This paper contains three applications of the semiadequacy properties for the edge Jones polynomial coefficients. We will deal with (a) the construction of infinitely many positive braid links which admit no minimal string positive braid representative, (b) a construction of minimal crossing almost alternating diagrams for almost all crossing numbers, and (c) the construction of infinitely many homogeneous links which are not alternative.
A famous theorem of Whitney asserts that spherical embeddings of any 2-connected 3-valent graph are interconvertible by flips. It is applied to characterize alternating and positive links with planar Seifert surfaces. Then we show that cellular embeddings of any 2-connected 3-valent planar graph G on any orientable compact surface S&neq;S>sup>2 are counterexamples to Whitney's theorem. Using this, we study special types of volume-maximizing sequences among links of given canonical Euler characteristic. We describe their maximal volume in terms of links associated to planar 3-connected 3-valent graphs. We investigate the relation between the volume of such links and the slN weight system of their graphs, coming from the theory of Vassiliev invariants.
Keywords: hyperbolic volume, triangulation, planar embedding, graph, link, Seifert surface, weight system.
We prove that if L is a two-bridge link and n≥ max(4,b(L)), then L admits infinitely many non-conjugate n-braid representatives. More precisely, it admits an exchangeable braid, all whose iterated positive exchanges are pairwise non-conjugate. We give similar results for some alternating pretzel links and alternating knots of given genus. The proofs exploit a connection between algebraic properties of the exchange move and combinatorial features of Seifert circles in link diagrams, which originates from the Vogel move. Another ingredient is the completion of the algebraic non-conjugacy result for iterated exchanged braids.
Keywords: braid group, exchange move, braid index, Seifert circle, alternating link, flype
We develop a method based on the Burau matrix to detect conditions on the linking num- bers of braid strands. Our main application is to iterated exchanged braids. Unless the braid permutation fixes both braid edge strands, we establish under some fairly generic conditions on the linking numbers a “subsymmetry” property; in particular at most two such braids can be mutually conjugate. As an ad- dition, we prove that the Burau kernel is contained in the commutator subgroup of the pure braid group. We discuss also some properties of the Burau image.
Keywords: exchange move, braid group, conjugacy, Burau matrix, linking number
Adapting the graph index to be used for braided surfaces, we gain realization results on string numbers of Bennequin surfaces. In particular, all fibered alternating links and pure alternating pretzel links have a minimal string Bennequin surface (as can be proved also for many cables of such knots). We also obtain a visual test for strong quasipositivity, which we call pseudo-positivity, and use it to classify strongly quasipositive pretzel knots.
Keywords: braid group, strongly quasipositive link, cable link, alternating link, Bennequin surface, positive link, pretzel link, genus, braid index, graph, alternative link
We prove that any two given links can be combined to give a strongly quasipositive link. This in particular implies that any link is a sublink of a strongly quasipositive link. We discuss also some complexity issues of the strongly quasipositive link constructed.
Keywords: braid group, strongly quasipositive link, alternating link, Bennequin surface, positive link
We obtain some fairly general conditions on the linking numbers and geometric properties of a link, under which it has infinitely many conjugacy classes of braid representatives if and only if it has one admitting an exchange move. We investigate a symmetry pattern of indices of conjugate iterated exchanged braids. We then develop a test based on the Burau matrix showing examples of knots admitting no minimal exchangeable braids, admitting non-minimal non-exchangeable braids, and admitting both minimal exchangeable and minimal non-exchangeable braids. This in particular proves that conjugacy, exchange moves and destabilization do not suffice to simplify braid representatives of a general link.
Keywords: exchange move, braid group, conjugacy, Burau matrix, Jones polynomial, hyperbolic link
We prove that under a fairly general condition (that the edge strands are not fixed by the braid permutation) an iterated exchange move gives infinitely many non-conjugate braid representatives of links. More precisely, almost all braids obtained by iterated positive exchange moves are pairwise non-conjugate. As a consequence, every link with no trivial components has infinitely many conjugacy classes of braid representatives if and only if it has one admitting an exchange move.
Keywords: exchange move, braid group, link, conjugacy, Conway polynomial.
We use the Chudnovski-Seymour Real Root Theorem for independence polynomials to obtain some statements about the coefficients and roots of the Alexander and Conway polynomial of some types of plumbing links, partially recovering and extending some recent results of Hirasawa-Murasugi, and addressing conjectures of Fox, Hoste and Liechti.
Keywords: polynomial root, alternating knot, Alexander polynomial, arboresecent link, 2-bridge link
Applying the concept of braiding sequences and the inequality between the signature and number of roots of the Alexander polynomial on the unit circle, we prove that only finitely many special alternating knots are (even algebraically) concordant, in that their concordance class determines their Alexander polynomial. We discuss some extensions of this result to positive and almost positive knots, and links.
Keywords: special alternating knot, positive knot, genus, Alexander polynomial, concordance, polynomial root
We prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies . This relates to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots. We extend our result to properties of zeros for Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links.
Keywords: alternating knot, Alexander polynomial, skein polynomial, rational link, 3-braid link, Montesinos knot, polynomial root, genus
A link diagram is said to be (orientedly) everywhere equivalent if all the diagrams obtained by switching one crossing represent the same (oriented) link. We classify such diagrams of two components.
Keywords: alternating link, Jones polynomial, Kauffman bracket, flype
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.
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 show that some knots are not slice.
It is known that the Brandt-Lickorish-Millett-Ho polynomial Q contains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable from Q is an open problem. We show that this is not so up to degree $\le 9$. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials, which are not mutants. Our calculations provide evidence to a negative answer to the question whether Vassiliev knot invariants of degree up to 10 are determined by the HOMFLY and Kauffman polynomial and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.
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.
