GOUSSAROV-POLYAK-VIRO CONJECTURE FOR DEGREE THREE CASE

作     者：Ito, Noboru Kotorii, Yuka Takamura, Masashi 

作者机构：Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo153-8914 Japan Mathematics Program Graduate school of Advanced Science and Engineering Hiroshima University 1-7-1 Kagamiyama Higashi-hiroshima City Hiroshima739-8521 Japan  1-4-1 Nihonbashi Chuo-ku Tokyo103-0027 Japan  RIKEN 2-1 Hirosawa Wako Saitama351-0198 Japan School of Social Informatics Aoyama Gakuin University 5-10-1 Fuchinobe Chuo-ku Kanagawa-ken Sagamihara-shi252-5258 Japan 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2019年

核心收录：

主　　题：Gaussian distribution 

摘      要：Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas have been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 23 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak-Viro Gauss diagram formula [19] is not a long virtual knot invariant;however, it is included in the list of 23 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [8, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). Copyright © 2019, The Authors. All rights reserved.