Radial index related to an intersection index

Bourgoin defined the notion of a twisted link. In a sense, it is a non-orientable version of a virtual link. Im, Lee and Lee defined a polynomial invariant of a virtual link by using the virtual intersection index. In this paper, we give an alternative definition of index polynomial by using indices of real crossings and extend it to a twisted links.

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ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216511009911 ◽

2012 ◽

Vol 21 (04) ◽

pp. 1250038

Author(s):

THOMAS FIEDLER ◽

ARNAUD MORTIER

Keyword(s):

Triple Point ◽

Simple Proof ◽

Intersection Index ◽

Vassiliev Invariant ◽

Triple Points ◽

New Formula

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.

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Physical meaning of the radial index of Laguerre-Gauss beams

Conference on Lasers and Electro-Optics ◽

10.1364/cleo_qels.2017.ftu1f.4 ◽

2017 ◽

Author(s):

William N. Plick ◽

Mario Krenn

Keyword(s):

Physical Meaning ◽

Radial Index

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Digital sorting of perturbed Laguerre-Gaussian modes by radial index

2020 International Conference on Information Technology and Nanotechnology (ITNT) ◽

10.1109/itnt49337.2020.9253172 ◽

2020 ◽

Author(s):

Yana Akimova ◽

Mikhail Bretsko ◽

Yuriy Egorov ◽

Alexander Volyar ◽

Svetlana Lapayeva

Keyword(s):

Radial Index

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Experimental Analysis of Multiplexing/demultiplexing Laguerre Gaussian Beams with Different Radial Index

Frontiers in Optics 2014 ◽

10.1364/fio.2014.fth4b.6 ◽

2014 ◽

Cited By ~ 2

Author(s):

Guodong Xie ◽

Yongxiong Ren ◽

Hao Huang ◽

Nisar Ahmed ◽

Long Li ◽

...

Keyword(s):

Experimental Analysis ◽

Gaussian Beams ◽

Radial Index

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Exciton scattering via algebraic topology

Journal of Topology and Analysis ◽

10.1142/s1793525319500110 ◽

2019 ◽

Vol 11 (02) ◽

pp. 251-272

Author(s):

Michael J. Catanzaro ◽

Vladimir Y. Chernyak ◽

John R. Klein

Keyword(s):

Algebraic Topology ◽

Index Theorem ◽

Intersection Theory ◽

Scattering Data ◽

Chemical Physics ◽

Electronic Excitations ◽

Intersection Index ◽

Finite Sum ◽

Exciton Scattering ◽

Special Case

This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group [Formula: see text] and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.

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