Classification of simple knots by Blanchfield duality

Abstract. For a differentiate knot, i.e. an imbedding S" c S"*2, one can associate a sequence of modules [Ag] over the ring Z[t, /"'], which are the source of many classical knot invariants. If X is the complement of die knot, and X -» X the canonical infinite cyclic covering, then Aq = // (JQ.In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of A,.In classical knot theory there are many "abelian" invariants which have proved useful in distinguishing knots, e.g. knot-polynomials, "elementary" ideals, homology and linking pairings in the finite cyclic branched coverings, ideal classes (see [F], [FS]). It is known (see [T]) that these invariants can all be extracted from a certain module A over the ring A = Z[t, t~l] and a "Hermitian" pairing on A taking values in Q (A)/A (Q(A) is the quotient field of A). The construction of A and < , > carries over to higher-dimensional knots and, in certain cases, are enough to classify the knot up to isotopy (see [LI [TI], [K]).In general, there is a finite collection A,,...,A" of such modules associated to an «-dimensional knot in (n + 2)-space, and < , ) exists on Ak when n = 2k -I. Our first purpose in this work will be to give an algebraic characterization of these objects. There is already a great deal known in this direction (see [K], [Ke], [L3], [G]). Our results, which will be complete except for some problems with the Z-torsion part of A,, will extend and reformulate these known results. For this purpose we will find it necessary to define a new pairing [, ] in the Z-torsion part of Ak, when n -2k.In the second part of this work, we will make an algebraic study of the modules and forms which have arisen from Part I. Our approach is to consider new modules and forms, derived from the original ones, over rings with a good structure theory: polynomial rings over fields, and rings of algebraic integers. The structure theory then classifies the derived object via invariants in these rings. These invariants include-most of the "classical" knot License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2 JEROME LEVINE invariants. Our main results will characterize in many cases the invariants which can arise from knots of a given dimension. For example, the rational knot modules (i.e. Ak® Q) with their product structure can often be completely characterized-this has consequences for knot cobordism realizability [L4]. In addition many integral invariants appear-including the ideal class invariants of [FS], but also many new ones-and are characterized.Some of the work in this paper in Part I is a redoing of known results, referred to above, in an effort to give a more unified and simple presentation of the entire subject. For example, we have been able to give a "coordinate free" formulation of some of the results of [Ke] and [G]. The middle-dimensional realization results of [K] are also given a different proof which has obvious applications to construction of m...

show abstract

Section: If a G A Has Finite Order Thenmentioning

confidence: 97%

Knot modules. I

Levine¹

1977

Trans. Amer. Math. Soc.

103

View full text Add to dashboard Cite

show abstract

“…(77 has a presentation with n generators and n -1 relations.) Since Hx(m') sa HX(P2) is a module of type K,by Proposition (8.2), this chain complex will become exact after ®A ß(A)-by (1.3). Therefore, by considerations of rank, d2 must be a monomorphism and H2 (P2) is free, we have the free resolution 0 -» C^P^ -» Ker dx -» H^P-^ -» 0 and, therefore, by Proposition (3.5), HX(P2) = Hx(m') is Z-torsion free.…”

Section: Consider the Commutative Diagrammentioning

confidence: 97%

Untitled

1977

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For a differentiate knot, i.e. an imbedding S" c S"*2, one can associate a sequence of modules [Ag] over the ring Z[t, /"'], which are the source of many classical knot invariants. If X is the complement of die knot, and X -» X the canonical infinite cyclic covering, then Aq = // (JQ.In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of A,.In classical knot theory there are many "abelian" invariants which have proved useful in distinguishing knots, e.g. knot-polynomials, "elementary" ideals, homology and linking pairings in the finite cyclic branched coverings, ideal classes (see [F], [FS]). It is known (see [T]) that these invariants can all be extracted from a certain module A over the ring A = Z[t, t~l] and a "Hermitian" pairing on A taking values in Q (A)/A (Q(A) is the quotient field of A). The construction of A and < , > carries over to higher-dimensional knots and, in certain cases, are enough to classify the knot up to isotopy (see [LI [TI], [K]).In general, there is a finite collection A,,...,A" of such modules associated to an «-dimensional knot in (n + 2)-space, and < , ) exists on Ak when n = 2k -I. Our first purpose in this work will be to give an algebraic characterization of these objects. There is already a great deal known in this direction (see [K], [Ke], [L3], [G]). Our results, which will be complete except for some problems with the Z-torsion part of A,, will extend and reformulate these known results. For this purpose we will find it necessary to define a new pairing [, ] in the Z-torsion part of Ak, when n -2k.In the second part of this work, we will make an algebraic study of the modules and forms which have arisen from Part I. Our approach is to consider new modules and forms, derived from the original ones, over rings with a good structure theory: polynomial rings over fields, and rings of algebraic integers. The structure theory then classifies the derived object via invariants in these rings. These invariants include-most of the "classical" knot License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2 JEROME LEVINE invariants. Our main results will characterize in many cases the invariants which can arise from knots of a given dimension. For example, the rational knot modules (i.e. Ak® Q) with their product structure can often be completely characterized-this has consequences for knot cobordism realizability [L4]. In addition many integral invariants appear-including the ideal class invariants of [FS], but also many new ones-and are characterized.Some of the work in this paper in Part I is a redoing of known results, referred to above, in an effort to give a more unified and simple presentation of the entire subject. For example, we have been able to give a "coordinate free" formulation of some of the results of [Ke] and [G]. The middle-dimensional realization results of [K] are also given a different proof which has obvious applications to construction of mo...

show abstract

“…First recall from [8] that knot modules have been characterised. Using this, we can deduce from [6, pp.…”

Section: Non-additivity Of the Nakanishi Indexmentioning

confidence: 99%