty$-level
preserving homeomorphism, then if follows that $Swl(\pi|J)$ is an annulus with
boundary $\mathcal{L}(\pi|J,0)\cup\mathcal{L}(\pi|J,\infty)$. Let
$A=j\circ\Theta^{-1}(Swl(\pi|J))$. Then A is an annulus in $S^{3}\times I$
with boundary
$j\circ\Theta^{-1}(\mathcal{L}(\pi|J,0)\cup\mathcal{L}(\pi|J,\infty))=(J\times\\{0\\})\cup(K\times\\{1\\})$.

Step 6: Recall that for every $t\in I$,
$j\circ\Theta^{-1}(\mathcal{L}(\pi|J,\lambda(t))\subset S^{3}\times\\{t\\}$.
Proposition 3 implies that there is a homeomorphism
$\Lambda:J\times[0,1)\rightarrow Swl(\pi|J)-\mathcal{L}(\pi|J,\infty)$ such
that $\Lambda(J\times\\{t\\})=\mathcal{L}(\pi|J,\lambda(t))$ for every
$t\in[0,1)$. Therefore, $j\circ\Theta^{-1}\circ\Lambda:J\times[0,1)\rightarrow
A-(K\times\\{1\\})$ is a homeomorphism such that
$j\circ\Theta^{-1}\circ\Lambda(J\times\\{t\\})\subset S^{3}\times\\{t\\}$ for
every $t\in[0,1)$. $\blacksquare$

##  4\. Proofs of Theorems 1 and 2 and Proposition 3

Proof of Theorem 1. Let $X$ be a compact metric space and let
$f,g:X\rightarrow S^{1}$ be homotopic maps.

Step 1: There is a homeomorphism $\Phi:(\Sigma X)\times
S^{1}\rightarrow(\Sigma X)\times S^{1}$ that carries $DblSwl(f)$ onto
$DblSwl(g)$.

For each $x\in X$, observe that $\mathcal{DblF}(f,x)\cup\mathcal{DblF}(g,x)$
is a _double helix_ in the _cylinder_ $(\mathbb{R}x)\times S^{1}\subset(\Sigma
X)\times S^{1}$. We will construct a map $\sigma:X\rightarrow\mathbb{R}$ such
that for each $x\in X$, a first-coordinate shift of $(\mathbb{R}x)\times
S^{1}$ through a distance of $-\sigma(x)$ slides $\mathcal{DblF}(f,x)$ onto
$\mathcal{DblF}(g,x)$. (See Figure 3.)

Figure 3. Double helix with vertical shift $-\sigma(x)$

Begin the construction of $\sigma$ by choosing a homotopy $h:X\times
I\rightarrow S^{1}$ such that $h_{0}=f$ and $h_{1}=g$. Using complex division
in $S^{1}$, define $k:X\times I\rightarrow S^{1}$ by $k(x,t)=h(x,t)/h(x,0)$.
Then for every $x\in X$, $k(x,0)=1$ and $k(x,1)f(x)=g(x)$. Let
$e:\mathbb{R}\rightarrow S^{1}$ be the exponential covering map $e(t)=e^{2\pi
it}$. The homotopy $k:X\times I\rightarrow S^{1}$ lifts to a homotopy
$\widetilde{k}:X\times I\rightarrow\mathbb{R}$ such that
$e\circ\widetilde{k}=k$ and $\widetilde{k}(x,0)=0$ for every $x\in X$. Define
$\sigma:X\rightarrow\mathbb{R}$ by $\sigma(x)=\widetilde{k}(x,1)$. Therefore,
for every $x\in X$,

$e^{2\pi
i\sigma(x)}f(x)=e\circ\sigma(x)f(x)=e\circ\widetilde{k}(x,1)f(x)=k(x,1)f(x)=g(x)$.

Since X is compact, there is a $b>0$ such that $\sigma(X)\subset(-b,b)$.

