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Relevant bibliographies by topics / Knot theory ; Braid theory ; Conjugacy classes

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Journal articles

Academic literature on the topic 'Knot theory ; Braid theory ; Conjugacy classes'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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Journal articles on the topic "Knot theory ; Braid theory ; Conjugacy classes"

1

Li, Weiping. "Casson-Lin's Invariant and Floer Homology." Journal of Knot Theory and Its Ramifications 06, no. 06 (1997): 851–77. http://dx.doi.org/10.1142/s0218216597000480.

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Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its

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2

Ali, Usman. "Conjugacy Classes of the 3-Braid Group." Algebra Colloquium 17, spec01 (2010): 829–40. http://dx.doi.org/10.1142/s1005386710000775.

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In this article, we describe the summit sets in the 3-braid group, the smallest element in a summit set, and we compute the Hilbert series corresponding to conjugacy classes. The results are related to the Birman–Menasco classification of knots with braid index three or less than three.

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3

XU, PEIJUN. "THE GENUS OF CLOSED 3-BRAIDS." Journal of Knot Theory and Its Ramifications 01, no. 03 (1992): 303–26. http://dx.doi.org/10.1142/s0218216592000185.

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The problem of finding the genus of a given link type of braid index 3 is solved by constructive methods. The results depend upon a new solution to the conjugacy problem in B3. In this solution, conjugacy classes are represented by shortest words in terms of cyclically symmetric elementary braids which are used as generators in the new presentation of B3. By related results of Bennequin [2] and of Birman and Menasco [4], the minimal spanning surfaces are described by these shortest words. An effective algorithm is given to find these shortest words starting with an arbitrary 3-braid representa

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4

Fukuda, Mizuki. "Irreducible SL(2, ℂ)-metabelian representations of branched twist spins". Journal of Knot Theory and Its Ramifications 28, № 02 (2019): 1950007. http://dx.doi.org/10.1142/s021821651950007x.

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An [Formula: see text]-branched twist spin is a fibered [Formula: see text]-knot in [Formula: see text] which is determined by a [Formula: see text]-knot [Formula: see text] and coprime integers [Formula: see text] and [Formula: see text]. For a [Formula: see text]-knot, Nagasato proved that the number of conjugacy classes of irreducible [Formula: see text]-metabelian representations of the knot group of a [Formula: see text]-knot is determined by the knot determinant of the [Formula: see text]-knot. In this paper, we prove that the number of irreducible [Formula: see text]-metabelian represen

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5

LIN, XIAO-SONG, and SAM NELSON. "ON GENERALIZED KNOT GROUPS." Journal of Knot Theory and Its Ramifications 17, no. 03 (2008): 263–72. http://dx.doi.org/10.1142/s0218216508006117.

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Generalized knot groups Gn(K) were introduced first by Wada and Kelly independently. The classical knot group is the first one G1(K) in this series of finitely presented groups. For each natural number n, G1(K) is a subgroup of Gn(K) so the generalized knot groups can be thought of as extensions of the classical knot group. For the square knot SK and the granny knot GK, we have an isomorphism G1(SK) ≅ G1(GK). From the presentations of Gn(SK) and Gn(GK), for n > 1, it seems unlikely that Gn(SK) and Gn(GK) would be isomorphic to each other. Curiously, we are able to show that for any finite g

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6

ANDERSEN, JØRGEN ELLEGAARD, and SØREN KOLD HANSEN. "ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT." Journal of Knot Theory and Its Ramifications 15, no. 04 (2006): 479–548. http://dx.doi.org/10.1142/s0218216506004555.

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We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL

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7

LI, WEIPING. "KNOT AND LINK INVARIANTS AND MODULI SPACE OF PARABOLIC BUNDLES." Communications in Contemporary Mathematics 03, no. 04 (2001): 501–31. http://dx.doi.org/10.1142/s0219199701000470.

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In this paper, we show that the representation variety of the fundamental group of a 2n-punctured S2 with different conjugacy classes in SU(2) along punctures is a symplectic stratified variety from the group cohomology point of view. The representation variety can be identified with the moduli space of s-equivalence classes of stable parabolic bundles over the 2n-punctured S2 with corresponding weights along punctures, and also can be identified with the moduli space of gauge equivalence classes of SU(2)-flat connections with prescribed holonomies along punctures. We obtain an invariant of li

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8

CONANT, JAMES, JACOB MOSTOVOY, and TED STANFORD. "FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND DOUBLED-DELTA MOVES." Journal of Knot Theory and Its Ramifications 19, no. 03 (2010): 355–84. http://dx.doi.org/10.1142/s0218216510007875.

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We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-

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9

Ishii, Atsushi. "The Markov theorems for spatial graphs and handlebody-knots with Y-orientations." International Journal of Mathematics 26, no. 14 (2015): 1550116. http://dx.doi.org/10.1142/s0129167x15501165.

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We establish the Markov theorems for spatial graphs and handlebody-knots. We introduce an IH-labeled spatial trivalent graph and develop a theory on it, since both a spatial graph and a handlebody-knot can be realized as the IH-equivalence classes of IH-labeled spatial trivalent graphs. We show that any two orientations of a graph without sources and sinks are related by finite sequence of local orientation changes preserving the condition that the graph has no sources and no sinks. This leads us to define two kinds of orientations for IH-labeled spatial trivalent graphs, which fit a closed br

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