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Knot polynomials of open and closed curves.

ABSTRACT: In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

SUBMITTER: Panagiotou E

PROVIDER: S-EPMC7482204 | biostudies-literature | 2020 Aug

REPOSITORIES: biostudies-literature

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Knot polynomials of open and closed curves.

Panagiotou Eleni E Kauffman Louis H LH

Proceedings. Mathematical, physical, and engineering sciences 20200805 2240

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a contin ...[more]

PMID: 32922152

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Project description:ObjectiveTying knots during suturing is one of the most challenging tasks in laparoscopic surgery. Therefore, measures aimed at ensuring both the ease and speed of knot tying not only benefit the surgeon but can also reduce operating time significantly. This study compared extracorporeal and intracorporeal knot tying techniques using a Szabo pelvic trainer model from the Gynaecological Endoscopic Surgical Education and Assessment program.DesignThe students tied intra- and extracorporeal knots using closed- and open-jaw knot pushers. Using an artificial tissue suturing pad in a certified Szabo pelvic trainer, students tied three knots using each technique according to block randomization. Task completion time, knot strength, knot-spread ability, and number of errors were recorded. The Wilcoxon test and mixed-effects models were used to analyze the results. After completing the exercises, participants answered a questionnaire concerning knot-tying techniques and their performance.SettingUniversity Hospital Basel, which provides tertiary-level clinical care.ParticipantsFifty-seven medical students with no experience in laparoscopy voluntarily signed up for this study.ResultsOpen and closed extracorporeal knot tying was significantly faster (p < 0.001, p < 0.001, respectively), more precise (p = 0.007, p = 0.003), and associated with reduced knot-spread ability (p < 0.001, p < 0.001) compared to intracorporeal knot tying. Open- and closed-jaw knot pushers were shown to be equal in terms of speed (p = 0.563), knot-spread ability (p = 0.49), and precision (p = 0.831). The study participants rated open (30 %) and closed (49 %) extracorporeal knot tying as more intuitive than intracorporeal (21 %) knot tying. Improved concentration was significantly correlated with tighter knots (p = 0.011).ConclusionsStudents achieved significantly better results using extracorporeal knot-tying techniques than intracorporeal ones, including greater speed, tighter knots, and optimized precision. These results suggest that beginners in the field of laparoscopy should be encouraged to practice extracorporeal knot-tying techniques.

Dataset Information

Knot polynomials of open and closed curves.

Publications

Knot polynomials of open and closed curves.

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