Knots and Physics

Kauffman, Louis H.

doi:10.1142/9789812384836

Cited by 322 publications

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“…Figure 3d shows the designed assembly complexes for two topoisomers of the Borromean rings (Ti-BR, which has a similar design to BR, as shown in Fig. 1k, but a different sequence), the torus-link cyclic [3]catenane (6 3 1 ) (Ti-C3C (left)) and the composite-link linear [3]catenane (2 2 1 #2 2 1 ) (Ti-L3C (right)). The topologies of these molecules can be identified readily by nickase cleavage.…”

Section: Resultsmentioning

confidence: 99%

Creating complex molecular topologies by configuring DNA four-way junctions

2016

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The realization of complex topologies at the molecular level represents a grand challenge in chemistry. This necessitates the manipulation of molecular interactions with high precision. Here we show that single-stranded DNA (ssDNA) knots and links can be created by utilizing the inherent topological properties that pertain to the DNA four-way junction, at which the two helical strands form a node and can be configured conveniently and connected for complex topological construction. Using this strategy, we produced series of ssDNA topoisomers with the same sequences. By finely designing the curvature and torsion, double-stranded DNA knots were accessed by hybridizing and ligating the complementary strands with the knotted ssDNA templates. Furthermore, we demonstrate the use of a constructed ssDNA knot both to probe the topological conversion catalysed by DNA topoisomerase and to study the DNA replication under topological constraint.

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Section: Resultsmentioning

confidence: 99%

Creating complex molecular topologies by configuring DNA four-way junctions

2016

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“…As it will be shown in Sec. 2-2-2, the reason for choosing equation (18) as the standard general form for representing the homogeneous quadratic equations (that could be also transformed to the ordinary representations of homogeneous quadratic equations (18-1) and , by linear transformations (18-3)) is not only its very simple algebraic structure, but also the simple linear homogeneous forms of the entries of square matrices A (expressed in terms of variables Moreover, as it is shown in the following, we may also assume certain Hermiticity and anti-Hermiticity conditions for the deriving square matrices A (in the corresponding systems of linear equations 0   M A equivalent to the quadratic equation (18)), without any restriction in the existence and procedure of derivation of these matrices. By adding these particular conditions, for a specific number of variables in equation 18, its equivalent matrix equation 0   M A could be determined uniquely.…”

Section: -2mentioning

confidence: 99%

On the Mathematical Origin of the Fundamental Laws of Nature: A New Algebraic-Axiomatic Approach