Define $\Phi:(\Sigma X)\times S^{1}\rightarrow(\Sigma X)\times S^{1}$ by

$\left\\{\begin{array}[]{l}\Phi(tx,z)=((t-\sigma(x))x,z)\text{ for
}(t,x)\in\mathbb{R}\times X\text{ and }z\in S^{1},\text{ and}\\\ \Phi=id\text{
on }\\{(-\infty)X,\infty X\\}\times S^{1}.\end{array}\right.$

$\Phi$ is the first-coordinate shift of $(\mathbb{R}x)\times S^{1}$ through a
distance of $-\sigma(x)$ mentioned above. It remains to show that $\Phi$ is a
homeomorphism of $(\Sigma X)\times S^{1}$ which slides $\mathcal{DblF}(f,x)$
onto $\mathcal{DblF}(g,x)$ for each $x\in X$.

$\Phi$ is continuous at points of $\\{(-\infty)X,\infty X\\}\times S^{1}$
because for every $x\in X$ and every $z\in S^{1}$,
$\Phi(([t,\infty]x)\times\\{z\\})\subset((t-b,\infty]x)\times\\{z\\}$ and
$\Phi(([-\infty,t]x)\times\\{z\\})\subset([-\infty,t+b)x)\times\\{z\\}$.

$\Phi$ is a homeomorphism because its inverse $\overline{\Phi}$ can be defined
explicitly by the equations

$\left\\{\begin{array}[]{l}\overline{\Phi}(tx,z)=((t+\sigma(x))x,z)\text{ for
}(t,x)\in\mathbb{R}\times X\text{ and }z\in S^{1},\text{ and}\\\
\overline{\Phi}=id\text{ on }\\{(-\infty)X,\infty X\\}\times
S^{1}.\end{array}\right.$

The verification that $\overline{\Phi}\circ\Phi=id=\Phi\circ\overline{\Phi}$
is straightforward.

For each $x\in X$, a typical point of $\mathcal{DblF}(f,x)$ has the form
$(tx,e^{2\pi it}f(x))$, and

$\begin{array}[]{l}\Phi((tx,e^{2\pi it}f(x))=((t-\sigma(x))x,e^{2\pi
it}f(x))=\\\ ((t-\sigma(x))x,e^{2\pi i(t-\sigma(x))}e^{2\pi
i\sigma(x)}f(x))=\\\ ((t-\sigma(x))x,e^{2\pi
i(t-\sigma(x))}g(x))\in\mathcal{DblF}(g,x).\end{array}$

Hence, $\Phi(\mathcal{DblF}(f,x))\subset\mathcal{DblF}(g,x)$. A similar
calculation shows

$\overline{\Phi}(\mathcal{DblF}(g,x))\subset\mathcal{DblF}(f,x)$

for every $x\in X$. Thus,

$\mathcal{DblF}(g,x)\subset\Phi(\mathcal{DblF}(f,x))$

for every $x\in X$. We conclude that

$\Phi(\mathcal{DblF}(f,x))=\mathcal{DblF}(g,x)$

for every $x\in X$.

We have shown that $\Phi:(\Sigma X)\times S^{1}\rightarrow(\Sigma X)\times
S^{1}$ is a homeomorphism with the property that
$\Phi|DblSwl(f):DblSwl(f)\rightarrow DblSwl(g)$ is a fiber-preserving
homeomorphism that fixes the $(-\infty)$\- and $\infty$-levels. Unfortunately,
we can’t conclude that $\Phi(Swl(f))$ and $Swl(g)$ are equal subsets of
$DblSwl(g)$. This completes Step 1.

Step 2: There is a homeomorphism $\Psi:(\Sigma X)\times
S^{1}\rightarrow(\Sigma X)\times S^{1}$ that maps $DblSwl(g)$ onto itself and
_twists_ $\Phi(Swl(f)$ onto $Swl(g)$.

Observe that for every $x\in X$, $\Phi(\mathcal{F}(f,x))$ and
$\mathcal{F}(g,x)$ are contained in the _helix_ $\mathcal{DblF}(g,x)$ which
lies in the _cylinder_ $(\mathbb{R}x)\times S^{1}$. For each $x\in X$, $\Psi$
will map this cylinder to itself with a _screw motion_ that preserves
$\mathcal{DblF}(g,x)$ and _twists_ $\Phi(\mathcal{F}(f,x))$ onto
$\mathcal{F}(g,x)$.

