On the skeins in the annulus and applications to invariants of 3-manifolds

Kawagoe, Kenichi

doi:10.1142/s0218216598000127

Cited by 9 publications

(18 citation statements)

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“…This is a special case of Turaev's result [11] that C is generated freely as a commutative algebra by X The skein of the annulus C ′ with two boundary points has been considered e.g. in [6] and [4]. We are using the version introduced by Morton in [10] where the two boundary points lie on different boundary components of the annulus.…”

Section: Aiston and Mortonmentioning

confidence: 99%

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Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Lukac¹

2005

Math. Proc. Camb. Phil. Soc.

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The Hecke algebra H n contains well known idempotents E λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja [5].A skein theoretical description of E λ was given by Aiston and Morton [2]. The closure of E λ becomes an element Q λ of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s λ as stated in theorem 8.2. But previous proofs of this fact as in [1] used results about quantum groups which were far beyond the scope of skein theory.Our elementary proof was motivated by [6] and uses only skein theory and basic algebra. The skein of a planar surfaceWe consider a planar surface F with designated n incoming and n outgoing boundary points for some integer n ≥ 0. The Homfly skein S(F ) is defined as the module of linear combinations of oriented tangles in F quotiented by regular isotopy (i.e. Reidemeister moves II and III), the two local skein relations in figure 1 where v and s are variables, and the relation that a disjoint simple closed curve can be removed from a diagram at the expense of multiplication with the scalar δ =

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Section: Aiston and Mortonmentioning

confidence: 99%

“…In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s λ as stated in theorem 8.2. But previous proofs of this fact as in [1] used results about quantum groups which were far beyond the scope of skein theory.Our elementary proof was motivated by [6] and uses only skein theory and basic algebra. …”

mentioning

confidence: 99%

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Lukac¹

2005

Math. Proc. Camb. Phil. Soc.

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“…Taking x = s, q = s 2 , v = 1 we get the common algebraist's version H n (q) of [4], while the endomorphism rings arising from the fundamental representation of sl(N) q and its standard R-matrix correspond to the choice x = s −1/N , v = s −N with s = e h/2 . The choice x = v gives the Homfly skein relation which is invariant under Reidemeister move I, and x = q r , s = q, v = a gives the version used by Kawagoe in [9].…”

Section: Variants Of the Hecke Algebrasmentioning

confidence: 99%

“…Kawagoe [9] and other authors have used a skein which is linearly isomorphic to A, based on the annulus with input and output on the same component. The significant advantage of A lies in its algebraic properties.…”

Section: Algebraic Properties Of Amentioning

confidence: 99%

Skein Theory and the Murphy Operators

Morton

2002

J. Knot Theory Ramifications

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The Murphy operators in the Hecke algebra H n of type A are explicit commuting elements whose sum generates the centre. They can be represented by simple tangles in the Homfly skein theory version of H n . In this paper I present a single tangle which represents their sum, and which is obviously central. As a consequence it is possible to identify a natural basis for the Homfly skein of the annulus, C.Symmetric functions of the Murphy operators are also central in H n . I define geometrically a homomorphism from C to the centre of each algebra H n , and find an element in C, independent of n, whose image is the mth power sum of the Murphy operators. Generating function techniques are used to describe images of other elements of C in terms of the Murphy operators, and to demonstrate relations among other natural skein elements.

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“…The work in [6,10] shows how to describe Q λ as the determinant of a matrix with entries drawn from a sequence of elements h n in C. In interpretations of C + as the representation ring of the quantum groups sl(N) q the elements h n correspond to the irreducible representations indexed by partitions with a single part n, and the determinants used give the irreducible representation corresponding to the partition λ in terms of the Jacobi-Trudy formula, [12, I(3.4)]. …”

Section: Introductionmentioning

confidence: 99%

A basis for the full Homfly skein of the annulus

Hadji¹,

Morton²

2006

Math. Proc. Camb. Phil. Soc.

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A basis, denoted {Q λ,µ }, for the full Homfly skein of the annulus C was introduced in [18], where λ and µ are partitions of integers n and p into k and k * parts respectively. The basis consists of eigenvectors of the two meridian maps on C; these maps are the linear endomorphisms of C induced by the insertion of a meridian loop with either orientation around a diagram in the annulus.In this paper we give an explicit expression for each Q λ,µ as the determinant of a (k * + k) × (k * + k) matrix whose entries are simple elements h n , h * n in the skein C. In the case p = 0 (µ = φ) the determinant gives the Jacobi-Trudy formula for the Schur function s λ of N variables as a polynomial in the complete symmetric functions h n of the variables, [12]. The Jacobi-Trudy determinants have previously been used by Kawagoe [6] and Lukac [10] in discussing the elements in the skein of the annulus represented by closed braids in which all strings are oriented in the same direction. Our results and techniques here form a natural extension of the work of Lukac.

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On the skeins in the annulus and applications to invariants of 3-manifolds

Abstract: In this paper, we relate Schur functions and a linear skein of annulus derived from the Homfly polynomail. Using this relations, we define topological invariants of 3-manifolds from the Homfly polynomial.

Cited by 9 publications

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Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Idempotents of the Hecke algebra become Schur functions in the skein of the annulus

Skein Theory and the Murphy Operators

A basis for the full Homfly skein of the annulus

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