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1.
We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numbers occur in the enumeration of fibered achiral and unknotting number one rational knots. Then we show how to enumerate rational knots of given crossing number depending on genus and/or signature. This allows to determine the asymptotical average value of these invariants among rational knots. We give also an application to the enumeration of lens spaces. 相似文献
2.
A. Stoimenow 《Expositiones Mathematicae》2010,28(2):133-178
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2. 相似文献
3.
A. Stoimenow 《Journal of Pure and Applied Algebra》2007,210(1):161-175
It is known that the linking form on the 2-cover of slice knots has a metabolizer. We show that several weaker conditions, or some other conditions related to sliceness, do not imply the existence of a metabolizer. We then show how the Rudolph-Bennequin inequality can be used indirectly to prove that some knots are not slice. 相似文献
4.
Alexander Stoimenow 《Journal of Geometry》2009,96(1-2):179-186
We use a refinement of an argument by Shinjo, and some study of the 3-strand Burau representation, to extend from knots to links her previous construction of infinite sequences of pairwise non-conjugate braids with the same closure of a non-minimal number of (and at least 4) strands. 相似文献
5.
Alexander Stoimenow 《Combinatorica》2016,36(5):557-589
We prove that a planar cubic cyclically 4-connected graph of odd χ < 0 is the dual of a 1-vertex triangulation of a closed orientable surface. We explain how this result is related to (and applied to prove at a separate place) a theorem about hyperbolic volume of links: the maximal volume of alternating links of given χ < 0 does not depend on the number of their components. 相似文献
6.
A. Stoimenow 《manuscripta mathematica》2003,110(2):203-236
We give inequalities for the coefficients and Mahler measure of the link polynomials in terms of the crossing number of a
link diagram.
Received: 15 January 2001 / Revised version: 1 August 2002 Published online: 24 January 2003
Supported by a DFG postdoc grant.
Mathematics Subject Classification (2000): Primary, Secondary 57M25 11B39, 11R04 相似文献
7.
A. Stoimenow 《Transactions of the American Mathematical Society》2002,354(10):3927-3954
We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular, we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots.
We give a counterexample to, and a corrected version of, a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply (from below) the braid index. A relation between (generalizations of) such examples and a conjecture of Jones that a minimal braid representation has unique writhe is discussed.
Finally, we give a counterexample to Morton's conjecture relating the genus and degree of the skein polynomial.
8.
A. Stoimenow 《Advances in Mathematics》2005,194(2):463-484
Tristram and Levine introduced a continuous family of signature invariants for knots. We show that any possible value of such an invariant is realized by a knot with given Vassiliev invariants of bounded degree. We also show that one can make a knot prime preserving Alexander polynomial and Vassiliev invariants of bounded degree. Finally, the Tristram-Levine signatures are applied to obtain a condition on (signed) unknotting number. 相似文献
9.
Alexander Stoimenow 《Comptes Rendus Mathematique》2009,347(13-14):809-811
We construct arbitrarily large (finite) families of hyperbolic non-mutant knots with equal colored Jones polynomial. To cite this article: A. Stoimenow, C. R. Acad. Sci. Paris, Ser. I 347 (2009). 相似文献
10.