Recall that $\sigma(X)\subset(-b,b)$. Hence, there is clearly a map
$\tau:\mathbb{R}\times X\rightarrow\mathbb{R}$ such that for each $x\in X$,
$\tau|\mathbb{R}\times\\{x\\}:\mathbb{R}\times\\{x\\}\rightarrow\mathbb{R}$ is
a homeomorphism such that $\tau(-\sigma(x))=0$ and $\tau(t,x)=t$ for
$t\in(-\infty,-b]\cup[b,\infty)$.

Define $\Psi:(\Sigma X)\times S^{1}\rightarrow(\Sigma X)\times S^{1}$ by

$\left\\{\begin{array}[]{l}\Psi(tx,z)=((\tau(t,x)x,e^{2\pi
i(\tau(t,x)-t)}z)\text{ for }(t,x)\in\mathbb{R}\times X\text{ and }z\in
S^{1},\text{ and}\\\ \Psi=id\text{ on }\\{(-\infty)X,\infty X\\}\times
S^{1}.\end{array}\right.$

$\Psi$ is continuous at points of $\\{(-\infty)X,\infty X\\}\times S^{1}$
because $\Psi=id$ on $\\{tx:|t|\geq b\text{ and }x\in X\\}\times S^{1}$.

$\Psi$ is a homeomorphism because its inverse $\overline{\Psi}$ can be defined
explicitly as follows. First observe that there is a map
$\overline{\tau}:\mathbb{R}\times X\rightarrow\mathbb{R}$ so that for every
$x\in X$, $t\mapsto\overline{\tau}(t,x):\mathbb{R}\rightarrow\mathbb{R}$ is
the inverse of the homeomorphism
$t\mapsto\tau(t,x):\mathbb{R}\rightarrow\mathbb{R}$. Now define
$\overline{\Psi}:(\Sigma X)\times S^{1}\rightarrow(\Sigma X)\times S^{1}$ by

$\left\\{\begin{array}[]{l}\overline{\Psi}(tx,z)=((\overline{\tau}(t,x)x,e^{2\pi
i(\overline{\tau}(t,x)-t)}z)\text{ for }(t,x)\in\mathbb{R}\times X\text{ and
}z\in S^{1},\text{ and}\\\ \overline{\Psi}=id\text{ on }\\{(-\infty)X,\infty
X\\}\times S^{1}.\end{array}\right.$

The verification that $\overline{\Psi}\circ\Psi=id=\Psi\circ\overline{\Psi}$
is straightforward.

Let $x\in X$. To prove that $\Psi(\mathcal{DblF}(g,x))=\mathcal{DblF}(g,x)$,
note that a typical point of $\mathcal{DblF}(g,x)$ has the form $(tx,e^{2\pi
it}g(x))$, and

$\begin{array}[]{l}\Psi((tx,e^{2\pi it}g(x))=(\tau(t,x)x,e^{2\pi
i(\tau(t,x)-t)}e^{2\pi it}g(x))=\\\ (\tau(t,x)x,e^{2\pi
i\tau(t,x)}g(x))\in\mathcal{DblF}(g,x).\end{array}$

Therefore, $\Psi(\mathcal{DblF}(g,x))\subset\mathcal{DblF}(g,x)$. A similar
calculation shows

$\overline{\Psi}(\mathcal{DblF}(g,x))\subset\mathcal{DblF}(g,x).$

Hence,

$\mathcal{DblF}(g,x)\subset\Psi(\mathcal{DblF}(g,x)).$

We conclude that

$\Psi(\mathcal{DblF}(g,x))=\mathcal{DblF}(g,x).$

We have shown that $\Psi:(\Sigma X)\times S^{1}\rightarrow(\Sigma X)\times
S^{1}$ is a homeomorphism with the property that
$\Psi|DblSwl(g):DblSwl(g)\rightarrow DblSwl(g)$ is a fiber-preserving
homeomorphism that fixes the $(-\infty)$\- and $\infty$-levels.

