Vassiliev Invariants of Framed Knots

In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k−1.

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Vassiliev invariants and a strange identity related to the Dedekind eta-function

Topology ◽

10.1016/s0040-9383(00)00005-7 ◽

2001 ◽

Vol 40 (5) ◽

pp. 945-960 ◽

Cited By ~ 89

Author(s):

Don Zagier

Keyword(s):

Dedekind Eta Function ◽

Vassiliev Invariants

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SATANIC AND THELEMIC CIRCLES ON KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821651350017x ◽

2013 ◽

Vol 22 (05) ◽

pp. 1350017 ◽

Cited By ~ 1

Author(s):

G. FLOWERS

Keyword(s):

Geometric Interpretation ◽

Second Order ◽

Geometric Properties ◽

Vassiliev Invariants ◽

Vassiliev Invariant ◽

Knot Diagrams

While Vassiliev invariants have proved to be a useful tool in the classification of knots, they are frequently defined through knot diagrams, and fail to illuminate any significant geometric properties the knots themselves may possess. Here, we provide a geometric interpretation of the second-order Vassiliev invariant by examining five-point cocircularities of knots, extending some of the results obtained in [R. Budney, J. Conant, K. P. Scannell and D. Sinha, New perspectives on self-linking, Adv. Math. 191(1) (2005) 78–113]. Additionally, an analysis on the behavior of other cocircularities on knots is given.

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Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

Topology ◽

10.1016/s0040-9383(01)00039-8 ◽

2003 ◽

Vol 42 (1) ◽

pp. 1-33 ◽

Cited By ~ 6

Author(s):

Vladimir Tchernov

Keyword(s):

Vassiliev Invariants

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SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216507005816 ◽

2007 ◽

Vol 16 (10) ◽

pp. 1295-1329

Author(s):

E. KALFAGIANNI ◽

XIAO-SONG LIN

Keyword(s):

Fundamental Group ◽

Lower Central Series ◽

Central Series ◽

Seifert Surface ◽

Vassiliev Invariants ◽

Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

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