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1. A Study of Gram Determinants in Knot Theory

Author

Ibarra, Dionne and Montoya-Vega, Gabriel

Subjects
Mathematics - Geometric Topology, Primary: 57M25 Secondary: 57M27
Abstract

Historically originated as a sub-field of topology, knot theory is an active area of mathematical investigation that has strong connections with a diverse set of scientific fields such as algebra, biology, and statistical mechanics. A popular and important concept in linear algebra, Gram determinants enjoy a connection with the mathematical theory of knots. In this article, we expose this concept and present several types of Gram determinants in what can be considered as a survey of the current Gram determinants of interest to knot theorists; examples are included to illustrate the definitions. In particular, we pay special attention to a recently defined determinant from a M\"obius band and we further study its structure. At the end, some speculation is presented regarding the closed formula for the Gram determinant of type $(Mb)_1$, a problem that arouses serious interest among knot theorists., Comment: 17 pages, 71 figures

Published
2024

2. A lower bound for the volumes of modular link complements

Author

Hui, Connie On Yu, Ibarra, Dionne, and Migueles, José Andrés Rodríguez

Subjects
Mathematics - Geometric Topology, Mathematics - Dynamical Systems, 57K10, 57K32 (primary) 37D40, 37E35 (secondary)
Abstract

Every finite collection of oriented closed geodesics in the modular surface has a canonically associated link in its unit tangent bundle coming from the periodic orbits of the geodesic flow. We study the volume of the associated link complement with respect to its unique complete hyperbolic metric. We provide the first lower volume bound that is linear in terms of the number of distinct exponents in the code words corresponding to the collection of closed geodesics., Comment: 20 pages, 8 figures

Published
2024

3. The Temperley-Lieb Algebra and the Artin Braid Group

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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4. Long Exact Sequence of Khovanov Homology and Torsion

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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5. Variations on Catalan Connections and the Children Pairing Game

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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6. Symmetrizers of Finite Groups and the Jones-Wenzl Idempotent

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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7. Goeritz and Seifert Matrices

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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8. From Fox 3-Coloring to the Yang-Baxter Operator and Its Homology

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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9. The Jones Polynomial and Kauffman Bracket Polynomial

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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10. History of Knot Theory from Gauss to Jones

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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11. The HOMFLYPT and the Two-Variable Kauffman Polynomial

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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12. The Witten-Reshetikhin-Turaev Invariants of 3-Manifolds

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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13. Khovanov Homology: A Categorification of the Jones Polynomial

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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14. Spin Structure and the Framing Skein Module of Links in 3-Manifolds

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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15. Gram Determinants of Type A and Generalized Type A

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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16. History of Knot Theory from Ancient Times to Gauss and His Student Listing

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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17. The Kauffman Bracket Skein Module and Algebra of Surface I-Bundles

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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18. Gram Determinants of Type B and Type Mb

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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19. Multiplicative Structure of the Kauffman Bracket Skein Algebra of the Thickened T-Shirt

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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20. Basics of Skein Modules

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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21. Plucking Polynomial of Rooted Trees and Its Use in Knot Theory

Author

Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Weeks, Deborah, Berestycki, Nathanaël, Series Editor, Casacuberta, Carles, Series Editor, Greenlees, John, Series Editor, MacIntyre, Angus, Series Editor, Sabbah, Claude, Series Editor, Süli, Endre, Series Editor, Przytycki, Józef H., Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, and Weeks, Deborah

Published
2024
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22. On a new Gram determinant from the M\'obius band

Author

Ibarra, Dionne and Montoya-Vega, Gabriel

Subjects
Mathematics - Geometric Topology, 57K10 (Primary), 57K31 (Secondary)
Abstract

Gram determinants earned traction among knot theorists after E. Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by N. Reshetikhin and V. Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by W. B. Raymond Lickorish's skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this article, we introduce a new version of such a determinant from the M\"obius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula., Comment: 15 pages, comments welcomed

Published
2023

23. Jones-Wenzl Idempotents in the Twisted $I$-bundle over the M\'obius band

Author

Ibarra, Dionne

Subjects
Mathematics - Geometric Topology, 57K10 (Primary), 57K31 (Secondary)
Abstract

The Jones-Wenzl idempotent plays a vital role in quantum invariants of $3$-manifolds and the colored Jones polynomial; it also serves as a useful tool for simplifying computations and proving theorems in knot theory. The relative Kauffman bracket skein module (RKBSM) for surface $I$-bundles and manifolds with marked boundaries have a well understood algebraic structure due to the work of J. H. Przytycki and T. T. Q. L\^e. It has been well documented that the RKBSM of the $I$-bundle of the annulus and the twisted $I$-bundle over the M\"obius band have distinct algebraic structures coming from the $I$-bundle structures. This paper serves as an introduction to studying the trace of Jones-Wenzl idempotents in the Kauffman bracket skein module (KBSM) of the twisted $I$-bundle of unorientable surfaces. We will give various results on Jones-Wenzl idempotents in the KBSM of the twisted $I$-bundle over the M\"obius band when it is closed through the crosscap of the M\"obius band. We will also uncover analog properties of Jones-Wenzl idempotents in the KBSM of the twisted $I$-bundle over the M\"obius band with the preservation of the $I$-bundle structure that differ from the KBSM of $Ann \times I$., Comment: 20 pages, comments welcomed