Again let $x\in X$. To prove that
$\Psi(\Phi(\mathcal{F}(f,x)))=\mathcal{F}(g,x)$, note that a typical point of
$\Phi(\mathcal{F}(f,x))$ has the form $(tx,e^{2\pi it}g(x)$ where
$t\geq-\sigma(x)$, and a typical point of $\mathcal{F}(g,x)$ has the form
$(tx,e^{2\pi it}g(x))$ where $t\geq 0$. Also note that $\tau$ maps
$[-\sigma(x),\infty)$ onto $[0,\infty)$. Since

$\begin{array}[]{l}\Psi((tx,e^{2\pi it}g(x))=(\tau(t,x)x,e^{2\pi
i(\tau(t,x)-t)}e^{2\pi it}g(x))=(\tau(t,x)x,e^{2\pi
i\tau(t,x)}g(x)),\end{array}$

then clearly $\Psi(\Phi(\mathcal{F}(f,x)))=\mathcal{F}(g,x)$.

It follows that $\Psi\circ\Phi(Swl(f))=Swl(g)$, completing Step 2.

We conclude that $\Psi\circ\Phi|Swl(f):Swl(f)\rightarrow Swl(g)$ is a fiber-
preserving homeomorphism that fixes the $\infty$-level. $\blacksquare$

Proof of Theorem 2. Let $Y$ be a compact metric space, $n$ a non-zero integer
and $f:Y\times S^{1}\rightarrow S^{1}$ the map satisfying $f(y,z)=z^{n}$. We
will construct a homeomorphism $\Theta:C(Y\times S^{1})\times S^{1}\rightarrow
C(Y\times S^{1})\times S^{1}$ that carries $Cyl(f)$ onto $Swl(f)$.

Define $\zeta:[0,\infty)\rightarrow S^{1}$ by $\zeta(t)=e^{-2\pi it/n}$, and
define $\Theta:C(Y\times S^{1})\times S^{1}\rightarrow C(Y\times S^{1})\times
S^{1}$ by

$\left\\{\begin{array}[]{l}\Theta(t(y,z),w)=(t(y,\zeta(t)z),w)\text{ for
}t\in[0,\infty),(y,z)\in Y\times S^{1}\text{ and }w\in S^{1},\text{ and}\\\
\Theta=id\text{ on }\\{\infty(Y\times S^{1})\\}\times
S^{1}.\end{array}\right.$

$\Theta$ is clearly level-preserving and fixes the $0$\- and $\infty$-levels.

We argue that $\Theta$ is continuous at points of $\\{\infty(Y\times
S^{1})\\}\times S^{1}$. For $a>0$, let
$V_{a}=\bigcup_{t\in(a,\infty]}t(Y\times S^{1})$; and for $w\in S^{1}$, let
$\mathcal{M}_{w}$ be a basis for the topology on $S^{1}$ at $w$. Then for each
$w\in S^{1}$, $\\{V_{a}\times M:a>0\text{ and }M\in\mathcal{M}_{w}\\}$ is a
basis for the topology on $C(Y\times S^{1})\times S^{1}$ at $(\infty(Y\times
S^{1}),w)$. Since for each $a>0$ and each $M\in\mathcal{M}_{w}$, $\Theta$ maps
$V_{a}\times M$ into itself, then $\Theta$ is continuous at $(\infty(Y\times
S^{1}),w)$.

$\Theta$ is a homeomorphism because its inverse $\overline{\Theta}$ can be
defined explicitly as follows. First define the map
$\overline{\zeta}:[0,\infty)\rightarrow S^{1}$ by $\overline{\zeta}(t)=e^{2\pi
it/n}$. Then define $\overline{\Theta}:C(Y\times S^{1})\times S^{1}\rightarrow
C(Y\times S^{1})\times S^{1}$ by

$\left\\{\begin{array}[]{l}\overline{\Theta}(t(y,z),w)=(t(y,\overline{\zeta}(t)z),w)\text{
for }t\in[0,\infty),(y,z)\in Y\times S^{1}\text{ and }w\in S^{1},\text{
and}\\\ \overline{\Theta}=id\text{ on }\\{\infty(Y\times S^{1})\\}\times
S^{1}\end{array}\right.$

The verification that
$\overline{\Theta}\circ\Theta=id=\Theta\circ\overline{\Theta}$ is
straightforward.

