Markov Trace Research Articles

The Kauffman bracket skein module of the lens spaces via unoriented braids

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces [Formula: see text], KBSM([Formula: see text]), for [Formula: see text]. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, [Formula: see text], which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, [Formula: see text], for knots and links in ST, via a unique Markov trace constructed on [Formula: see text]. The universal invariant [Formula: see text] is equivalent to the KBSM(ST). For passing now to the KBSM([Formula: see text]), we impose on [Formula: see text] relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in [Formula: see text] but not in ST, and which reflect the surgery description of [Formula: see text], obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM([Formula: see text]). We first present the solution for the case [Formula: see text], which corresponds to obtaining a new basis, [Formula: see text], for KBSM([Formula: see text]) with [Formula: see text] elements. We note that the basis [Formula: see text] is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case [Formula: see text], we first show how the new basis [Formula: see text] of KBSM([Formula: see text]) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case [Formula: see text]. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.

Cost-effectiveness of imaging strategies to diagnose and select patients with non-obstructive coronary artery disease for statin treatment in the United Kingdom

Abstract Funding Acknowledgements Type of funding sources: Public grant(s) – EU funding. Main funding source(s): The project leading to this publication has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 668142. Background Patients with non-obstructive coronary artery disease (NOCAD) are at a higher risk of cardiovascular events compared to those with normal arteries. Plaque rupture is associated with increased adverse events and statin therapy seems to be beneficial for plaque stabilisation. Coronary Computed Tomography Angiography (CCTA) is currently the non-invasive imaging modality of choice for the morphological evaluation of NOCAD in the United Kingdom (UK). However, CCTA provides limited information regarding the vulnerability of plaques to rupture and the selection of patients for preventive statin treatment. Currently being tested on patients, Spectral Photon-Counting CT (SPCCT) may provide increased accuracy for vulnerable plaque detection and, in turn, improved selection of patients for statin treatment. Purpose We investigated the potential cost-effectiveness of SPCCT (compared to a set of CCTA-based strategies) in identifying NOCAD patients with rupture-prone plaques for preventive statin treatment. Methods A decision tree and a Markov trace were developed to model the expected outcomes (costs and quality-adjusted life-years (QALYs)) for a hypothetical UK cohort of 50-year-old male patients with stable chest pain and no history of CAD. Input data were obtained from the literature. Deterministic and probabilistic sensitivity analyses were performed. The impact of a pairwise variation of SPCCT sensitivity and specificity was analysed. Furthermore, five competing imaging strategies were compared in terms of their lifetime costs and effects: 1) CCTA and treat NOCAD based on imaging results, 2) CCTA and treat all NOCAD, 3) CCTA and do not treat NOCAD, 4) SPCCT with high specificity and treat NOCAD based on imaging results, and 5) SPCCT with high sensitivity and treat NOCAD based on imaging results. Results Our deterministic and probabilistic results showed that an improved imaging test would add value compared to CCTA. While increased specificity (to 95%) is favorable at a lower willingness to pay (WTP) (up to ∼£9,000 per QALY), increased sensitivity (to 95%) is more likely to be favorable at a higher WTP (∼£9,000 to £120,000 per QALY). The role of a CCTA-treat-none strategy and a CCTA-treat-all strategy is minimal and potential only at really low (<£2,000 per QALY) and high (>£120,000 per QALY) WTP, respectively. The uncertainty around these results is highly correlated to the uncertainty around the long-term risk for NOCAD patients to experience myocardial infarction or stroke. Conclusion An improved imaging test based on higher sensitivity in identifying rupture-prone coronary plaques in NOCAD patients seems to have value in guiding the decision of preventive statin treatment in the UK. However, additional data regarding the efficacy of statins and of combined treatments for NOCAD patients are needed before the cost-effectiveness of SPCCT can be precisely estimated in this population.

Tied links in the solid torus

We introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in [Formula: see text] previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the bt-algebra of type [Formula: see text] and the Markov trace defined on this.

