scholarly journals Non-Hermitian Hopf-link exceptional line semimetals

2019 ◽  
Vol 99 (8) ◽  
Author(s):  
Zhesen Yang ◽  
Jiangping Hu
Keyword(s):  
Hopf Link

scholarly journals The SU(2) Casson–Lin invariant of the Hopf link

2016 ◽  
Vol 285 (2) ◽  
pp. 283-288
Author(s):  
Hans Boden ◽  
Christopher Herald
Keyword(s):  
Hopf Link

scholarly journals Triple-crossing number and moves on triple-crossing link diagrams

2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Triple Points ◽  
Definition Of ◽  
Hopf Link

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

scholarly journals ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE

2010 ◽  
Vol 19 (02) ◽  
pp. 187-289 ◽  
Author(s):  
JÓZEF H. PRZYTYCKI ◽  
KOUKI TANIYAMA
Keyword(s):  
Partial Answer ◽  
Link Diagram ◽  
Link Diagrams ◽  
Left Handed ◽  
Trefoil Knot ◽  
Trivial Link ◽  
Hopf Link

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

scholarly journals THE HOMFLY POLYNOMIAL OF THE DECORATED HOPF LINK

2003 ◽  
Vol 12 (03) ◽  
pp. 395-416 ◽  
Author(s):  
HUGH R. MORTON ◽  
SASCHA G. LUKAC
Keyword(s):  
Power Series ◽  
Symmetric Function ◽  
Vandermonde Matrix ◽  
Homfly Polynomial ◽  
Quantum Invariant ◽  
Hopf Link

The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements Qλ, depending on partitions λ. We show how the 2-variable Homfly invariant <λ, μ> of the Hopf link arising from decorations Qλ and Qμ can be found from the Schur symmetric function sμ of an explicit power series depending on λ. We show also that the quantum invariant of the Hopf link coloured by irreducible sl(N)q modules Vλ and Vμ, which is a 1-variable specialisation of <λ, μ>, can be expressed in terms of an N × N minor of the Vandermonde matrix (qij).

scholarly journals Smooth concordance of links topologically concordant to the Hopf link

2011 ◽  
Vol 44 (3) ◽  
pp. 443-450 ◽  
Author(s):  
Jae Choon Cha ◽  
Taehee Kim ◽  
Daniel Ruberman ◽  
Sašo Strle
Keyword(s):  
Hopf Link

2-Adjacency between some special links and pretzel links

2018 ◽  
Vol 27 (12) ◽  
pp. 1850062
Author(s):  
Zhi-Xiong Tao
Keyword(s):  
Whitehead Link ◽  
Hopf Link

This paper studies 2-adjacency between a 3-strand pretzel link and one of the Hopf link, the Solomon’s link and the Whitehead link by using the results that have been obtained about 2-adjacency between knots or links and their polynomials and etc. This paper shows that of all 3-strand pretzel links, only ordinary pretzel links are 2-adjacent to the Hopf link or the Solomon’s link or the Whitehead link. Conversely, these special links are not 2-adjacent to any other 3-strand pretzel links, except for themselves, respectively.

HOPF-BRAID GROUPS

1998 ◽  
Vol 07 (08) ◽  
pp. 1107-1117
Author(s):  
VINCENT MOULTON
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
Motion Group ◽  
Three Sphere ◽  
Group A ◽  
Complex Lines ◽  
Group Of Motions ◽  
Hopf Link

In this note we define the Hopf-braid group, a group that is directly related to the group of motions of n mutually distinct lines through the origin in [Formula: see text], which is better known as the braid group of the two-sphere. It is also related to the motion group of the Hopf link in the three-sphere. This relationship is provided by considering the link of a union of complex lines through the origin in [Formula: see text] (i.e. the intersection of the lines with the unit 3-sphere centered at the origin in [Formula: see text]). Through the study of this group we also illustrate some of the connections between the field of knots and braids and that of hyperplane arrangements.

scholarly journals Links with non-trivial Alexander polynomial which are topologically concordant to the Hopf link

2019 ◽  
Vol 371 (8) ◽  
pp. 5379-5400 ◽  
Author(s):  
Min Hoon Kim ◽  
David Krcatovich ◽  
JungHwan Park
Keyword(s):  
Alexander Polynomial ◽  
Hopf Link
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