# Tagged Questions

Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.

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### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that If $K$ is a virtual knot whose underlying Gauss ...
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### Can knot diagrams be monotonically simplified using under moves?

It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger ...
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### Question about general torus knot lengths [closed]

Can anyone send me a reference about calculating the length of a thin string wound on a p,q torus with major radius R and minor radius r in terms of p, q, R and r? This may appear trivial but I have ...
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### Does the limit in the Volume conjecture converge?

The Volume conjecture says that if $J_n(q)$ are the colored Jones polynomials of a knot $K \subset S^3$, then $$\lim_{N \to \infty} \frac{ 2 \pi}N \left\vert J_N(e^{2\pi i / N})\right\vert = vol(K)$$ ...
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### Reference on representations of knot groups

Recently, I was studying the knot group and I want to learn some more material about it (e.g. its representations). "Knots" by Burde and Zieschang discusses some material but it is not entirely ...
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### Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
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### Quandle colorings under Reidemeister moves

Let $D$ be a knot diagram and $Q$ a quandle. We use $c$ to denote a fixed coloring of $D$ with $Q$. If $D'$ is another knot diagram of the same knot, and $R_1$ is a sequence of Reidemeister moves ...
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### Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...
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### Understanding a program for computing Khovanov homology

I would like to understand how a computer program for computing Khovanov homology works. The particular program I have in mind is by John Baldwin: https://web.math.princeton.edu/~baldwinj/Kh.cpp The ...
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### Classification of tangles?

Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't ...
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I am interested in the following situation: given a braid $B$, it induces a link $L$ in a pretty straightforward way ("glue" the endpoints, like here). For a braid $B$, we know how to represent it in ...
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I'm working through McMullen's paper "The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology" and have a question concerning the following setup: Given a link complement $(X, ... 0answers 213 views ### The relations between some 3-components links and trefoil knots [closed] It is intuitive to see that the 3-components links (under Alexander–Briggs notations)$6^3_1, 6^3_2, 6^3_3$are closely related to each other; in a sense by doing a cut-gluing or sew-gluing surgery, ... 2answers 147 views ### Why is a braided left autonomous category also right autonomous? In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ... 1answer 118 views ### Practical application of lattice knots I am looking for examples of practical applications of lattice knots. Any help? 0answers 204 views ### The Markov trace via Bott-Samelson fibers? Let$H_n$be the Hecke algebra of GL(n), i.e., the algebra over$\mathbb{Q}(q)$with generators$T_1, \ldots, T_{n-1}$which satisfy the braid relations and also$T^2 = (q-1) T + q$. Recall the ... 1answer 176 views ### Do we get a instanton$S^{3}$if we do$1/n$surgery on a knot in$S^{3}$? Consider the following question: If$K\subset S^{3}$is a nontrivial knot. Let$Y$be the manifold obtained by doing$1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of$Y$... 1answer 196 views ### Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation' Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ... 0answers 159 views ### Does the shortest path between two braids pass through string links? One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links. This means that braids$B_1$and$B_2$, considered inside a cube$I^3$, ... 0answers 110 views ### Jones polynomial of 2-knots Question: is it possible to define the Jones polynomial for knotted surfaces (or$S^2$for simplicity) in$R^4$? Jones polynomial has several definitions (see How many definitions are there of the ... 1answer 498 views ### Is this knot invariant already treated somewhere in the literature? Fix a knot type$K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$ We can turn$Y_K$into a metric space by considering the distance induced by ... 2answers 637 views ### Are Fourier series of length 2 'asymmetric enough' to generate all crossing patterns? - A reformulation of the Fourier-(1,1,2) knot question Given$N$pairs of distinct real numbers$t_i, t'_i \in [0,1]$,$i = 1,\ldots,N$, we ask if there is a function$f(x) = \cos(2\pi mx+\alpha) + \gamma\cdot \cos(2\pi nx+\beta)$, with$m, n \in ...
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Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that: ...
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### Ribbon knot presentations

Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and ...
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### What properties of knots lead Lord Kelvin to hypothesize that atoms were knots in the ether?

I've often heard that Lord Kelvin was one of the first people to study knot theory, as he hypothesized that atoms were knots in the ether. I assume that he had some compelling evidence for this fact. ...
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### Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with ...
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### Bitangent locus of torus knots

Anyone know how to compute the bitangent locus of a space curve, e.g. a torus knot (pick whatever parametrization you like)? Specifically, what is the set of normal vectors (in the two-sphere) of ...
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### Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$ passing through one another in the following sense. (Caveat lector: This question is not of general interest! It is also long.) $H_1$ is ...