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Bulletin of the American Mathematical Society | 1985

A new polynomial invariant of knots and links

P. Freyd; D. Yetter; Jim Hoste; W. B. R. Lickorish; K. Millett; A. Ocneanu

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation


Journal of Knot Theory and Its Ramifications | 2004

A FORMULA FOR THE A-POLYNOMIAL OF TWIST KNOTS

Jim Hoste; Patrick D. Shanahan

The fundamental group of a 2-bridge knot has a particularly nice presentation, having only two generators and a single relation. For certain families of 2-bridge knots, such as the torus knots, or the twist knots, the relation takes on an especially simple form. Exploiting this form, we derive a formula for the A-polynomial of twist knots. Our methods extend to at least one other infinite family of (non-torus) 2-bridge knots. Using these formulae we determine the associated Newton polygons. We further prove that the A-polynomials of twist knots are irreducible.


Journal of Knot Theory and Its Ramifications | 1993

THE (2, ∞)-SKEIN MODULE OF LENS SPACES: A GENERALIZATION OF THE JONES POLYNOMIAL

Jim Hoste; Jozef H. Przytycki

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.


Transactions of the American Mathematical Society | 1990

Dichromatic link invariants

Jim Hoste; Mark E. Kidwell

We investigate the skein theory of oriented dichromatic links in S3. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new 1 2 invariant as an invariant of oriented monochromatic links in S x D , and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.


Journal of Knot Theory and Its Ramifications | 2001

TRACE FIELDS OF TWIST KNOTS

Jim Hoste; Patrick D. Shanahan

In this paper we compute the trace field for the family of hyperbolic twist knots. We describe this field as a simple extension ℚ(z0) where z0 is a specified root of a particular irreducible polynemial Φn(z)∈ℤ[z]. As a consequence, we find that the degree of the trace field is precisely two less than the-minimal crossing number of a twist knot.


Topology and its Applications | 1984

The Arf invariant of a totally proper link

Jim Hoste

Abstract Suppose L is an oriented link in S 3 such that each pair of components of L link each other an even number of times. Then the Arf invariant of L is equal to the sum (mod 2) of the Arf invariants of all sublinks of L plus a certain coefficient of the Conway polynomial of L . This result extends the formula recently given by Murasugi in the case when L has two components.


Experimental Mathematics | 2009

Sampling Lissajous and Fourier Knots

Adam Boocher; Jay Daigle; Jim Hoste; Wenjing Zheng

A Lissajous knot is one that can be parameterized as , where the frequencies n x , n y , and n z are relatively prime integers and the phase shifts ϕ x , ϕ y , and ϕ z are real numbers. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems that allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots. A Fourier-(i, j, k) knot is similar to a Lissajous knot except that the x, y, and z coordinates are now each described by a sum of i, j, and k cosine functions, respectively. According to Lamm, every knot is a Fourier-(1, 1, k) knot for some k. By randomly searching the set of Fourier-(1, 1, 2) knots we find that all 2-bridge knots with up to 14 crossings are either Lissajous or Fourier-(1, 1, 2) knots. We show that all twist knots are Fourier-(1, 1, 2) knots and give evidence suggesting that all torus knots are Fourier-(1, 1, 2) knots. As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.


Handbook of Knot Theory | 2005

The Enumeration and Classification of Knots and Links

Jim Hoste

Publisher Summary This chapter describes the theoretical and practical aspects of link classification, with special emphasis on the mathematics involved in recent, large-scale link tabulations. A link is a disjoint union of knots in S3, considered up to homeomorphisms taking one link to another. In the case of links having multiple components, it is important to consider which components are chosen to be connected together. But additional, more subtle, problems can exist because of ones choice of link equivalence. A crossing in a link diagram is nugatory if there is a circle in the projection plane that meets the diagram transversely only at that crossing. A nugatory crossing can clearly be removed by a flype (or by a single Type I Reidemeister move). A diagram that has no nugatory crossings is called reduced. A link is alternating if it is represented by a diagram whose crossings alternate, over-under-over-under and so on, as traveling around the components takes place. Once a table of prime knots and links has been found, these can then be connected and summed together in all possible ways to obtain the composite links.


Algebraic & Geometric Topology | 2010

Epimorphisms and boundary slopes of 2–bridge knots

Jim Hoste; Patrick D. Shanahan

In this article we study a partial ordering on knots in S 3 where K1 K2 if there is an epimorphism from the knot group of K1 onto the knot group of K2 which preserves peripheral structure. If K1 is a 2‐bridge knot and K1 K2 , then it is known that K2 must also be 2‐bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2‐bridge knot Kp=q , produces infinitely many 2‐bridge knots Kp0=q0 with Kp0=q0 Kp=q . After characterizing all 2‐bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, Kp0=q0 is either a torus knot or has 5 or more distinct boundary slopes. We also prove that 2‐bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of 2‐bridge knots with Kp0=q0 Kp=q arise from the Ohtsuki‐Riley‐Sakuma construction. 57M25


Journal of Knot Theory and Its Ramifications | 2005

COMMENSURABILITY CLASSES OF TWIST KNOTS

Jim Hoste; Patrick D. Shanahan

In this paper we prove that if MK is the complement of a non-fibered twist knot K in

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Jozef H. Przytycki

George Washington University

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Adam Boocher

University of California

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Alissa S. Crans

Loyola Marymount University

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Blake Mellor

Loyola Marymount University

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Laura Zirbel

Loyola Marymount University

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Wenjing Zheng

University of California

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