http://hdl.handle.net/123456789/1326 Thesis submitted by MP -2015 batch as part of their course 2023-05-14T16:20:57Z http://hdl.handle.net/123456789/2127 Title: Skeleton ideals of graphs and their associated invariants Authors: Lather, Gargi Abstract: Parking functions are multifaceted objects with applications in many areas of math- ematics. For a graph G on n + 1 vertices with a designated vertex as root, Postnikov and Shapiro associated a G-parking function ideal in the standard polynomial ring over a field with variables corresponding to the non-root vertices of G. The standard monomials of this ideal, given by the G-parking functions, are in bijective correspondence with the spanning tree of G. Recently, Dochtermann introduced and investigated the k-skeleton ideals, which are certain parameter-dependent subideals of the G-parking function ideal. We have studied the homological and combinatorial properties of these k-skeleton ideals. We have calculated all the multigraded Betti numbers of k-skeleton ideals of complete graphs. We give alternative proof for calculating the number of standard monomials of the k-skeleton ideal of complete multigraphs via Steck determinant evaluation. Dochtermann conjectured the existence of a bijective correspondence between the set of the spherical parking functions of the complete graph and the set of uprooted trees on the vertex set {1, 2, . . . , n}, preserving degree and surface inversions. We have proved this conjecture. Our proof involves the use of a modified version of the depth-first-search algorithm. We also give an extension of this map for the case of general simple graphs and show that this map is always an injection but not necessarily a surjection. For many classes of graphs, we explicitly describe the image of this extension map and compute the cardinality of the associated set of spherical parking functions. Dochtermann also conjectured that for a simple graph, the number of standard monomials of the 1-skeleton ideal is bounded below by the determinant of the reduced signless Laplacian of the graph. We extended this conjecture in a general framework of positive semidefinite matrices over nonnegative integers and obtained necessary and sufficient conditions for which the equality holds. 2022-05-01T00:00:00Z http://hdl.handle.net/123456789/1707 Title: Structural aspects of planar braid groups Authors: Nanda, Neha; Singh, Mahender Abstract: Artin braid groups are celebrated objects which appear in and affix several areas of mathemat- ics and theoretical physics. A geometric interpretation given by Artin in his pioneering work in the 1920s, which captures the behaviour of intertwined strings in the Euclidean 3-space, has led to a deeply rooted connection with links in the 3-space. Since then the theory has been ramified by topologists and algebraists both. This naturally leads to a question of how the strings would intertwine if considered on a plane, and how it can be signified algebraically. The thesis explores this direction and presents a detailed investigation of structural aspects of planar braid groups and their (higher genus) virtual analogues. Study of certain isotopy classes of a finite collection of immersed circles (called doodles on surfaces) without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory with the genus zero case corresponding to the classical knot theory. In the case of doodles on the 2-sphere, the role of groups is played by a class of right-angled Coxeter groups called twin groups. For the higher genus case in the virtual setting, the role of groups is played by a new class of groups called virtual twin groups. We give a topological description of virtual twin groups and establish Alexander and Markov theorems for oriented virtual doodles. This paves a way for constructing invariants for doodles on surfaces. We investigate structural aspects of (pure) virtual twin groups in detail. More precisely, we obtain a presentation of the pure virtual twin group and deduce that it is an irreducible right-angled Artin group. We then prove that pure virtual twin groups can be written as iterated semidirect products of infinite rank free groups. Consequently, it follows that pure virtual twin groups have trivial centers, which confirms a well-known conjecture about triviality of centers of irreducible non-spherical Artin groups. We also compute the automorphism group of pure virtual twin groups in full generality and give applications to twisted conjugacy. We investigate the conjugacy problem in twin groups and derive a formula for the number of conjugacy classes of involutions, which, quite interestingly, is related to the well-known Fibonacci sequence. We also investigate z-classes in twin groups and derive a recursive formula for the number of z-classes of involutions. Finally, we determine automorphism groups of twin groups and give applications to twisted conjugacy 2021-07-28T00:00:00Z http://hdl.handle.net/123456789/1703 Title: Algebraic structures in knot theory Authors: Singh, Manpreet; Singh, Mahender Abstract: Knot theory is the study of embedded circles in the 3-sphere. A central problem in the subject is to develop computational invariants that can distinguish two knots. One such almost complete invariant that surfaced independently in the works of Matveev and Joyce in 1982 is what is called a link quandle, which is basically a minimal algebraic structure that encodes the three Reidemeister moves of planar diagrams of links in the 3-sphere. One of the fundamental results is that two non-split tame links have isomorphic link quandles if and only if there is a homeomorphism of the 3-sphere that maps one link onto the other, not necessarily preserving the orientations of the ambient space and that of links. Many classical topological, combinatorial and geometric knot invariants such as the knot group, the knot coloring, the Conway polynomial, the Alexander polynomial and the volume of the complement in the 3-sphere of a hyperbolic knot can be retrieved from the knot quandle. Thus, understanding of knot quandles is of fundamental importance for the classification problem for knots. The first and major component of the thesis is a fusion of ideas from combinatorial group theory into the theory of quandles. More precisely, we introduce residual finiteness and or- derability in quandles. One of our main results is that every link quandle is residually finite, a proof of which uses the idea of subgroup separability in fundamental groups of 3-manifolds. As immediate consequences of this result, it follows that the word problem is solvable for link quandles, and that every link admits a non-trivial coloring by a finite quandle. We also develop a general theory of orderability of quandles with a focus on link quandles and give some gen- eral constructions of orderable quandles. We prove that knot quandles of many fibered prime knots are right-orderable, whereas link quandles of many non-trivial torus links are not right- orderable. We prove that link quandles of certain non-trivial positive (or negative) links are not bi-orderable, which includes some alternating knots of prime determinant and alternating Montesinos links. The results show that orderability of link quandles behave quite differently than that of corresponding link groups. Viewing classical knots as knots in the thickened 2-sphere, it is natural to explore knot theory in thickened surfaces of higher genera. This idea led to what is now known as virtual knot theory, a subject pioneered by Kauffman in 1999 with a completely different set-up. Though many invariants from the classical knot theory extend to the virtual setting, a lot is still unknown, and the second component of the thesis focuses on this theme. We define virtually symmetric representations of virtual braid groups by automorphism groups. We prove that many known representations of these groups such as the generalized Artin representation, the Silver-Williams representation, the Boden-Dies representation and the Wada representation are equivalent to virtually symmetric representations. We use one such representation to define new virtual link groups which are extensions of link groups known due to Kauffman. Finally, we introduce marked Gauss diagrams as a generalization of Gauss diagrams and extend the definition of virtual link groups to marked Gauss diagrams 2021-07-28T00:00:00Z