• 제목/요약/키워드: torus knots

검색결과 18건 처리시간 0.024초

PRIMITIVE/SEIFERT KNOTS WHICH ARE NOT TWISTED TORUS KNOT POSITION

  • Kang, Sungmo
    • 호남수학학술지
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    • 제35권4호
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    • pp.775-791
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    • 2013
  • The twisted torus knots and the primitive/Seifert knots both lie on a genus 2 Heegaard surface of $S^3$. In [5], J. Dean used the twisted torus knots to provide an abundance of examples of primitive/Seifert knots. Also he showed that not all twisted torus knots are primitive/Seifert knots. In this paper, we study the other inclusion. In other words, it shows that not all primitive/Seifert knots are twisted torus knot position. In fact, we give infinitely many primitive/Seifert knots that are not twisted torus knot position.

NEW FAMILIES OF HYPERBOLIC TWISTED TORUS KNOTS WITH GENERALIZED TORSION

  • Keisuke, Himeno;Masakazu, Teragaito
    • 대한수학회보
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    • 제60권1호
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    • pp.203-223
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    • 2023
  • A generalized torsion element is an obstruction for a group to admit a bi-ordering. Only a few classes of hyperbolic knots are known to admit such an element in their knot groups. Among twisted torus knots, the known ones have their extra twists on two adjacent strands of torus knots. In this paper, we give several new families of hyperbolic twisted torus knots whose knot groups have generalized torsion. They have extra twists on arbitrarily large numbers of adjacent strands of torus knots.

DEHN SURGERIES ON MIDDLE/HYPER DOUBLY SEIFERT TWISTED TORUS KNOTS

  • Kang, Sungmo
    • 대한수학회보
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    • 제57권1호
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    • pp.1-30
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    • 2020
  • In this paper, we classify all twisted torus knots which are middle/hyper doubly Seifert. By the definition of middle/hyper doubly Seifert knots, these knots admit Dehn surgery yielding either Seifert-fibered spaces or graph manifolds at a surface slope. We show that middle/hyper doubly Seifert twisted torus knots admit the latter, that is, non-Seifert-fibered graph manifolds whose decomposing pieces consist of two Seifert-fibered spaces over the disk with two exceptional fibers.

TWISTED TORUS KNOTS WITH GRAPH MANIFOLD DEHN SURGERIES

  • Kang, Sungmo
    • 대한수학회보
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    • 제53권1호
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    • pp.273-301
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    • 2016
  • In this paper, we classify all twisted torus knots which are doubly middle Seifert-fibered. Also we show that all of these knots possibly except a few admit Dehn surgery producing a non-Seifert-fibered graph manifold which consists of two Seifert-fibered spaces over the disk with two exceptional fibers, glued together along their boundaries. This provides another infinite family of knots in $S^3$ admitting Dehn surgery yielding such manifolds as done in [5].

ON THE INTERSECTION OF TWO TORUS KNOTS

  • Lee, Sang-Youl;Lim, Yong-Do
    • East Asian mathematical journal
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    • 제16권1호
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    • pp.61-69
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    • 2000
  • We show that the intersection of two standard torus knots of type (${\lambda}_1$, ${\lambda}_2$) and (${\beta}_1$, ${\beta}_2$) induces an automorphism of the cyclic group ${\mathbb{Z}}_d$, where d is the intersection number of the two torus knots and give an elementary proof of the fact that all non-trivial torus knots are strongly invertiable knots. We also show that the intersection of two standard knots on the 3-torus $S^1{\times}S^1{\times}S^1$ induces an isomorphism of cyclic groups.

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PRIMITIVE POLYNOMIAL RINGS

  • Kwon, Mi-Hyang;Kim, Chol-On;Huh, Chan
    • East Asian mathematical journal
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    • 제16권1호
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    • pp.71-79
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    • 2000
  • We show that the intersection of two standard torus knots of type (${\lambda}_1$, ${\lambda}_2$) and (${\beta}_1$, ${\beta}_2$) induces an automorphism of the cyclic group ${\mathbb{Z}}_d$, where d is the intersection number of the two torus knots and give an elementary proof of the fact that all non-trivial torus knots are strongly invertiable knots. We also show that the intersection of two standard knots on the 3-torus $S^1{\times}S^1{\times}S^1$ induces an isomorphism of cyclic groups.

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EXAMPLES OF KNOTS IN S3 ADMITTING SEIFERT-FIBERED SURGERIES OVER S2 WITH FOUR EXCEPTIONAL FIBERS

  • Kang, Sungmo
    • 호남수학학술지
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    • 제40권3호
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    • pp.591-600
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    • 2018
  • In [4] Miyazaki and Motegi constructed one family of knots in $S^3$ which admits Dehn surgery producing a Seifert-fibered space over $S^2$ with four exceptional fibers. On the other hand, in [3] using doubly hyper Seifert twisted torus knots, the author constructed six families of knots in $S^3$ which admit Dehn surgery yielding a Seifert-fibered space over $S^2$ with four exceptional fibers. It is questioned in [3] whether or not the family of the knots constructed in [4] belongs to one of the six families of the knots in [3]. In this paper, we give the positive answer for this question.

On Minimal Unknotting Crossing Data for Closed Toric Braids

  • Siwach, Vikash;Prabhakar, Madeti
    • Kyungpook Mathematical Journal
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    • 제57권2호
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    • pp.331-360
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    • 2017
  • Unknotting numbers for torus knots and links are well known. In this paper, we present a new approach to determine the position of unknotting number crossing changes in a toric braid such that the closure of the resultant braid is equivalent to the trivial knot or link. Further we give unknotting numbers of more than 600 knots.

ON THE 2-BRIDGE KNOTS OF DUNWOODY (1, 1)-KNOTS

  • Kim, Soo-Hwan;Kim, Yang-Kok
    • 대한수학회보
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    • 제48권1호
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    • pp.197-211
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    • 2011
  • Every (1, 1)-knot is represented by a 4-tuple of integers (a, b, c, r), where a > 0, b $\geq$ 0, c $\geq$ 0, d = 2a+b+c, $r\;{\in}\;\mathbb{Z}_d$, and it is well known that all 2-bridge knots and torus knots are (1, 1)-knots. In this paper, we describe some conditions for 4-tuples which determine 2-bridge knots and determine all 4-tuples representing any given 2-bridge knot.