Mirror-curve codes for knots and links (original) (raw)
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Mirror-Curves and Knot Mosaics
2011
Inspired by the paper on quantum knots and knot mosaics [23] and grid diagrams (or arc presentations), used extensively in the computations of Heegaard-Floer knot homology , we construct the more concise representation of knot mosaics and grid diagrams via mirrorcurves. Tame knot theory is equivalent to knot mosaics [23], mirror-curves, and grid diagrams . Hence, we introduce codes for mirror-curves treated as knot or link diagrams placed in rectangular square grids, suitable for software implementation. We provide tables of minimal mirror-curve codes for knots and links obtained from rectangular grids of size 3 × 3 and p × 2 (p ≤ 4), and describe an efficient algorithm for computing the Kauffman bracket and L-polynomials directly from mirror-curve representations.
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Complementary Regions of Knot and Link Diagrams
Annals of Combinatorics, 2011
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a projection to the sphere such that the number of edges of each complementary face of the projection comes from the given sequence. In this paper, it is proved that the following infinite sequences are all universal for knots and links: (3, 5, 7,. . .), (2, n, n + 1, n + 2,. . .) for all n ≥ 3 and (3, n, n + 1, n + 2,. . .) for all n ≥ 4. Moreover, the following finite sequences are also universal for knots and links: (3, 4, 5) and (2, 4, 5). It is also shown that every knot has a projection with exactly two odd-sided faces, which can be taken to be triangles, and every link of n components has a projection with at most n odd-sided faces if n is even and n + 1 odd-sided faces if n is odd.
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Proceedings of the 22nd International Conference on Distributed Multimedia Systems, 2016
Knots occur in many areas of science and art. The mathematical field of Knot Theory studies an idealised form of knots by viewing them as closed loops in 3-space. They can be formally studied via knot drawings which are well-behaved projections of the knot onto the 2-D plane. Equivalence of knots in 3-space (ambient isotopy) can be encapsulated via sequences of diagram rewriting rules, called Reidemeister moves, but finding such sequences demonstrating isotopy of two knots can be immensely challenging. Whilst there are some sophisticated tools available for some knot theoretic tasks, there is limited (free) tool support for certain knot creation and interaction tasks, which could be useful for lecturers and students within University courses. We present KnotSketch, a tool with multiple functionalities including the ability to: (i) read off a form of Gauss code for a user sketched diagram; (ii) generate a diagram from such a code; (iii) regenerate a knot diagram via a different projection, thereby producing examples of equivalent knot diagrams that may look very different; (iv) interaction capabilities to quickly alter the knot via crossing changes and smooth the curves of the sketched diagram; (v) export facilities to generate svg images of the constructed knots. We evaluate KnotSketch via a case study demonstrating examples of intended usage within an educational setting. Furthermore, we performing a preliminary user study to evaluate the general usability of the tool.
Journal of Knot Theory and Its Ramifications, 2007
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Introductory Lectures on Knot Theory
Series on Knots and Everything, 2011
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