PCV60 Preliminary Budget IMPACT Analysis of Advanced Imaging and Mechanical Thrombectomy for ACUTE Ischaemic Stroke Beyond 6 Hours from Onset in the United Kingdom

Mechanical thrombectomy (MT) informed by advanced imaging (ADIM) is cost-effective up to 24 hours from acute ischaemic stroke (AIS) onset in the United Kingdom (UK). Our goal is to assess the financial impact of commissioning ADIM and MT within and beyond 6 hours from onset in the UK (compared to conventional imaging and MT within 6 hours only). We used the decision tree and Markov trace of a recently published cost-effectiveness analysis. We considered 87,078 as the average annual number of AIS patients admitted to the hospital. All suspected stroke patients were considered to be diagnosed by ADIM which was modelled with a perfect accuracy. Given the lack of evidence regarding the prior probability for an AIS patient to benefit from late MT, 3 scenarios were simulated. The UK NHS perspective was used and costs were based on the year 2018. Acute care costs and post-stroke costs indexed on the level of dependence were included. Assuming prior probabilities of 10%, 20% and 30%, the annual number of MTs performed beyond 6 hours would be 6,382, 12,765 and 19,147, respectively. At a 3-year time horizon, preliminary analyses show that the financial impact of ADIM + MT beyond 6 hours (versus conventional care within 6 hours) would be £113,320,807 (+3%), £200,898,224 (+5%) and £288,475,021 (+7%), respectively. Considering an annual NHS budget for new stroke cases of £1.6 billion, this would suggest an annual budget increase of 2.4%, 4.2% and 6%, respectively. A more precise estimate of the prior probability to benefit from late MT is needed to ascertain our preliminary results. Although ADIM and MT up to 24 hours from onset is cost-effective, it is important to assess whether the NHS can afford this intervention for all eligible patients and widespread ADIM for all suspected stroke patients countrywide.

Cost-effectiveness of Mechanical Thrombectomy More Than 6 Hours After Symptom Onset Among Patients With Acute Ischemic Stroke

Two 2018 randomized controlled trials (DAWN and DEFUSE 3) demonstrated the clinical benefit of mechanical thrombectomy (MT) more than 6 hours after onset in acute ischemic stroke (AIS). Health-economic evidence is needed to determine whether the short-term health benefits of late MT translate to a cost-effective option during a lifetime in the United States. To compare the cost-effectiveness of 2 strategies (MT added to standard medical care [SMC] vs SMC alone) for various subgroups of patients with AIS receiving care more than 6 hours after symptom onset. This economic evaluation study used the results of the DAWN and DEFUSE 3 trials to populate a cost-effectiveness model from a US health care perspective combining a decision tree and Markov trace. The DAWN and DEFUSE 3 trials enrolled 206 international patients from 2014 to 2017 and 182 US patients from 2016 to 2017, respectively. Patients were followed until 3 months after stroke. The clinical outcome at 3 months was available for 29 subgroups of patients with AIS and anterior circulation large vessel occlusions. Data analysis was conducted from July 2018 to October 2019. MT with SMC in the extended treatment window vs SMC alone. Expected costs and quality-adjusted life-years (QALYs) during lifetime were estimated. Deterministic results (incremental costs and effectiveness, incremental cost-effectiveness ratios, and net monetary benefit) were presented, and probabilistic analyses were performed for the total populations and 27 patient subgroups. In the DAWN study, the MT group had a mean (SD) age of 69.4 (14.1) years and 42 of 107 (39.3%) were men, and the control group had a mean (SD) age of 70.7 (13.2) years and 51 of 99 (51.5%) were men. In the DEFUSE 3 study, the MT group had a median (interquartile range) age of 70 (59-79) years, and 46 of 92 (50.0%) were men, and the control group had a median (interquartile range) age of 71 (59-80) years, and 44 of 90 (48.9%) were men. For the total trial population, incremental cost-effectiveness ratios were $662/QALY and $13 877/QALY based on the DAWN and DEFUSE 3 trials, respectively. MT with SMC beyond 6 hours had a probability greater than 99.9% of being cost-effective vs SMC alone at a willingness-to-pay threshold of $100 000/QALY. Subgroup analyses showed a wide range of probabilities for MT with SMC to be cost-effective at a willingness-to-pay threshold of $50 000/QALY, with the greatest uncertainty observed for patients with a National Institute of Health Stroke Scale score of at least 16 and for those aged 80 years or older. The results of this study suggest that late MT added to SMC is cost-effective in all subgroups evaluated in the DAWN and DEFUSE 3 trials, with most results being robust in probabilistic sensitivity analyses. Future MT evidence-gathering could focus on older patients and those with National Institute of Health Stroke Scale scores of 16 and greater.

