When Thoughts Get Tangled: What Knot Theory Can Tell Us About the Mind

Have you ever worried about something, tried to stop worrying, and then started worrying about why you cannot stop?

We often describe experiences like this as “getting stuck in a loop” or having “tangled thoughts”. But what if these expressions could inspire a mathematical way of studying thoughts, emotions and behaviour? That question brought us together across mathematics and psychology. In our article, we explore an unexpected connection between mathematics and the human mind.

What does a mathematical knot have to do with a thought?

Knot theory studies closed loops in three-dimensional space. Imagine tying a knot in a piece of string and joining its ends. You can stretch, bend or twist the string, but you cannot undo a genuine knot without cutting it or allowing the string to pass through itself.

What matters is its structure, not its appearance. Two knots can look different yet mathematically equivalent. Conversely, two loops that look similar may have fundamentally different structures.

Psychology has long studied recurring processes: worry that triggers more worry, avoidance that brings temporary relief but maintains anxiety, or habits that repeatedly return us to familiar responses. We are not proposing that knot theory should replace these psychological explanations. Instead, we are asking whether the structure of such processes might reveal something that conventional measurements overlook.

From worrying to worrying about worrying

Consider someone feeling anxious about climate change. Their first thoughts may concern future disasters and uncertainty. This worry might initially feel useful, as though anticipating potential problems provides a sense of control. Then comes another layer: Why am I unable to stop thinking about this? How can I carry on with everyday life while feeling so worried?

They may avoid the news, try to suppress their thoughts, or seek reassurance. These responses can offer short-term relief, yet sometimes help maintain the cycle.

Psychology already explains many such mechanisms. Our contribution is to ask what happens when we represent the changing relationship among three dimensions: primary worry, worry about worrying (often called meta-worry), and efforts to control or avoid worry.

Imagine recording these three quantities repeatedly. At every moment, they determine a point in a three-dimensional mathematical space. As they change, the point moves, drawing a trajectory.

Under suitable assumptions, a recurrent trajectory can be represented as a closed curve. We can then investigate its topology: does the curve form a simple loop, or does it have a more intricate knot structure?

There is a crucial qualification. A recurring psychological cycle is not automatically a mathematical knot. The construction of a closed curve, and the interpretation of any resulting knot, require careful justification.

Same statistics, different shapes. This is where our theoretical example becomes interesting.

We construct two hypothetical trajectories involving worry, meta-wrry, and control responses. Their corresponding variables have the same meaning and standard deviations. These statistics cannot distinguish them.

Yet their trajectories differ topologically. One forms an unknotted loop. The other forms a trefoil, the simplest nontrivial knot.

The lesson is not that someone who experiences anxiety has a literal trefoil in their mind, or that a more complicated knot indicates more severe distress. Our examples are mathematical constructions, not clinical measurements. Rather, they show that two processes can share the same averages and variability while differing in how their components evolve together over time. How a process is organised may matter alongside how intensely it is experienced.

Could we find such structures in real psychological data?

That remains an open question.

Researchers can increasingly collect repeated measurements of emotions, thoughts and behaviour over days or weeks. Topological data analysis and dynamical systems theory already provide methods for investigating patterns in time-dependent data. Knot theory might add another perspective, but moving from an elegant mathematical example to reliable psychological insight will not be straightforward.

What counts as a meaningful recurrent trajectory? How should we close a curve when real observations do not return exactly to their starting point? Would its classification remain stable in the presence of noise or missing measurements? Would knot-based descriptions tell us anything beyond existing statistical and psychological methods?

These questions need to be answered empirically. The framework may prove useful in some settings but not others.

One particularly intriguing possibility concerns psychological change. Imagine that someone reports feeling less anxious after treatment. That matters. But has the organisation of the worry–response cycle changed as well? Could similar reductions in symptom intensity conceal different underlying dynamics?

We are not claiming that knots can diagnose mental health conditions, predict relapse or guide treatment. Such applications require rigorous validation. At present, the framework offers a way to formulate new research questions.

An invitation to think differently

Mathematics and psychology are not as distant as they might first appear. Nearly a century ago, psychologist Kurt Lewin drew on topological ideas to describe people in relation to their goals and environments. Our work revisits this interdisciplinary spirit, focusing specifically on the organisation of psychological trajectories.

Sometimes, understanding a problem means looking beyond its individual parts and asking how those parts are connected, how they move together and what makes a pattern persist.

Perhaps studying the shape of a psychological process will one day tell us something new about change. Connecting mathematical modeling with pervasive contemporary challenges such as navigating climate anxiety may offer a conceptual lens that could eventually help inform long-term public mental health discussions.

For now, Knotted Minds is a beginning—an invitation to mathematicians and psychologists to explore what happens when two seemingly distant disciplines start asking questions together.