S ep 2 00 9 From Goeritz matrices toquasi - alternating links
نویسنده
چکیده
Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow as to place the main topic in a historical context. For example, we mention that the Goeritz matrix was preceded by the Kirchhoff matrix of an electrical network. The network complexity extracted from the matrix corresponds to the determinant of a link. We assume basic knowledge of knot theory and graph theory, however, we offer a short introduction under the guise of a historical perspective.
منابع مشابه
Invariants of Knots and Links via Integral Matrices
This is a brief summary of the recent works joint with Dr. Sang Youl Lee, on the Seifert matrices and the (modi ed) Goeritz matrices of knots and links and their invariants: the Alexander polynomial, the Minkowski unit, the signature, the nullity, and the determinant of a knot and a link. We introduce the relationship between the modi ed Goeritz matrices of 2-peroidc links and those of their fa...
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An explicit expression for the numbers A(n, r; 3) describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, A(n, r; 3)'s are represented as 1-fold sums which can also be written in terms of terminating 4 F 3 series of argument 1/4.
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In [11], Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux [9] defined the key of an alternating sign matrix by recursively removing all −1’s in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Mor...
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In [12], Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all −1’s in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreove...
متن کاملar X iv : m at h - ph / 0 40 40 45 v 2 2 5 N ov 2 00 4 On the refined 3 - enumeration of alternating sign matrices
An explicit expression for the numbers A(n, r; 3) describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, A(n, r; 3)'s are represented as 1-fold sums which can also be written in terms of terminating 4 F 3 series of argument 1/4.
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