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  1. Abstract We give a new, algebraically computable formula for skein modules of closed 3-manifolds via Heegaard splittings. As an application, we prove that skein modules of closed 3-manifolds are finite-dimensional, resolving in the affirmative a conjecture of Witten. 
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    Free, publicly-accessible full text available April 1, 2024
  2. null (Ed.)
    We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$ . 
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  4. ABSTRACT

    The anterior bundle of the medial collateral ligament (AMCL) resists the loads that arise at the elbow during overhand throwing and has commonly been divided into posterior and anterior bands. While these anterior and posterior bands have been thought to bear the load at different flexion angles, any transition of the load distribution between the two bands is poorly understood and has not considered laxity (slack). This study considers the AMCL as three bands and quantifies the mechanical response to vertical distraction, simulating valgus‐load joint opening, through the sequential superposition of the band responses after the elimination of inherent laxity. Eight cadaveric elbow specimens were used for the study. The intact AMCL of each specimen was tested under vertical distraction in a specialized load frame at four elbow flexion angles and then subsequently retested after two longitudinal transections. The greatest laxity at full extension and full flexion belonged to the posterior (1.9 mm) and anterior (2.4 mm) band, respectively. At the lesser and higher flexion angles, the greatest structural stiffness belonged to the anterior and middle band. The overall AMCL was the most structurally stiff at 60°, with approximately 150 N of force required for 2% elongation. This study shows that the different bands of the AMCL may have different load bearing properties at different flexion angles, causing each band to support different proportions of an imposed load. The presence of the laxity may impose a load‐bearing delay, causing load‐bearing in each band to begin asynchronously. © 2019 Orthopaedic Research Society. Published by Wiley Periodicals, Inc. J Orthop Res 37:2027–2034, 2019

     
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