Published
2023

24. A note on a short proof of the parallelizability of orientable $3$-manifolds

Author

Ibarra, Dionne

Subjects
Mathematics - Geometric Topology, 57K10 (Primary), 57K50, 57R25 (Secondary)
Abstract

We survey, complete, and modify a proof, involving knot theory, of Stiefel's theorem that all orientable $3$-manifolds are parallelizable. The completion of the proof is done by using the relationship between the tangent bundle and normal bundle of manifolds with non-trivial boundary and on stably parallelizable and parallelizable manifolds. We end with a remark on $7$-manifolds and present J. Korba\v{s}' example of a non-parallelizable $7$-manifold., Comment: 7 pages, 1 figure, comments welcomed

Published
2022

33. Basics of Skein Modules

Author

Przytycki, Józef H., primary, Bakshi, Rhea Palak, additional, Ibarra, Dionne, additional, Montoya-Vega, Gabriel, additional, and Weeks, Deborah, additional

Published
2023
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39. Goeritz and Seifert Matrices

Author

Przytycki, Józef H., primary, Bakshi, Rhea Palak, additional, Ibarra, Dionne, additional, Montoya-Vega, Gabriel, additional, and Weeks, Deborah, additional

Published
2023
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41. On framings of links in 3-manifolds

Author

Bakshi, Rhea Palak, Ibarra, Dionne, Montoya-Vega, Gabriel, Przytycki, Józef H., and Weeks, Deborah

Subjects
Mathematics - Geometric Topology, Mathematics - Algebraic Topology, Primary: 57M27. Secondary: 55A25, 57M25
Abstract

We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough's work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules., Comment: 12 pages, 9 figures

Published
2020
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42. A Generalization of the Gram determinant of type A

Author

Bakshi, Rhea Palak, Ibarra, Dionne, Mukherjee, Sujoy, and Przytycki, Józef H.

Subjects
Mathematics - Geometric Topology, Mathematics - Combinatorics, Primary: 57M27. Secondary: 57M25, 05A19
Abstract

The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type $Mb$ evaluated in the Klein bottle., Comment: 17 pages, 19 figures

Published
2019
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43. Schur Multipliers and Second Quandle Homology

Author

Bakshi, Rhea Palak, Ibarra, Dionne, Mukherjee, Sujoy, Nosaka, Takefumi, and Przytycki, Józef H.

Subjects
Mathematics - Geometric Topology, Mathematics - Group Theory, Mathematics - K-Theory and Homology, Primary: 19C09, Secondary: 20J06, 57M27
Abstract

We define a map from second quandle homology to the Schur multiplier and examine its properties. Furthermore, we express the second homology of Alexander quandles in terms of exterior algebras. Additionally, we present a self-contained proof of its structure and provide some computational examples., Comment: 16 pages

Published
2018
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46. On a state model for the SO(2n) Kauffman polynomial

Author

Caprau, Carmen, Heywood, David, and Ibarra, Dionne

Subjects
Mathematics - Geometric Topology, 57M27, 57M15
Abstract

F. Jaeger presented the two-variable Kauffman polynomial of an unoriented link L as a weighted sum of HOMFLY-PT polynomials of oriented links associated with L. Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the sl(n)-link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger's work, and construct a state summation model for the SO(2n) Kauffman polynomial., Comment: 14 pages, many figures

Published
2013
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49. On a state model for the $\mathrm{SO}(2n)$ Kauffman polynomial

Author

Caprau, Carmen, Heywood, David, and Ibarra, Dionne

Subjects
graphs, Kauffman polynomial, 57M27, Mathematics::Quantum Algebra, 57M15, Mathematics::Representation Theory, Mathematics::Geometric Topology, invariants for knots and links
Abstract

François Jaeger presented the two-variable Kauffman polynomial of an unoriented link [math] as a weighted sum of HOMFLY-PT polynomials of oriented links associated with [math] . Murakami, Ohtsuki and Yamada (MOY) used planar graphs and a recursive evaluation of these graphs to construct a state model for the [math] -link invariant (a one-variable specialization of the HOMFLY-PT polynomial). We apply the MOY framework to Jaeger’s work, and construct a state summation model for the [math] Kauffman polynomial.

Published
2014

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