Mathematics
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Item Relationship between the kauffman bracket polynomials of torus knots: (T3,3n+1) and (T3,3n+2)(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Dissanayake, A.A.A.W.N.; Almeida, S.V.A.Knot theory is a branch of topology that studies mathematical knots. New knot invariants were the foundation for the work of many mathematicians. A knot invariant is a property for a knot , where is the same for any projection of . A knot polynomial is one such knot invariant. Knot polynomials are polynomials that are assigned to knot projections based on the mathematical properties of the knots. This study is restricted to polynomials of torus knots, knots that lie on an unknotted torus, without crossing over or under themselves as they lie on the torus. Every torus knot is a (, )-torus knot, where and are two relatively prime integers that are represented by the symbol ,. Most of the research done on this particular area of Knot Theory, has focused on finding polynomial representations such as Kauffman Bracket polynomial, the Bracket polynomial for the (2, )-torus knot and polynomial representations such as Alexander polynomial, Conway polynomial and Jones polynomial for the (, )-torus knot. With the exception of the complete solution to the Alexander, Conway, and Jones polynomials of ,, the problem of determining the polynomial for , is almost solved. The study is an attempt to solve the computation problem for the Kaufmann Bracket Polynomial of ,. The work will provide a relationship between Kauffman Bracket polynomials of the torus knots’ , and , for .Item Maximal embedding genus of 3-edge connected harary graphs(Faculty of Science, University of Kelaniya Sri Lanka, 2023) Withanaarachchi, W.A.K.D.H.; Almeida, S.V.A.; Wijesiri, G.S.One of the most prominent problems of topological graph theory is to determine the type of surface a nonplanar graph can be embedded. Almost complete results have been obtained for 4-edge connected graphs. The methods that were used to obtain specific results (finding the maximum and minimum genus embedding) for 4-edge connected graphs do not generalise for 3-edge connected graphs. Graph embedding is an important representational technique that aims to maintain the structure of a graph while learning low-dimensional representations of its vertices. The aim of this research project was to study the embedding of 3-edge connected Harary graphs H3,n. Specifically to complete the problem of maximal embeddings of 3-edge connected Harary graphs. The result is proved using Jungerman’s study, which showed that for any graph, is upper-embeddable if and only if it has a spanning tree T such that has at most one component with an odd number of edges. More specifically, a spanning tree for each graph was observed by dividing all 3-edge connected Harary graphs into two groups: odd number of vertices and even number of vertices. The pattern of a set of deleting edges and corresponding spanning trees was generalised in both cases. It was proved that H3,n is upper-embeddable, and the maximum genus of H3,n is given by for each n, by analysing the odd components of the complement of the corresponding spanning trees.