# Questions tagged [knot-invariants]

For properties of knots that remain unaffected by Reidemeister moves

255 questions
Filter by
Sorted by
Tagged with
1answer
57 views

### Image of elements of $S_n$ in the Temperley-Lieb algebra

Consider the algebra $A_n$ generated by $u_1,\ldots,u_{n-1}$ subject to relations $u_i^2=-2u_i$, $u_iu_{j}u_i=u_i$ for $|i-j|=1$ and $u_iu_j=u_ju_i$ for $|i-j| \geqslant 2$. The algebra $A_n$ is the ...
1answer
67 views

### What is a slice knot?

so I've recently come across Lisa Piccirillo's proof on the Conway Knot Problem, and I want to learn more about it, but I can't really understand what is meant by whether or not a knot is "slice&...
0answers
35 views

### Computation of Colored HOMFLY Polynomials

I am trying to understand the colored HOMFLY polynomials. The theoretic description Anna Aiston gave in her PhD thesis is really nice, but what about the computation? I would like to understand the ...
0answers
39 views

### Application of higher-dimensional knot theory

There are quite a few applications to classical (1-dimensional) knot theory, such as knotted proteins, DNA,... Are there any potential applications of 2-dimensional knot theory? E.g. knotted surfaces ...
0answers
28 views

### Analogy to Van kampen theorem for Quandles

This is from thesis of David joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory Please help me to understand the $\lim$ $AQ(U_{i}$,$U_{i} \cap K)$as given on page 42 of the ...
1answer
33 views

### Knots with infinite crossing

For me knots are embedding of $S^1$ in $\mathbb{R}^3$. I have following questions: Will knots have infinite crossing? If so, Why are we considering only knots with finite crossing ? Can someone ...
1answer
154 views

### What is the degree of an n-fold branched cover over a trefoil?

The order-2 cyclic branched cover over a trefoil has degree 6, meaning the preimage of any point off the trefoil has cardinality six. (You can find a wonderful video of this here, made by Moritz ...
1answer
52 views

### Questions on the symmetry of of the alexander polynomial and the rank of the Seifert matrix

For my bachelor thesis I am using the book "Lectures on the topology of 3-manifolds. an introduction to the Casson invariant"(1999) by Nikolai Saveliev. Regarding the Alexander Polynomial as ...
1answer
147 views

0answers
96 views

### Is this knotted graph knotted?

Above, I've drawn a knotted 3-valent graph. I suspect that it's not isotopic to the "unknotted" version below, but I'm not sure. Is it? I know about fundamental groups of knot complements, ...
0answers
26 views

### Is it true that the generators of the link quandle corresponding to arcs in a reduced diagram of a link are pairwise distinct?

A reduced diagram of a link $L$ is a diagram $D$ without a nugatory crossing (isthmi) and it is such that the number of crossings cannot be reduced using a finite sequence of Reidemeister moves that ...
0answers
28 views

### Looking for formulas of colored jones polynomial

I have found equations for the colored jones polynomial of the trefoil knot as a function of $n$, where $n$ is the coloring. I have also found equations for the colored jones polynomial of $(p,m)$ ...
0answers
25 views

### What is the point of the differing appearances of the HOMFLY skein?

What is the point of the differing appearances of the HOMFLY skein? For example, can somebody explain to me why one may prefer the relation $tP_P(q,t)-t^{-1}P_-(q,t)=(q-q^{-1})P_0(q,t)$ to the ...
0answers
63 views

### Homology of a submanifold complement

Let $X$ be a closed, connected, orientable smooth n-manifold and let $Y\subset X$ be a smooth closed m- submanifold. Express homology of the complement $H_*(X\smallsetminus Y;\mathbb{Z})$ in ...
1answer
72 views

### Pronunciation of HOMFLY polynomial

How should HOMFLY polynomial be pronounced? It is an acronym, but clearly chosen so that it could be pronounced as a word.
1answer
83 views

### Definition of equivalence of knots

I've just started reading An Introduction to Knot Theory from W. B. Raymond Lickorish and I've come up with the following definition: Definition. Two knots $K_1$ and $K_2$ are equivalent if there ...
1answer
151 views

### Fixed points of strong inversion of a knot

(Edit: added definition of strong inversion.) (2nd edit: added motivation for question at the end.) Definition: A knot $K$ in $S^3$ is strongly invertible if there is an involution of $(S^3,K)$ which ...
1answer
39 views

### Is the surface bounding the complement of an unlink in 3-space incompressible?

Let $L$ be an unlink of $n>1$ components in $\mathbb{S}^3$. Let $N=\mathbb{S}^3-L$. Is the boundary of $N$ incompressible?
1answer
40 views

### Can we apply Reidemeister moves to self crossings of one component only of a “trivial” link and keep the others unchanged?

Suppose $L=L_1 \cup L_2 \cup L_3$ be a classical link of three components. Suppose $L$ is an unlink, that is $L$ can be splitted into three simple closed curves. Assume that $L$ has a diagram in 2-...
0answers
18 views

### Is there a way to get from a polynomial to a link with the polynomial as its Jones polynomial?

Sorry for the convoluted wording of my question. Suppose that I have some polynomial $P$. Is there an algorithm for getting a link $L$ such that $V(L)=P$? Is there an algorithm for getting the ...
0answers
34 views

### How can one find a knot with a given Alexander polynomial?

Suppose that I'm given a polynomial $P$ such that $P$ is the Alexander polynomial of some knot. Is there an algorithm to come up with a projection of a knot $K$ such that $\Delta(K)=P$? Are there ...
1answer
77 views

### 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 ...
2answers
67 views

### 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 ...
1answer
71 views