On the Signatures of Torus Knots

Maciej Borodzik; Krzysztof Oleszkiewicz

Displaying similar documents to “On the Signatures of Torus Knots”

Elementary moves for higher dimensional knots

Dennis Roseman (2004)

Fundamenta Mathematicae

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For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in $ℝ^{n + 2}$ (or $^{n + 2}$ ), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

Representations of (1,1)-knots

Alessia Cattabriga, Michele Mulazzani (2005)

Fundamenta Mathematicae

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We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG₂(T). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω:PMCG₂(T) → MCG(T) ≅ SL(2,ℤ), which is a free group of rank two, to the class of all (1,1)-knots in a fixed lens space. The second representation is parametric:...

Some non-trivial PL knots whose complements are homotopy circles

Greg Friedman (2007)

Fundamenta Mathematicae

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We show that there exist non-trivial piecewise linear (PL) knots with isolated singularities $S^{n - 2} \subset S ⁿ$ , n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally flat, and topological locally flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

Quandles and symmetric quandles for higher dimensional knots

Seiichi Kamada (2014)

Banach Center Publications

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A symmetric quandle is a quandle with a good involution. For a knot in ℝ³, a knotted surface in ℝ⁴ or an n-manifold knot in $ℝ^{n + 2}$ , the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle presentation, and show how to get a presentation of a knot symmetric quandle from a diagram.

Legendrian and transverse twist knots

John B. Etnyre, Lenhard L. Ng, Vera Vértesi (2013)

Journal of the European Mathematical Society

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In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m (5_{2})$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston-Bennequin number of the twist knot $K_{- 2 n}$ with crossing number $2 n + 1$ . In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{- 2 n}$ has exactly $⌈\frac{n^{2}}{2}⌉$ Legendrian representatives with maximal Thurston–Bennequin...

Brunnian local moves of knots and Vassiliev invariants

Akira Yasuhara (2006)

Fundamenta Mathematicae

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K. Habiro gave a neccesary and sufficient condition for knots to have the same Vassiliev invariants in terms of $C_{k}$ -moves. In this paper we give another geometric condition in terms of Brunnian local moves. The proof is simple and self-contained.

$θ$ -curves inducing two different knots with the same $2$ -fold branched covering spaces

Soo Hwan Kim, Yangkok Kim (2003)

Bollettino dell'Unione Matematica Italiana

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For a knot $K$ with a strong inversion $i$ induced by an unknotting tunnel, we have a double covering projection $Π : S^{3} \to S^{3} / i$ branched over a trivial knot $Π (fix (i))$ , where $fix (i)$ is the axis of $i$ . Then a set $Π (fix (i) \cup K)$ is called a $θ$ -curve. We construct $θ$ -curves and the $Z_{2} \oplus Z_{2}$ cyclic branched coverings over $θ$ -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.

Knot theory programs KNOT2000 (K2K) + LinKnot(K2K)+LinKnot(K2KC)

S. Jablan, R. Sazdanovic (2003)

Visual Mathematics

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Minimizing the presentation of a knot group.

Dugopolski, Mark J. (1985)

International Journal of Mathematics and Mathematical Sciences

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Mp-small summands increase knot width.

Hendricks, Jacob (2004)

Algebraic & Geometric Topology

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Unknotting number and knot diagram.

Yasutaka Nakanishi (1996)

Revista Matemática de la Universidad Complutense de Madrid

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This note is a continuation of a former paper, where we have discussed the unknotting number of knots with respect to knot diagrams. We will show that for every minimum-crossing knot-diagram among all unknotting-number-one two-bridge knot there exist crossings whose exchange yields the trivial knot, if the third Tait conjecture is true.

A proof of Tait’s Conjecture on prime alternating $-$ achiral knots

Nicola Ermotti, Cam Van Quach Hongler, Claude Weber (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait’s Conjecture on alternating $-$ achiral knots: Let $K$ be a prime alternating $-$ achiral knot. Then there exists a minimal projection $Π$ of $K$ in $S^{2} \subset S^{3}$ and an involution $ϕ : S^{3} \to S^{3}$ such that: 1) $ϕ$ reverses the orientation of $S^{3}$ ; 2) $ϕ (S^{2}) = S^{2}$ ; 3) $ϕ (Π) = Π$ ; 4) $ϕ$ has two fixed points on $Π$ and hence...

Torus knots that cannot be untied by twisting.

Mohamed Ait Nouh, Akira Yasuhara (2001)

Revista Matemática Complutense

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We give a necessary condition for a torus knot to be untied by a single twisting. By using this result, we give infinitely many torus knots that cannot be untied by a single twisting.

Unsolved problems in virtual knot theory and combinatorial knot theory

Roger Fenn, Denis P. Ilyutko, Louis H. Kauffman, Vassily O. Manturov (2014)

Banach Center Publications

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This paper is a concise introduction to virtual knot theory, coupled with a list of research problems in this field.

Lissajous knots and billiard knots

Vaughan Jones, Józef Przytycki (1998)

Banach Center Publications

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We show that Lissajous knots are equivalent to billiard knots in a cube. We consider also knots in general 3-dimensional billiard tables. We analyse symmetry of knots in billiard tables and show in particular that the Alexander polynomial of a Lissajous knot is a square modulo 2.