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(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error-correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of \mathbb{Z}_2 ℤ 2 gauge theory with fermionic charges, in \mathbb{Z}_2 \times \mathbb{Z}_2 ℤ 2 × ℤ 2 gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ( D_n D n ) and alternating ( A_6 A 6 ) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an H^4 H 4 cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of the interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D A_6 A 6 gauge theory.
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Knotty Knits are Tangles on Tori
In this paper we outline a topological framework for constructing 2-periodic knitted stitches and an algebra for joining stitches together to form more complicated textiles. Our topological framework can be constructed from certain topological “moves" which correspond to “operations" that knitters make when they create a stitch. In knitting, unlike Jacquard weaves, a set of n loops may be combined in topologically nontrivial ways to create n stitches. We define a swatch as a mathematical construction that captures the topological manipulations a hand knitter makes. Swatches can capture the topology of all possible 2-periodic knitted motifs: standard patterns such as garter and ribbing, cables in which stitches connect one row of loops to a permutation of those same loops on the next row much like operators of a braid group, and lace or pieces with shaping which use increases and decreases to disrupt the underlying square lattice of stitches.
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- Award ID(s):
- 1847172
- PAR ID:
- 10230429
- Date Published:
- Journal Name:
- Proceedings of Bridges 2020
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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null (Ed.)Automatic knitting machines are robust, digital fabrication devices that enable rapid and reliable production of attractive, functional objects by combining stitches to produce unique physical properties. However, no existing design tools support optimization for desirable physical and aesthetic knitted properties. We present KnitGIST (Generative Instantiation Synthesis Toolkit for knitting), a program synthesis pipeline and library for generating hand- and machine-knitting patterns by intuitively mapping objectives to tactics for texture design. KnitGIST generates a machine-knittable program in a domain-specific programming language.more » « less
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Knitting creates complex, soft fabrics with unique texture properties that can be used to create interactive objects.However, little work addresses the challenges of designing and using knitted textures computationally. We present KnitPick: a pipeline for interpreting hand-knitting texture patterns into KnitGraphs which can be output to machine and hand-knitting instructions. Using KnitPick, we contribute a measured and photographed data set of 472 knitted textures. Based on findings from this data set, we contribute two algorithms for manipulating KnitGraphs. KnitCarving shapes a graph while respecting a texture, and KnitPatching combines graphs with disparate textures while maintaining a consistent shape. KnitPick is the first system to bridge the gap between hand- and machine-knitting when creating complex knitted textures.more » « less
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Abstract Entanglement of collections of filaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBCs), which create an infinite periodic system whose global entanglement may be impossible to capture and is repetitive. We introduce two new methods to assess topological entanglement in PBC: the Periodic Jones polynomial and the Cell Jones polynomial. These tools capture the grain of geometric/topological entanglement in a periodic system of open or closed chains, by using a finite link as a representative of the global system. These polynomials are topological invariants in some cases, but in general are sensitive to both the topology and the geometry of physical systems. For a general system of 1 closed chain in 1 PBC, we prove that the Periodic Jones polynomial is a recurring factor, up to a remainder, of the Jones polynomial of a conveniently chosen finite cutoff of arbitrary size of the infinite periodic system. We apply the Cell Jones polynomial and the Periodic Jones polynomial to physical PBC systems such as 3D realizations of textile motifs and polymer melts of linear chains obtained from molecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective geometric/topological entanglement complexityin such systems of physical relevance.more » « less