To prove that $\Theta(Cyl(f))=Swl(f)$, note that a typical point of $Cyl(f)$
has the form $(t(x,z),f(x,z))$, a typical point of $Swl(f)$ has the form
$(t(x,z),e^{2\pi it}f(x,z))$, and

$\begin{array}[]{l}\Theta(t(x,z),f(x,z))=(t(x,\zeta(t)z),f(x,z))=\\\
(t(x,\zeta(t)z),z^{n})=(t(x,\zeta(t)z),(\zeta(t))^{-n}(\zeta(t)z)^{n})=\\\
(t(x,\zeta(t)z),e^{2\pi it}(\zeta(t)z)^{n})=(t(x,\zeta(t)z),e^{2\pi
it}f(x,\zeta(t)z))\in Swl(f).\end{array}$

Therefore, $\Theta(Cyl(f))\subset Swl(f)$. A similar calculation shows
$\overline{\Theta}(Swl(f))\subset Cyl(f)$. Hence,
$Swl(f)\subset\Theta(Cyl(f))$. We conclude that $\Theta(Cyl(f))=Swl(f)$.

It follows that $\Theta|Cyl(f):Cyl(f)\rightarrow Swl(f)$ is a level-preserving
homeomorphism that fixes the $0$\- and $\infty$-levels. $\blacksquare$

Proof of Proposition 3. A compact metric space $X$, a map $f:X\rightarrow
S^{1}$ and a homeomorphism $\lambda:[0,1)\rightarrow[0,\infty)$ are given.
Define $\Lambda:X\times[0,1)\rightarrow Swl(f)-\mathcal{L}(f,\infty)$ by
$\Lambda(x,t)=(\lambda(t)x,e^{2\pi i\lambda(t)}f(x))$. We show that $\Lambda$
is a homeomorphism by exhibiting its inverse. Let
$q:X\times[0,\infty]\rightarrow CX$ denote the quotient map $q(x,t)=tx$. Then
$q|X\times[0,\infty):X\times[0,\infty)\rightarrow CX-\\{\infty X\\}$ is a
homeomorphism; let $r:CX-\\{\infty X\\}\rightarrow X\times[0,\infty)$ denote
its inverse. Let $p:(CX-\\{\infty X\\})\times S^{1}\rightarrow CX-\\{\infty
X\\}$ denote projection. Define $\tau:(CX-\\{\infty X\\})\times
S^{1}\rightarrow X\times[0,1)$ by $\tau=(id_{X}\times\lambda^{-1})\circ r\circ
p$. It is easily verified that $\tau\circ\Lambda=id_{X\times[0,1)}$ and
$\Lambda\circ(\tau|(Swl(f)-\mathcal{L}(f,\infty)))=id_{Swl(f)-\mathcal{L}(f,\infty)}$.
Hence, $\tau|(Swl(f)-\mathcal{L}(f,\infty))$ is the inverse of $\Lambda$.
$\blacksquare$

## References

  * [1] Fredric D Ancel and Craig R Guilbault  “Mapping swirls and pseudo-spines of compact 4-manifolds”  In _Topology and its Applications_ 71.3 Elsevier, 1996, pp. 277–293
  * [2] R H Bing  “A simple closed curve that pierces no disk”  In _J. Math. Pures Appl.(9)_ 35, 1956, pp. 337–343
  * [3] Robert J Daverman  “Decompositions of manifolds”  Academic Press, 1986
  * [4] Jonathan Arthur Hillman  “Algebraic invariants of links”  World Scientific, 2012
  * [5] Yukio Matsumoto  “Wild embeddings of piecewise linear manifolds in codimension two”  In _Geometric Topology_ Elsevier, 1979, pp. 393–428
  * [6] Barry Mazur  “A note on some contractible 4-manifolds”  In _Annals of Mathematics_ JSTOR, 1961, pp. 221–228
  * [7] Sergey A Melikhov  “Topological isotopy and Cochran’s derived invariants”  In _arXiv preprint arXiv:2011.01409_ , 2020
  * [8] Edwin E Moise  “Geometric topology in dimensions 2 and 3”  Springer Science & Business Media, 2013