Twist Sequences and Vassiliev Invariants

Trapp, Rolland

doi:10.1142/9789812796172_0011

Cited by 9 publications

(17 citation statements)

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“…But note that these are extended Vassiliev invariants. Thus, direct calculations as in [16,53] will not prove that they are not Vassiliev invariants. See also [8].…”

Section: Concluding Questions and Problemsmentioning

confidence: 99%

Application of Braiding Sequences. II. Polynomial Invariants of Positive Knots

Stoimenow¹

2016

Proceedings of the Edinburgh Mathematical Society

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We apply the concept of braiding sequences to link polynomials to show polynomial growth bounds on the derivatives of the Jones polynomial evaluated on S 1 and of the Brandt-Lickorish-MillettHo polynomial evaluated on [−2, 2] on alternating and positive knots of given genus. For positive links, boundedness criteria for the coefficients of the Jones, HOMFLY and Kauffman polynomials are derived. (This is a continuation of the paper 'Applications of braiding sequences. I': Commun. Contemp. Math. 12(5) (2010), 681-726.)

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“…But note that these are extended Vassiliev invariants. Thus, direct calculations as in [16,53] will not prove that they are not Vassiliev invariants. See also [8].…”

Section: Concluding Questions and Problemsmentioning

confidence: 99%

Application of Braiding Sequences. II. Polynomial Invariants of Positive Knots

Stoimenow¹

2016

Proceedings of the Edinburgh Mathematical Society

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“…A knot or link invariant v is called a Vassiliev invariant if v is a Vassiliev invariant of degree n for some nonnegative integer n. R. Trapp showed that a Vassiliev invariant of degree n has a polynomial growth of degree less than or equal to n on a twist sequence of knots ( [12]). In [8], Kauffman introduced the bracket polynomial of links by using state models.…”

Section: F In Ite T Y P E Invariants O F V Irtu Al K N Otsmentioning

confidence: 99%

Finite Type Invariants and the Kauffman Bracket Polynomials of Virtual Knots

Jeong¹,

Park²,

Yeo³

2014

Kyungpook mathematical journal

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A b st r a c t . In [9], Kauffman introduced virtual knot theory and generalized many classi cal knot invariants to virtual ones. For example, he extended the Jones polynomials Vx(t) of classical links to the /-polynomials fa {A) of virtual links by using bracket polynomials. In [4], M. Goussarov, M. Polyak and O. Viro introduced finite type invariants of virtual knots. In this paper, we give a necessary condition for a virtual knot invariant to be of finite type by using t(a\, ■ -, am/-sequences of virtual knots. Then we show that the higher derivatives (a) of the / -polynomial fx (A) of a virtual knot K at any point a are not of finite type unless n < 1 and a = 1.

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“…In an attempt to create an alternative to the (defects of the) classical approach and generalizing some ideas of Dean [12], Trapp [35] and Stanford [27], in [28] I introduced the notion of a braiding sequence. It offered a simple direct understanding of the behaviour of Vassiliev invariants on special knot classes, something, which was never worked out using the classical approach.…”

Section: Introductionmentioning

confidence: 99%

“…In latter case it is assumed to be evident from the preceeding discussion/references; else (and anyway) I'm grateful for any feedback. x jPj (see [28] and [35]), and this polynomial is called braiding polynomial of v on this braiding sequence.…”

Section: Introductionmentioning

confidence: 99%

“…Now for each such scheme/braiding sequence, the braiding polynomial of deg n is uniquely determined by its values on tuples (x 1 ; : : : ; x k ) with 0 x i ; ∑ k i=1 x i n (see remark below), and all braids corresponding to such (x i ) k i=1 have 11n crossings. To finish with, use the result of D. Welsh [38], as quoted in [1, p. 49], that the number of knots with n crossings is exponentially bounded (above) in n. x i n, can be observed from the multivariable Newton formula (see [35] and loc. cit.)…”

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confidence: 99%

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