State invariants of 2-bridge knots

2018 ◽  
Vol 27 (02) ◽  
pp. 1850016
Author(s):  
Cynthia L. Curtis ◽  
Vincent Longo
Keyword(s):  
Alexander Polynomial ◽  
Bilinear Forms ◽  
Essential Surfaces ◽  
Essential State ◽  
Boundary Slopes

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally, we realize the boundary slopes of essential surfaces as differences of signatures of the knot.

A slope conjecture for links

2015 ◽  
Vol 24 (14) ◽  
pp. 1550077
Author(s):  
R. van der Veen
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Essential Surfaces ◽  
Torus Links ◽  
Boundary Slopes

The slope conjecture [S. Garoufalidis, The degree of a q-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011) 4–27] gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this paper, we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating or more generally adequate links. We also verify the conjecture for torus links.

scholarly journals The slope conjecture for Montesinos knots

2020 ◽  
Vol 31 (07) ◽  
pp. 2050056 ◽  
Author(s):  
Stavros Garoufalidis ◽  
Christine Ruey Shan Lee ◽  
Roland van der Veen
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Essential Surfaces ◽  
Sum Formula ◽  
Boundary Slopes

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

scholarly journals A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus

2009 ◽  
Vol 147 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Tao Li ◽  
Ruifeng Qiu ◽  
Shicheng Wang
Keyword(s):  
Essential Surfaces ◽  
Boundary Slopes

scholarly journals Repeated boundary slopes for 2-bridge knots

2015 ◽  
Vol 24 (14) ◽  
pp. 1550068 ◽  
Author(s):  
Cynthia L. Curtis ◽  
William Franczak ◽  
Randoplh J. Leiser ◽  
Ryan J. Manheimer
Keyword(s):  
Infinite Family ◽  
Essential Surfaces ◽  
Branched Surfaces ◽  
Boundary Slopes

We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

Bilinear forms on non-homogeneous Sobolev spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 995-1026
Author(s):  
Carme Cascante ◽  
Joaquín M. Ortega
Keyword(s):  
Sobolev Spaces ◽  
Bilinear Form ◽  
Positive Measure ◽  
Bilinear Forms ◽  
Homogeneous Sobolev Spaces

AbstractIn this paper, we show that if {b\in L^{2}(\mathbb{R}^{n})}, then the bilinear form defined on the product of the non-homogeneous Sobolev spaces {H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})}, {0<s<1}, by(f,g)\in H_{s}^{2}(\mathbb{R}^{n})\times H_{s}^{2}(\mathbb{R}^{n})\to\int_{% \mathbb{R}^{n}}(\mathrm{Id}-\Delta)^{\frac{s}{2}}(fg)(\mathbf{x})b(\mathbf{x})% \mathop{}\!d\mathbf{x}is continuous if and only if the positive measure {\lvert b(\mathbf{x})\rvert^{2}\mathop{}\!d\mathbf{x}} is a trace measure for {H_{s}^{2}(\mathbb{R}^{n})}.

THIN POSITION FOR TANGLES

2003 ◽  
Vol 12 (01) ◽  
pp. 117-122
Author(s):  
DAVID BACHMAN ◽  
SAUL SCHLEIMER
Keyword(s):  
Essential Surfaces ◽  
Thin Position

If a tangle, [Formula: see text] , has no planar, meridional, essential surfaces in its exterior then thin position for K has no thin levels.

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