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INTRODUCTION: In practice, the use of a whip stitch versus a locking stitch in anterior cruciate ligament (ACL) graft preparation is based on surgeon preference. Those who prefer efficiency and shorter stitch time typically choose a whip stitch, while those who require improved biomechanical properties select a locking stitch, the gold standard of which is the Krackow method. The purpose of this study was to evaluate a novel suture needle design that can be used to perform two commonly used stitch methods, a whip stitch, and a locking stitch, by comparing the speed of graft preparation and biomechanical properties. It was hypothesized that adding a locking mechanism to the whip stitch would improve biomechanical performance but would also require more time to complete due to additional steps required for the locking technique. METHODS: Graft preparation was performed by four orthopaedic surgeons of different training levels where User 1 and User 2 were both attendings and User’s 3 and 4 were both fellows. A total of 24 matched pair cadaveric knees were dissected and a total of 48 semitendinosus tendons were harvested. All grafts were standardized to the same size. Tendons were randomly divided into 4 groups (12 tendons per group) such that each User performed analogous stitch on matched pair: Group 1, User 1 and User 3 performed whip stitches; Group 2, User 1 and User 3 performed locking stitches; Group 3, User 2 and User 4 performed whip stitches; Group 4, User 2 and User 4 performed locking stitches. For instrumentation, the two ends of tendon grafts were clamped to a preparation stand. A skin marker was used to mark five evenly spaced points, 0.5 cm apart, as a guide to create a 5-stitch series. The stitches were performed with EasyWhip, a novel two-part suture needle which allows one to do both a traditional whip stitch and a locking whip stitch, referred to as WhipLock (Figure 1). The speed for graft preparation was timed for each User. Biomechanical testing was performed using a servohydraulic testing machine (MTS Bionix) equipped with a 5kN load cell (Figure 2). A standardized length of tendon, 10 cm, was coupled to the MTS actuator by passing it through a cryoclamp cooled by dry ice to a temperature of -5°C. All testing samples were pre-conditioned to normalize viscoelastic effects and testing variability through application of cyclical loading to 25-100 N for three cycles. The samples were then held at 89 N for 15 minutes. Thereafter, the samples were loaded to 50-200 N for 500 cycles at 1 Hz. If samples survived, they were ramped to failure at 20 mm/min. Displacement and force data was collected throughout testing. Metrics of interest were peak-to-peak displacement (mm), stiffness (N/mm), ultimate failure load (N) and failure mode. Data are presented as averages and standard deviations. A Wilcoxon signed-rank test was used to evaluate the groups for time to complete stitch and biomechanical performance. Statistical significance was set at P = .05. RESULTS SECTION: In Group 1, the time to complete the whip stitch was not significantly different between User 1 and User 3, where the average completion time was 1 min 13 sec. Similarly, there were no differences between Users when performing the WhipLock (Group 2) with an average time of 1 min 49 sec. In Group 3 (whip stitch), User 2 took 1 min 48 sec to complete the whip stitch, whereas User 4 took 1 min 25 sec (p=.033). The time to complete the WhipLock stitch (Group 4) was significantly different, where User 2 took 3 min and 44 sec, while User 4 only took 2 min 3 sec (p=.002). Overall, the whip stitch took on average 1 min 25 sec whereas the WhipLock took 2 min 20 sec (p=.001). For whip stitch constructs, no differences were found between Users and all stitches were biomechanically equivalent. Correspondingly, for WhipLock stitches, no differences were found between Users and all suture constructs were likewise biomechanically equivalent. Averages for peak-to-peak displacement (mm), stiffness (N/mm), and ultimate failure load (N) are presented in Table 1. Moreover, when the two stitch methods were compared, the WhipLock constructs significantly increased stiffness by 25% (p <.001), increased ultimate failure load by 35% (p<.001) and reduced peak-to-peak displacement by 55% (p=.001). The common mode of failure for grafts with whip stitch failed by suture pullout from tendon (18/24), where a few instances occurred by suture breakage (6/24). Tendon grafts with WhipLock stitch commonly failed by suture breakage (22/24), where two instances of combined tendon tear and suture breakage were noted. DISCUSSION: The WhipLock stitch significantly increased average construct stiffness and ultimate failure load, while significantly reducing the peak-to- peak displacement compared to the whip stitch. These added strength benefits of the WhipLock stitch took 55 seconds more to complete than the whip stitch. No statistically significant difference in biomechanical performance was found between the Users. Data suggests equivalent stitch performance regardless of the time to complete stitch and surgeon training level. SIGNIFICANCE/CLINICAL RELEVANCE: Clinically, having a suture needle device available which can be used to easily perform different constructs including one with significant strength advantages regardless of level of experience is of benefit.more » « less
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