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Abstract We automate the process of machine learning correlations between knot invariants. For nearly 200 000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture. 

Abstract We show that there exist split, orientable, 2‐component surface‐links in with non‐isotopic splitting spheres in their complements. In particular, for non‐negative integers with , the unlink consisting of one component of genus and one component of genus contains in its complement two smooth splitting spheres that are not topologically isotopic in . This contrasts with link theory in the classical dimension, as any two splitting spheres in the complement of a 2‐component split link are isotopic in . 

For every integerg\ge 2we construct 3-dimensional genus-g1-handlebodies smoothly embedded inS^{4}with the same boundary, and which are defined by the same cut systems of their boundary, yet which are not isotopic rel. boundary via any locally flat isotopy even when their interiors are pushed intoB^{5}. This proves a conjecture of Budney–Gabai for genus at least 2. 

Previous research on student thinking about experimental measurement and uncertainty has primarily focused on students’ procedural reasoning: Given some data, what should students calculate or do next? This approach, however, cannot tell us what beliefs or conceptual understanding leads to students’ procedural decisions. To explore this relationship, we first need to understand the range of students’ beliefs and conceptual understanding of measurement. In this work, we explored students’ philosophical beliefs about the existence of a true value in experimental measurement. We distributed a survey to students from 12 universities in which we presented two viewpoints on the existence of a true definite position resulting from an experiment, asking participants to indicate which view they agreed with more and asking them to explain their choice. We found that participants, both students and experts, varied in their beliefs about the existence of a true definite position and discussed a range of concepts related to quantum mechanics and the experimental process to explain their answers, regardless of whether or not they agreed with the existence of a true value. From these results, we postulate that students who exhibit similar procedural reasoning may hold widely varying philosophical views about measurement. We recommend that future work investigates this potential relationship and whether and how instruction should attend to these philosophical views in addition to students’ procedural decisions. 

Uncertainty is an important concept in physics laboratory instruction. However, little work has examined how students reason about uncertainty beyond the introductory (intro) level. In this work we aimed to compare intro and beyond-intro students’ ideas about uncertainty. We administered a survey to students at 10 different universities with questions probing procedural reasoning about measurement, student-identified sources of uncertainty, and predictive reasoning about data distributions. We found that intro and beyond-intro students answered similarly on questions where intro students already exhibited expert-level reasoning, such as in comparing two data sets with the same mean but different spreads, identifying limitations in an experimental setup, and predicting how a data distribution would change if more data were collected. For other questions, beyond-intro students generally exhibited more expertlike reasoning than intro students, such as when determining whether two sets of data agree, identifying principles of measurement that contribute to spread, and predicting how a data distribution would change if better data were collected. Neither differences in institutions, student majors, lab courses taken, nor research experience were able to fully explain the variability between intro and beyond-intro student responses. These results call for further research to better understand how students’ ideas about uncertainty develop beyond the intro level. 

We use deep neural networks to machine learn correlations betweenknot invariants in various dimensions. The three-dimensional invariantof interest is the Jones polynomial J(q) J ( q ) ,and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t) Kh ( q , t ) ,smooth slice genus g g ,and Rasmussen’s s s -invariant.We find that a two-layer feed-forward neural network can predict s s from \text{Kh}(q,-q^{-4}) Kh ( q , − q − 4 ) with greater than 99% 99 % accuracy. A theoretical explanation for this performance exists in knottheory via the now disproven knight move conjecture, which is obeyed byall knots in our dataset. More surprisingly, we find similar performancefor the prediction of s s from \text{Kh}(q,-q^{-2}) Kh ( q , − q − 2 ) ,which suggests a novel relationship between the Khovanov and Leehomology theories of a knot. The network predicts g g from \text{Kh}(q,t) Kh ( q , t ) with similarly high accuracy, and we discuss the extent to which themachine is learning s s as opposed to g g ,since there is a general inequality |s| ≤2g | s | ≤ 2 g .The Jones polynomial, as a three-dimensional invariant, is not obviouslyrelated to s s or g g ,but the network achieves greater than 95% 95 % accuracy in predicting either from J(q) J ( q ) .Moreover, similar accuracy can be achieved by evaluating J(q) J ( q ) at roots of unity. This suggests a relationship with SU(2) S U ( 2 ) Chern—Simons theory, and we review the gauge theory construction ofKhovanov homology which may be relevant for explaining the network’sperformance.