# Questions tagged [knot-theory]

For questions on knot theory, the study of mathematical knots and their properties.

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### Higher homotopy groups of a string complement in a cylinder.

We call the image of a smooth embedding $f:\coprod_{i=1}^n [0,1]\times\{i\} \hookrightarrow \mathbb{D}^2 \times [0,1]$ a string if $f(0,i)$ and $f(1,i)$ have the second coordinates $0$ and $1$ ...
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### Simple examples of colored HOMFLY polynomial?

I'm trying to find some simple examples of using the colored HOMFLY polynomial and was wondering if anyone could give me a reference. I really only need an extremly basic definition of the colored ...
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### Where can I get an introduction to the Colored HOMFLY Polynomial?

Where can I get an free introduction to the Colored HOMFLY Polynomial? It would be great If I could get an exposition of the invariant with regards to rational links, but I'm not holding my breath. I ...
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### Is the Mazur swindle a supertask?

The Mazur swindle can be used to show that any knot with an inverse under connected sum must be isotopic to the unknot. The isotopy, as I understand it, involves instantiating a pair of the knot and ...
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### Unusual skein relation in HOMFLY polynomial

If I take the HOMFLY(PT) polynomial defined by $$l \,P(L_+) + l^{-1}\,P(L_-) + m\,P(L_0) = 0,$$ I have looked at expressions of the form (knots that are the same except inside a small disk, where ...
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### Given a sequence of head to tail vectors forming a closed loop, how can I determine if they form a knot? [closed]

Consider if we have some sequence of vectors placed head to tail which form a closed loop. How can one determine whether they form a loop? We assume that it is given that the vectors close, that is ...
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### Distortion of the Unknot

In Mikhail Gromov's "Filling Riemannian Manifolds" he defines the distortion of a knot $K$ embedded in $S^3$ as \delta (K) := \inf_{\gamma \in K} \sup_{x,y \in \gamma} \frac{d_{\gamma}(x,y)}{||x-y||}...
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### Parametric curve end tip orientation

If I have a parametric curve equation, say for a trefoil ($3_1$) knot $x = 15(t^3-3t)$ $y = 15(-t^4+4t^2)$ $z= -15(0.2t^5-2t)$ Suppose I plot this only between t=(-2,0). Say the coordinate at the tip ...
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### Tying a portion of a knot diagram

I am having some trouble with a paper by Livingston (Infinite Order Amphicheiral Knots, Algebraic and Geometric Topology 1, 2001, 231-241). He starts with a knot $K$, and constructs a new knot "...
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### computing the Kauffman bracket with the given relation

My Problem: Use the relation to compute the bracket of the link diagram $D_n$ with $n$ components: My attempt: It seems to me that raising the given equation to the $n^{th}$ power is the most ...
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### Algebraically slice implies slice

I am studying an article by Livingston ("New examples of non-slice, algebraically slice knots", Proceeding of the AMS, 2001) about an example of an infinite class of knots which are algebraically ...
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### Compute $H_{*}(M, \mathbb{Z})$ of gluing 2 solid tori.

Let $M$ be the space obtained by gluing two solid tori $D^2 \times S^1$ and $S^1 \times D^2$ together via the identity map of their boundaries. Compute $H_{*}(M, \mathbb{Z}.)$ A Hint: Use Mayer ...
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### Conway Notation Question For a Knot Diagram

my question is fairly simple. I have been staring at this knot diagram for a while and cannot understand why its Conway Notation is 2 -2 2 -2 2 4 (according to Adam's The Knot Book). The way I ...
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### Alexander polynomial of any knot evaluated at 1 is $\pm$ 1

I'm supposed to prove for a knot theory homework assignment that the Alexander polynomial of any knot (as opposed to link) is $\pm1$. From examples, I'm pretty convinced that this is true, but I have ...
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### A regular Morse map $f:C_K\to S^1$

A circle-valued Morse map $f:C_K \to S^1$ on the complement of a knot $K$ is said to be regular if there is a $C^\infty$ trivialisation $\Phi : N(K)\to K\times B^2(0,\epsilon)$ of a tubular ...
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### Why do we study representation of knot groups?

In my independent study class in Knot Theory, my professor said that solving the relation in knot groups is an undecidable problem, and he said we study representations of knot groups because then ...
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### Homotopy rel boundary of disks in knot complement $\times I$

Let $K$ and $J$ be disjoint knots in $S^3$ and let $D_1$ and $D_2$ be two immersed disks in $(S^3 - K) \times I$ with boundary $J$. Is it true that $D_1$ and $D_2$ are homotopic through a homotopy ...
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### A wild knot and its complement

I have been reading about wild knots in $\mathbb{R^3}$ that have nonsimply connected complements. I'm a bit confused here and I added a picture of a wild curve with two circles in its complement. Is ...
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### Baffling bit of notation

So I'm reading a paper about deriving polynomial invariants for links in an arbitrary surface, and I'm stuck on some under explained notation. The full quote states that something is a 'right module ...
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### Conjugacy classes with linking number 1 in the complement of a knot?

Let $K$ be a knot in $S^3$ and let $lk: \pi_1(S^3 - K) \to H_1(S^3-K;\mathbb{Z})$ denote the abelianization map. Let $1 \in H_1(S^3-K;\mathbb{Z})$ denote a choice of generator. Is it possible that ...