Keywords: unknotting number, Vassiliev invariant, signature, Alexander polynomial.
We show that the coefficients of the Conway polynomial of special alternating links satisfy certain inequalities.
Keywords: Conway polynomial, special alternating link, signature, irreducible polynomial, discriminant.
Bennequin showed that any link of braid index 3 has a minimal genus Seifert surface placed naturally on a 3-braid, and Rudolph that any link has such a surface for some, not necessarily minimal, braid representation. With explicit examples, we close the gap between both results and show that Bennequin's theorem does not extend to knots of braid index 4.
We use a recent method of Sundberg-Thistlethwaite to improve Welsh's upper bound on the rate of growth of the number of links of given crossing number and to show that the number of polynomials of alternating links drops off exponentially in the crossing number compared to the number of such links.
We show how the Alexander/Conway link polynomial occurs in the context of planar even valence graphs, refining the notion of the number of their spanning trees. Then we apply knot theory to deduce several statements about this graph polynomial, in particular estimates for its coefficients and relations between congruences of the number of vertices and number of spanning trees of the graph.
We prove for rational knots a conjecture of Adams et al. that an alternating unknotting number one knot has an alternating unknotting number one diagram. We use this then to show a refined signed version of the Kanenobu--Murakami theorem on unknotting number one rational knots. Together with a similar refinement of the linking form condition of Montesinos-Lickorish and the HOMFLY polynomial, we prove a condition a knot to be -trivadjacent, improving the previously known condition on the degree-2-Vassiliev invariant. We finally show several partial cases of the conjecture that the knots with everywhere -trivial knot diagrams are exactly the trivial, trefoil and figure eight knots. (A knot diagram is called everywhere -trivial, if it turns into an unknot diagram by switching any set of of its crossings.)
Keywords: knot diagram, crossing change, unknotting number, trivadjacent knots, HOMFLY polynomial.
We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Lickorish-Millett value of the Jones polynomial to show that any achiral knot with determinant divisible by 3 does not have unknotting number one, and Jones' value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 816, 949 and 6 further new entries in Kawauchi's tables, confirming about 20 others. Another method, recovering most of these results, is developed by applying and extending the linking form criterion of Montesinos and Lickorish. This leads to several conjectured relations between the Jones value of Q and the linking form.
Keywords: Jones polynomial, Goeritz matrix, double branched cover, linking form, Brandt-Lickorish-Millett-Ho polynomial, unknotting number.
We prove inequalities for the determinant of alternating links in terms of their hyperbolic volume, conjectured non-rigorously (here) by Dunfield.
Keywords: alternating knots, determinant, spanning tree, hyperbolic volume.
We prove that any non-trivial primitive Vassiliev invariant expressible as a polynomial in the coefficients of the Conway polynomial vanishes only on finitely many twist knots.
We classify all knot diagrams of genus two and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic volume for weak genus two knots. We also study the values of the link polynomials at roots of unity, extending denseness results of Jones. Using these values, examples of knots with unsharp Morton (weak genus) inequality are found.
Keywords: genus, Seifert algorithm, alternating knots, positive knots, unknot diagrams, homogeneous knots, Jones, Brandt-Lickorish-Millett-Ho and HOMFLY polynomial, 3-move conjecture, hyperbolic volume.
We collect some examples showing that some Vassiliev invariants are not obtainable from the HOMFLY and Kauffman polynomials in the real sense, namely, that they distinguish knots not distinguishable by the HOMFLY and/or Kauffman polynomial.
Keywords: Vassiliev invariants, HOMFLY polynomial, Kauffman polynomial, chirality.
We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.
Keywords: positive knots, Seifert surfaces, Gauss diagrams, genus, unknotting number, alternating knots, Vassiliev invariants.
We give an example showing that the Fiedler solid torus degree 3 Gauss sum invariants can be used to detect mutation of links.
Using the Fiedler-Polyak-Viro Gauss diagram formulas we study the Vassiliev invariants of degree and math>3 on almost positive knots. As a consequence we show that the number of almost positive knots of given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, \em{inter alia} to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property .
We prove a relation in the algebra of the Jones Vassiliev invariants, giving a new restriction to the values of the Jones polynomial on knots, and the non-existence of another family of such relations.
We prove, that Jones polynomials of positive knots have positive degree and extend this result to -almost positive knots.
We prove that if a positive knot is alternating, then all its alternating diagrams are positive.
Introducing a way to modify knots using -trivial rational tangles, we show that knots with given values of Vassiliev invariants of bounded degree can have arbitrary unknotting number (extending a recent result of Ohyama, Taniyama and Yamada). The same result is shown for 4-genera and finite reductions of the homology group of the double branched cover. Closer consideration is given to rational knots, where it is shown that the number of -trivial rational knots of at most crossings is for any asymptotically at least for any .
Using the recent Gauss diagram formulas for Vassiliev invariants of Polyak-Viro-Fiedler and combining these formulas with the Bennequin inequality, we prove several inequalities for positive knots relating their Vassiliev invariants, genus and degrees of the Jones polynomial. As a consequence, we prove that for any of the polynomials of Alexander/Conway, Jones, HOMFLY, Brandt-Lickorish-Millett-Ho and Kauffman there are only finitely many positive knots with the same polynomial and no positive knot with trivial polynomial.
We also discuss an extension of the Bennequin inequality, showing that the unknotting number of a positive knot is not less than its genus, which recovers some recent unknotting number results of A'Campo, Kawamura and Tanaka, and give applications to the Jones polynomial of a positive knot.
Keywords: k-equivalence, braids, alternating knots, invertibility, chirality, mutation, Vassiliev invariants, closed 3-braids, Gauß sums, n-trivial knots, Jones polynomial, algebraic knots, positive knots and braids.
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