On Numerical Invariant of Graph

2021 ◽  
Vol 19 (1) ◽  
pp. 57-75
Author(s):  
Rohit Μ. Patne ◽  
Gajanan R. Avachar
Keyword(s):  
Numerical Invariant

PROPERTIES OF VARIETIES DETERMINED BY THE DEGREES OF PROPER HYPERSUBSTITUTIONS

2008 ◽  
Vol 01 (01) ◽  
pp. 53-68 ◽  
Author(s):  
Klaus Denecke ◽  
Rattana Srithus
Keyword(s):  
Positive Integer ◽  
Numerical Invariant

The degree of proper hypersubstitutions is a numerical invariant of a variety. The main problem related to the degree of proper hypersubstitutions is to characterize all varieties of a given type τ which have, for a given positive integer n, this n as degree of proper hypersubstitutions. In this paper we answer to this question for solid varieties and certain integers as degrees and then for not necessarily solid varieties of type τ = (n) satisfying some additional conditions.

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

KAWACHI'S INVARIANT FOR NORMAL SURFACE SINGULARITIES

1998 ◽  
Vol 09 (05) ◽  
pp. 623-640 ◽  
Author(s):  
VLADIMIR MAŞEK
Keyword(s):  
Linear Systems ◽  
A Priori ◽  
Normal Surface ◽  
Numerical Invariant ◽  
Normal Surfaces ◽  
Log Canonical ◽  
Surface Singularities ◽  
Canonical Surface ◽  
Quick Proof ◽  
Canonical Singularities

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic Gorenstein or rational (without assuming a priori that they are ℚ-Gorenstein). In Sec. 2 we prove effective results (stated in terms of Kawachi's invariant) regarding global generation of adjoint linear systems on normal surfaces with boundary. Such results can be used in proving effective estimates for global generation on singular threefolds. The theorem of Ein–Lazarsfeld and Kawamata, which says that the minimal center of log-canonical singularities is always normal, explains why the results proved here are relevant in that situation.

scholarly journals Monotonic invariants under blowups

2020 ◽  
Vol 31 (12) ◽  
pp. 2050093
Author(s):  
Zhenjian Wang
Keyword(s):  
General Framework ◽  
Plane Curve ◽  
Curve Singularity ◽  
Numerical Invariant ◽  
Plane Curve Singularity ◽  
New Perspective

We prove that the numerical invariant [Formula: see text] of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by Dimca and Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

scholarly journals Valuative stability of polarised varieties

2022 ◽  
Author(s):  
Ruadhaí Dervan ◽  
Eveline Legendre
Keyword(s):  
Recent Progress ◽  
Fano Variety ◽  
Numerical Invariant

AbstractFujita and Li have given a characterisation of K-stability of a Fano variety in terms of quantities associated to valuations, which has been essential to all recent progress in the area. We introduce a notion of valuative stability for arbitrary polarised varieties, and show that it is equivalent to K-stability with respect to test configurations with integral central fibre. The numerical invariant governing valuative stability is modelled on Fujita’s $$\beta $$ β -invariant, but includes a term involving the derivative of the volume. We give several examples of valuatively stable and unstable varieties, including the toric case. We also discuss the role that the $$\delta $$ δ -invariant plays in the study of valuative stability and K-stability of polarised varieties.

An affine index polynomial invariant and the forbidden move of virtual knots

2016 ◽  
Vol 25 (07) ◽  
pp. 1650040 ◽  
Author(s):  
Migiwa Sakurai
Keyword(s):  
Polynomial Invariant ◽  
Numerical Invariant ◽  
Virtual Knots ◽  
The Difference

Kauffman defines an affine index polynomial invariant for virtual knots. The invariant is induced from a numerical invariant called an [Formula: see text]-writhe. In this paper, we provide the difference of the values obtained from invariants between two virtual knots which can be transformed into each other by a single forbidden move. As a result, we make it possible for many virtual knots to determine the unknotting numbers by forbidden moves.

scholarly journals CROSSING NUMBER OF LINKS FORMED BY EDGES OF A TRIANGULATION

2003 ◽  
Vol 12 (03) ◽  
pp. 281-286 ◽  
Author(s):  
SIMON A. KING
Keyword(s):  
Exponential Function ◽  
Upper Bound ◽  
Crossing Number ◽  
Numerical Invariant

We study the crossing number of links that are formed by edges of a triangulation [Formula: see text] of S3 with n tetrahedra. We show that the crossing number is bounded from above by an exponential function of n2. In general, this bound can not be replaced by a subexponential bound. However, if [Formula: see text] is polytopal (resp. shellable) then there is a quadratic (resp. biquadratic) upper bound in n for the crossing number. In our proof, we use a numerical invariant [Formula: see text], called polytopality, introduced by the author.

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