Phylogenetics is the study of how species are related through evolution. These relationships are traditionally represented using branching diagrams called phylogenetic trees. However, certain evolutionary processes, such as hybridization or horizontal gene transfer, cannot be represented by trees alone. Therefore, phylogenetic networks - which allow additional connections between branches - are used to capture more complex evolutionary histories.
As phylogenetic networks become more complex, it becomes harder to determine whether a particular network can be uniquely reconstructed from observed DNA sequence data. This leads to the concept of identifiability. A network is said to be identifiable if, in theory, it can be uniquely determined from the data it generates. If two different networks produce the same data under a given evolutionary model, they are not identifiable from that data. Identifiability is essential for developing reconstruction methods that aim to infer evolutionary relationships from DNA.
This thesis investigates a specific class of phylogenetic networks, known as trinets. A trinet is a small sub-network that describes the evolutionary relationship between just three species. Trinets are useful building blocks for understanding and reconstructing larger phylogenetic networks. The main question studied in this work is whether these trinets are identifiable. We investigate whether specific types of trinets can be distinguished from simpler tree structures, using a mathematical approach based on phylogenetic invariants. These are special algebraic expressions that relate to the probabilities of observing certain DNA patterns. By evaluating such invariants, we may be able to detect whether the data came from a network or from a tree.
This thesis provides new insights into the identifiability of trinets. It shows that a specific type of trinet can be distinguished from a simple tree using an invariant. This is an important step toward understanding how more complex evolutionary networks can be identified from biological data from genes.