Simulation Study to Identify the Characteristics of Markov Chain Properties

Markov models have a long tradition in modeling and simulation of dynamic systems. In this article, we look at certain properties of a discrete-time Markov chain, including entropy, trace, and second-largest eigenvalue to better understand their role for time-series analysis. We simulate a number of possible input signals, fit a discrete-time Markov chain, and explore properties with the help of Sobol indices, partial correlation coefficients, and the Morris elementary effect screening method. Our analysis suggests that the presence of a trend, periodicity, and autocorrelation impact entropy, trace, and second-largest eigenvalue to varying degrees but not independently of each other and with Markov chain parameter settings as other influencing factors. The properties of interest are promising to distinguish time-series data, as evidenced for the entropy measure by recent results in the analysis of cell development for Xenopus laevis in cell biology.

Framization of a Temperley-Lieb algebra of type B

We extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra of type B. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type B to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

Classification of Thurston relation subfactor planar algebras

Bisch and Jones proposed the classification of planar algebras by simple generators and relations. They investigated with the second author the classification of planar algebras generated by 2-boxes. In this paper, we classify singly-generated Thurston-relation planar algebras, defined as subfactor planar algebras generated by a 3-box satisfying a relation proposed by Dylan Thurston. Our main result shows that such subfactor planar algebras are either the E_6 subfactor planar algebras or belong to a two-parameter family of planar algebras arising from the representations of type A quantum groups. We introduce a new method for determining positivity of the Markov trace of planar algebras in this family.

A braids and ties algebra of type B

We introduce an algebra of braids and ties (or bt-algebra) of type B. In analogy to the construction of the bt-algebra of type A, we define this bt–algebra of type B through a framization of the Hecke algebra of type B. We find a basis for it, a faithful tensorial representation, and we prove that it supports a Markov trace, from which we derive invariants of classical links in the solid torus.

Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

Kauffman type invariants for tied links

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.

A framization of the Hecke algebra of type B

We introduce a framization of the Hecke algebra of type B. For this framization, we construct a faithful tensorial representation and two linear bases. We also construct a Markov trace on such an algebra, and from this trace we derive isotopy invariants for framed and classical knots and links in the solid torus.

Framization of the Temperley–Lieb algebra

We propose a framization of the Temperley-Lieb algebra. The framization is a procedure that can briefly be described as the adding of framing to a known knot algebra in a way that is both algebraically consistent and topologically meaningful. Our framization of the Temperley-Lieb algebra is defined as a quotient of the Yokonuma-Hecke algebra. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra to pass through to the quotient algebra. Using this we construct 1-variable invariants for classical knots and links, which, as we show, are not topologically equivalent to the Jones polynomial.

On the link invariants from the Yokonuma–Hecke algebras

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

Markov Trace on the Algebra of Braids and Ties

We prove that the so–called algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones recipe we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant for classical knots is an extension of the Homflypt polynomial and the invariant for singular knots is an extension of an invariant of singular knots defined by the second author and S. Lambropoulou.

Markov Traces on Affine and Cyclotomic Yokonuma–Hecke Algebras

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace.

Markov trace on a tower of affine Temperley–Lieb algebras of type Ã

We define a tower of affine Temperley–Lieb algebras of type Ã. We prove that there exists a unique Markov trace on this tower, this trace comes from the Markov–Ocneanu–Jones trace on the tower of Temperley–Lieb algebras of type A. We define an invariant of special kind of links as an application of this trace.

A classification of SU(d)-type C*-tensor categories

Kazhdan and Wenzl classified all rigid tensor categories with fusion ring isomorphic to the fusion ring of the group SU(d). In this paper we consider the C*-analogue of this problem. Given a rigid C*-tensor category 𝒞 with fusion ring isomorphic to the fusion ring of the group SU(d), we can extract a constant q from 𝒞 such that there exists a *-representation of the Hecke algebra Hn(q) into 𝒞. The categorical trace on 𝒞 induces a Markov trace on Hn(q). Using this Markov trace and a representation of Hn(q) in [Formula: see text] we show that 𝒞 is equivalent to a twist of the category [Formula: see text]. Furthermore a sufficient condition on a C*-tensor category 𝒞 is given for existence of an embedding of a twist of [Formula: see text] in 𝒞.

G-LINKS INVARIANTS, MARKOV TRACES AND THE SEMI-CYCLIC Uq𝔰𝔩(2)-MODULES

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin–Turaev link invariant construction to tangles with a flat connection in a principal G-bundle of the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to 𝔰𝔩(2), defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

A cubic defining algebra for the Links–Gould polynomial

We define a finite-dimensional cubic quotient of the group algebra of the braid group, endowed with a (essentially unique) Markov trace which affords the Links–Gould invariant of knots and links. We investigate several of its properties, and state several conjectures about its structure.