# Tagged Questions

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### Kontsevich integral : state of the art

The Kontsevich integral is known to be a universal Vassiliev invariant. It is still an open question whether it is a complete knot invariant, i.e. whether it distinguishes a given knot from all other ...
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### More questions about high-dimensional knot invariants

In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about ...
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### Invariants of high-dimensional knots

In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about ...
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### What are the homotopy classes of two-component links in $\mathbb{RP}^3$?

This question comes from an unanswered question on Math Stack Exchange. A two-component link in $\mathbb{RP}^3$ is any embedding $S^1\uplus S^1\to \mathbb{RP}^3$. Two such links are homotopic if ...
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### Semidirect product decomposition of the Borromean rings group

Let $X=S^3\setminus B$ be the link complement of the Borromean rings. Then $G=\pi_1(X)$ has a presentation of the form  G = \langle \; a,b,c \mid [a,[b^{-1},c]],\; [b,[c^{-1},a]], \; ...
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### Visualize Fourth Homotopy Group of $S^2$

I know $\pi_4(S^2)$ is $\mathbb{Z}_2$. However, I don't know how to visualize it. For example, it is well known that $\pi_3(S^2)=\mathbb{Z}$ can be understood by Hopf Fibration. Elements in ...
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### Knot diagrams, sets of moves and equivalence relations

Short version: Does anyone study equivalence classes generated by a given set of "moves" (in the sense of, but not limited to, Reidemeister moves) on the set of knot diagrams? Yes, I understand that ...
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### Homology and homotopy type for knot complements

I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a ...
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### High-dimensional ribbon knots

Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the ...
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### Source on the proof that codimension 2 is sufficient for knottings?

Hi all. I'm not even sure that this is a theorem, but a while ago I heard a topologist friend of mine (whom I haven't been able to reach) saying that given a continuous embedding $f:M^{n}\to N^{r}$ ...
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Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be a embedding map and $i_{*}:\pi(T^{2})\rightarrow \pi(S^{1}\times S^{2})$ be the homomorphism induced by $i$. If $i(T^{2})$ is separable in $S^{1}\times ... 0answers 152 views ### Finding a ribbon graph for a mapping class group action Turaev defines TQFT$(T, \tau)$in his book "Quantum invariants of knots and 3-manifolds". He uses it to define an action of a mapping class group of a d-surface$\Sigma$. This action$\epsilon$is ... 0answers 229 views ### Knots that turn around an axis [closed] Take a thick cord (the alim cord of your laptop or your mouse cord for example) and wrap it around your hand (or finger) turning always in the same direction but possibly knotting it. Then try to ... 0answers 219 views ### subset embedding gives trefoil knot [closed] Let$X$be a topological space and$E_n(X)$the space of finite sets of cardinality$\leq n$. It is a theorem of Bott that$E_3(S^1)=S^3$. What is the idea to show that the embedding ... 1answer 572 views ### Link of singularities For an isolated plane curve singularity, given by homogeneous equation$F=0 \subset \mathbb{C}^2$, one consider the curve$(F=0) \cap S^3 \subset S^3$, and we call it the link of singularity. some ... 1answer 529 views ### Length of shortest possible knot Consider a line L in R^3 in the shape of a trefoil knot. Consider the surface S that is the union of all unit circles that have centers on this line and whose tangent vectors are all perpendicular to ... 1answer 218 views ### Simplified Jones trace invariant for links Jones (1985) defines a simplified trace invariant for knots by$W_K(t)=\frac{1-V_K(t)}{(1-t^3)(1-t)}$. Then, e.g., the Arf invariant for$K$is$Arf(K)=W_K(i)$. Does this work for oriented links as ... 1answer 221 views ### Are braid links proper links? Are braid links proper links? Or are the concepts involved unrelated? 5answers 927 views ### Can surfaces be interestingly knotted in five-dimensional space? It's possible this question is trivial, in which case it will be answered quickly. In any case, I realized that it's a basic question the answer to which I should know but do not. Everybody loves ... 1answer 881 views ### Genera and the Milnor Conjecture on the Unknotting Number of a Torus Knot Let$f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface$V_{f, \kappa} = ...
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Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with $H_{*}(M;Q)=H_{*}(S^3;Q)$ As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's ...
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### 3-manifold with torus boundary with trivial “peripheral ideal”?

Given a 3-manifold $M$, one can define the Kauffman bracket skein module $K_t(M)$ as the $C$-vector space with basis "links (including the empty link) in $M$ up to ambient isotopy," modulo the skein ...