A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.
Survey of Turk's head knots and links properties.
The minimum number of colors is a challenging knot invariant since, by definition, its calculation requires taking the minimum over infinitely many minima. In this article we estimate and in some cases calculate the minimum number of colors for the Turk's head knots on three strands.
Study knots with large character varieties.
We compute the Kauffman bracket polynomial of the three-lead Turk's head, the chain sinnet and the figure-eight chain shadow diagrams. Each of these knots can in fact be constructed by repeatedly concatenating the same 3-tangle, respectively, then taking the closure. The bracket is then evaluated by expressing the stat…
The m,n Turk's Head Knot, THK(m,n), is an "alternating (m,n) torus knot." We prove the Harary-Kauffman conjecture for all THK(m,n) except for the case where m \geq 5 is odd and n \geq 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that non…
Research classifies knots based on sliceness and amphichirality.
We derive a factorization of the Alexander polynomial of the 4-strand Turk's head knot using hypergeometric representations.
The paper calculates minimal ribbonlength for various knots.
We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.
A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a previous paper, it was proved that every knot and link has a quadruple crossing…
New findings on Jones polynomial for 4-strand braids.
In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.
We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We further show that the head and tail functions only depend on the reduced checkerbo…
We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the …
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques from skein theory.
The use of synthetic data generated by Generative Adversarial Networks (GANs) has become quite a popular method to do data augmentation for many applications. While practitioners celebrate this as an economical way to get more synthetic data that can be used to train downstream classifiers, it is not clear that they re…
Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four-strand braids. Incorporating the properties of four-point con…
We show that the Artin pure braid group on at least four strands is not residually free. Our results also show that the pure braid group on at least three strands has corank two.
The colored Jones polynomial is a series of one variable Laurent polynomials J(K,n) associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of J(K,n) are independent of n when n is sufficiently large. Computation of sample…
Building machine translation (MT) test sets is a relatively expensive task. As MT becomes increasingly desired for more and more language pairs and more and more domains, it becomes necessary to build test sets for each case. In this paper, we investigate using Amazon's Mechanical Turk (MTurk) to make MT test sets chea…
In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston norms of spanning surfaces for the individual link components, and we explicitly…
GWHD dataset offers 4,700 high-res images of wheat heads.
Multi-head attention outperforms single-head in in-context linear regression tasks.
The Kinoshita graph is the most famous example of a Brunnian theta graph, a nontrivial spatial theta graph with the property that removing any edge yields an unknot. We produce a new family of diagrams of spatial theta graphs with the property that removing any edge results in the unknot. The family is parameterized by…
New theory shows how multi-head attention reduces variance and decorrelates outputs.
Geometric theory of projection heads in self-supervised learning.
When might human input help (or not) when assessing risk in fairness domains? Dressel and Farid (2018) asked Mechanical Turk workers to evaluate a subset of defendants in the ProPublica COMPAS data for risk of recidivism, and concluded that COMPAS predictions were no more accurate or fair than predictions made by human…
Improved language models with talking-heads attention.
New approach improves multi-head attention by making heads less similar.
Paper introduces regularization for multi-head attention to spot keywords.
Deep forecasting models show output heads significantly improve performance on fat-tailed financial returns.
Study estimates 163 million online freelancers globally.
We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a…
Attention mechanisms have become ubiquitous in NLP. Recent architectures, notably the Transformer, learn powerful context-aware word representations through layered, multi-headed attention. The multiple heads learn diverse types of word relationships. However, with standard softmax attention, all attention heads are de…
One-layer transformers can't solve induction heads task efficiently.
This work proposes a collaborative multi-head attention layer to reduce model size without sacrificing accuracy.
Multi-headed ensembles boost model performance with faster training.
New approach reduces model size for Transformer architectures.
Study on multi-head softmax attention dynamics for in-context learning.
Transformer-MGK replaces redundant heads with Gaussian key mixtures, improving efficiency and performance.
Minimalistic model captures head direction system properties.
Develops a mean-field theory for multi-head self-attention under cross-entropy training.
Investigates the benefits of multi-head attention in Transformers, deriving convergence and generalization guarantees.
Proposes a new neural head for asymmetric representation learning.
New deep learning model generates accurate personalized human head models for electromagnetic dosimetry.
Investigates optimal parameter allocation in Transformers for efficiency and expressivity.
We demystify attention patterns in multi-head softmax models for linear data.