Zahedi¹

2016

Preprint

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In this article*, as a new mathematical approach to origin of the laws of nature, using a new basic algebraic axiomatic (matrix) formalism based on the ring theory and Clifford algebras (presented in Sec.2), “it is shown that certain mathematical forms of fundamental laws of nature, including laws governing the fundamental forces of nature (represented by a set of two definite classes of general covariant massive field equations, with new matrix formalisms), are derived uniquely from only a very few axioms”; where in agreement with the rational Lorentz group, it is also basically assumed that the components of relativistic energy-momentum can only take rational values. In essence, the main scheme of this new mathematical axiomatic approach to fundamental laws of nature is as follows. First based on the assumption of rationality of D-momentum, by linearization (along with a parameterization procedure) of the Lorentz invariant energy-momentum quadratic relation, a unique set of Lorentz invariant systems of homogeneous linear equations (with matrix formalisms compatible with certain Clifford, and symmetric algebras) is derived. Then by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of these determined systems of linear equations, a set of two classes of general covariant massive (tensor) field equations (with matrix formalisms compatible with certain Clifford, and Weyl algebras) is derived uniquely as well. Each class of the derived general covariant field equations also includes a definite form of torsion field appeared as generator of the corresponding field’ invariant mass. In addition, it is shown that the (1+3)-dimensional cases of two classes of derived field equations represent a new general covariant massive formalism of bispinor fields of spin-2, and spin-1 particles, respectively. In fact, these uniquely determined bispinor fields represent a unique set of new generalized massive forms of the laws governing the fundamental forces of nature, including the Einstein (gravitational), Maxwell (electromagnetic) and Yang-Mills (nuclear) field equations. Moreover, it is also shown that the (1+2)-dimensional cases of two classes of these field equations represent (asymptotically) a new general covariant massive formalism of bispinor fields of spin-3/2 and spin-1/2 particles, corresponding to the Dirac and Rarita–Schwinger equations. As a particular consequence, it is shown that a certain massive formalism of general relativity – with a definite form of torsion field appeared originally as the generator of gravitational field’s invariant mass – is obtained only by first quantization (followed by a basic procedure of minimal coupling to space-time geometry) of a certain set of special relativistic algebraic matrix equations. In addition, in agreement with recent astronomical data, a new particular form of massive particle is identified (as the gauge boson of U(1) symmetry group) with invariant mass: m_γ ≈ 4.90571×10^-50 kg, which is generated by a definite coupled torsion field of the background space-time geometry. Moreover, based on the definite mathematical formalism of this axiomatic approach, along with the C, P and T symmetries (represented basically by the corresponding quantum operators) of the fundamentally derived field equations, it has been concluded that the universe could be realized solely with the (1+2) and (1+3)-dimensional space-times (where this conclusion, in particular, is based on the T-symmetry of these equations). It is shown that CPT is the only combined form of C, P, and T symmetries of interacting fields. In addition, on the basis of these discrete symmetries of derived field equations, it has been also shown that only left-handed particle fields (along with their complementary right-handed fields) could be coupled to the corresponding (any) source currents. Furthermore, it has been shown that the metric of background space-time is diagonalized for the uniquely derived fermion field equations (defined and expressed solely in (1+2)-dimensional space-time), where this property generates a certain set of additional symmetries corresponding uniquely to the SU(2)_L⊗U(2)_R symmetry group for spin-1/2 fermion fields (representing “1+3” generations of four fermions, including a group of eight leptons and a group of eight quarks), and also the SU(2)_L⊗U(2)_R and SU(3) gauge symmetry groups for spin-1 boson fields coupled to the spin-1/2 fermionic source currents. Hence, along with the known elementary particles, eight new elementary particles, including four new charge-less right-handed spin-1/2 fermions (including two leptons and two quarks), a spin-3/2 fermion, and also three new spin-1 (massive) bosons are predicted uniquely by this axiomatic approach. As a particular result, based on the definite formulation of the derived Maxwell (and Yang-Mills) field equations, it has been also concluded that magnetic monopoles could not exist in nature.* *Note: "See also new published version of this article at: https://inspirehep.net/record/1387680/ ”.Comments: 89 Pages. A summary of a submitted and accepted research project (On "Foundations of Physics"), Ramin Zahedi, Japan, 2012 - 2015.This article has been invited and presented at the following international conferences: - The 2016 SIAM International Conference on Mathematical Aspects of Materials Science, Philadelphia, USA, 2016. (https://www.siam.org/meetings/ms16)- The 17th International Conference on Quantum Foundations: Quantum and Beyond, International Centre for Mathematical Modeling in Physics, Engineering and Cognitive Sciences (ICMM), Linnaeus University, Sweden, 2016. (https://lnu.se/en/qb/) -The 4th International Conference on New Frontiers in Physics, CERN Organized and Supported Conference (Europe), ICNFP-2015. (https://indico.cern.ch/e/icnfp2015/) - The 2016 International Conference on Algebraic Geometry and Mathematical Physics, University of Tromsø, Norway, 2016. (https://site.uit.no/)- The XXXVII Max Born International Symposium, International Conference on Non-commutative Geometry, Quantum Symmetries and Quantum Gravity (II), Wroclaw University, Poland, 2016. (http://ift.uni.wroc.pl/~mborn37/) - The 2016 GRavitational-wave Astronomy International Conference in Paris, Institute d‘Astrophysique de Paris (IAP), The University of Paris VI - Sorbonne University, CNRS, LERU, EUA, Paris, France, 'Supported by the European Union's 7th Framework Program: FP7/PEOPLE-2011-CIG,‘ 2016. (http://www.iap.fr/vie_scientifique/ateliers/GravitationalWave/2016/scripts/abstract.aspx) - The 22nd Internnational Australian Institute of Physics Congress (AIP), University of Queensland, Brisbane, Australia, 2016. (http://appc-aip2016.org.au) - The 21st International Conference on General Relativity and Gravitation, Columbia University, New York, USA, 2016. (http://www.gr21.org).External URLs (including preprints and earlier published versions of this research article): https://cds.cern.ch/record/1980381/ , https://inspirehep.net/record/1387680/, https://indico.cern.ch/event/344173/session/22/contribution/422/attachments/1140145/1646101/R.a.Zahedi--OnDiscretePhysics-Jan.2015-signed.pdf, http://eprints.lib.hokudai.ac.jp/dspace/handle/2115/59279/, https://core.ac.uk/display/41879427, http://www.gr21.org/GR21FullAbstractBooklet.pdf#section*.177, https://dx.doi.org/10.17605/OSF.IO/ZCHTP, https://dumas.ccsd.cnrs.fr/TDS-MACS/hal-01476703 , https://indico.cern.ch/event/344173/contribution/1740565/attachments/1140145/1726912/R.A.Zahedi--Forces.of.Nature.Laws-Jan.2015-signed.pdf ,http://mts-ncomms.nature.com/ncomms_files/2017/07/10/00136200/00/136200_0_art_file_2445656_bsw9c4.pdf .

