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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Ozsvath-Szabo orientation convention for Seifert fibred spaces

I am confused by the orientation convention that Ozsvath and Szabo use in On Heegaard Floer homology and Seifert Fibered Surgeries and would appreciate if someone clarifies this for me. On page 15 ...
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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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10answers
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Are there any very hard unknots?

Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any ...
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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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Proof of the Reidemeister theorem

While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can ...
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287 views

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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878 views

Is there a table of (fibred knot) monodromies?

Background/motivation I'm working on contact topology (in dimension three): a fundamental theorem of Giroux gives us a bijection between contact structures (up to isotopy) and open books (up to ...
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1answer
69 views

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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240 views

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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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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129 views

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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2answers
306 views

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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1answer
172 views

The Jones polynomial at specific values of $t$.

I've been calculating some Jones polynomials lately and I was just curious if there was a physical meaning to evaluating the Jones polynomial at a particular value of $t$. For example, if I take the ...
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1answer
238 views

Kauffman's state model for the Alexander polynomial, via representation theory

I've been reading Oleg Viro's paper on "quantum relatives of the Alexander polynomial" (arXiv:math/0204290), which, among other more general things, derives state-sum formulas for the Alexander ...
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1answer
130 views

Parity of knot signatures

I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally ...
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1answer
91 views

Computable link invariants

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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What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...
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2answers
487 views

Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$, there exists some graph $G$ whose every embedding into $\mathbb{R}^3$ (or into $\mathbb{S}^3$) contains a cycle that realizes $K$? I know the ...
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3answers
240 views

Heegaard Floer Homology of double branched cover

The question is the following: Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...
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1answer
196 views

Knot invariants in 3-manifolds that are not $\mathbb{R}^3$ or $S^3$ or $B^3$?

This is just a reference request; I have no sharp mathematical question. Inspired by the $(3+)$-year old MO question, In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?, I would ...
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342 views

How to tell if two or more knots are linked

Given a number of knots, I would like to know if they are linked. I know that the linking number can tell if two knots are linked. There is any method that completely solves this problem?
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1answer
108 views

Khovanov $sl_2$ homology of a connected sum of some torus knots

Let $T_{p,q}$ be the (p,q) torus knot. Could anybody possibly compute either unreduced or reduced Khovanov $\mathfrak{sl}(2)$ homology of the connected sum $T_{2,3} \sharp T_{3,4}$ of the (2,3) and ...
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270 views

Do homeomorphic complements give homotopic knots?

Maybe this question is too trivial for a research site but there are so many notions of equivalence of knots that I am lost in literature. The question that interests me is the following. A knot is a ...
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336 views

Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings. Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$? Here, ...
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Crossing bound implies Reidemeister move bound?

In 1998 Galatolo established an upper bound on the number of Reidemeister moves needed to convert a diagram $D$ of the unknot into a trivial loop diagram. The upper bound is a function of $n$, the ...
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97 views

Alexander Invariant of Torus knot

I am very interested in knot theory, especially in knot groups and knot polynomials. As is well known, it is easy to calculate the Alexander polynomial from the fundamental group $\pi_{1}(K)$ of a ...
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3answers
199 views

Measures of entangledness of an open curve

Let $\gamma$ be a simple (non-self-intersecting) open curve in $\mathbb{R}^3$. I am seeking a measure of its degree of "entangledness," some measure that accords with the intuition one senses with a ...
5
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0answers
121 views

Software for BMW algebra calculations?

Does software exist for computations in the BMW algebra? For example, I'd like to be able to express elements in a basis of "totally descending tangles" as in a paper of Morton–Wassermann. At ...
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Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state: Any reduced diagram of an alternating link has the fewest possible crossings. Any two reduced ...
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The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group

My question is about the Alexander polynomial of a slice knot. For a slice knot $K$, Fox-Millnor and Terasaka proved that $$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$ for some polynomial $f(t) \in ...
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1answer
104 views

Closed formula or program for computing Tristram-Levine signatures of torus knots?

I need to compute a series of Tristram-Levine signatures for a family of torus knots. I was wondering if this has already been done or whether there is a good way to streamline the computation. I am ...
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1answer
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Relation between the Alexander module of a link and intermediate free abelian covers

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, ...
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Applications of knot theory

An answer of André Henriques' inspired the following closely related CW question. Parts of the following is extracted from his answer and my comments. I regularly teach a knot theory class. ...
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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, ...
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1answer
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One question about the quandle

Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston ...
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Freely Periodic map of $(S^{3} , K) $ and a fixed loop in the induced isomorphism of $\pi_{1} ( S^{3} \backslash K )$

Let $K$ be a link in $S^{3}$ and $f: S^{3} \rightarrow S^{3} $ a freely periodic map of order $n$ with $f(K) = K$. Let $\psi_{f} : \pi_{1} ( S^{3} \backslash K ) \rightarrow \pi_{1} ( S^{3} ...
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2answers
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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 ...
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1answer
114 views

Practical application of lattice knots

I am looking for examples of practical applications of lattice knots. Any help?
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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 $Y$ ...
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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 ...
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1answer
569 views

monodromy defects and Chern-Simons

In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to ...
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1answer
615 views

isotopy doesn't make sense (Milnor)

hello, I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots For each $c$ and $p$ in $\mathbb{R}^{n-1}$ such that $\|c-p\| > < r$, there is ...
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1answer
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Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1

I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...
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2answers
543 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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Stick knot questions: simple but may not be easy

I have a few questions about nonplanar "stick circuits" (or hexagons and higher $n$-gons) that you might be able to help with: (I know that $n=6$ is the minimum number of points to form a stick ...
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108 views

Minimal piecewise-linear knot diagram [closed]

I'm looking for an answer to the following question: Given a knot in $\mathbb{R^{3}}$ can we find a piecewise-linear diagram of it wich is minimal (has a minimal number of verticies)?
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Connected Sum Decomposition of a Knot

Given a composite knot, is it possible to decompose it in prime knots by an algorithm that runs in polynomial time?
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146 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 ...