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“…The first type of knot polynomial was introduced by J.W. Alexander in 1923, but it was only in 1969 that J. Conway showed that the Alexander polynomial of a knot can be derived by using a recursive relation (called skein relation) based on elementary computations performed on each crossing site of the knot diagram (Kauffman 1987(Kauffman , 2001. Other polynomials followed suit, including the Jones polynomial, introduced by V.F.R.…”

Section: Tackling Topological Complexity By Knot Polynomials: the Jonmentioning

confidence: 99%

“…and orientation does not matter. A physical interpretation of the skein relation is given in terms of a correspondence between field change and field line curvature and curl (see Kauffman 2001). Some simple configurations and their computed Jones polynomial are shown in figure 3.…”

Section: Tackling Topological Complexity By Knot Polynomials: the Jonmentioning

confidence: 99%

The Jones polynomial as a new invariant of topological fluid dynamics

Ricca¹,

Liu²

2014

Fluid Dyn. Res.

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A new method based on the use of the Jones polynomial, a well-known topological invariant of knot theory, is introduced to tackle and quantify topological aspects of structural complexity of vortex tangles in ideal fluids. By re-writing the Jones polynomial in terms of helicity, the resulting polynomial becomes then function of knot topology and vortex circulation, providing thus a new invariant of topological fluid dynamics. Explicit computations of the Jones polynomial for some standard configurations, including the Whitehead link and the Borromean rings (whose linking numbers are zero), are presented for illustration. In the case of a homogeneous, isotropic tangle of vortex filaments with same circulation, the new Jones polynomial reduces to some simple algebraic expression, that can be easily computed by numerical methods. This shows that this technique may offer a new setting and a powerful tool to detect and compute topological complexity and to investigate relations with energy, by tackling fundamental aspects of turbulence research. |

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Knots and Physics

Cited by 322 publications

References 0 publications

Creating complex molecular topologies by configuring DNA four-way junctions

Creating complex molecular topologies by configuring DNA four-way junctions

On the Mathematical Origin of the Fundamental Laws of Nature: A New Algebraic-Axiomatic Approach

The Jones polynomial as a new invariant of topological fluid dynamics

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