# Tagged Questions

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### Knots indistinguishable by HOMFLY

Is there any list (incomplete of course) of knots, that have similar HOMFLY polynomials? I am mainly interested in torus knots.
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### Can distinct open knots correspond to the same closed knot?

A topological ("closed") knot is an embedding of a circle in $\mathbb{R}^3$. It's possible for a knot to be distinct from the unknot because there are no free ends to move around and untie the knot. ...
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Consider three 2-torus ($S^1*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked ...
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### How to visulize surface link in four dimension?

I am now facing a problem with "surface link" in four dimension. I have heard that three 2-torus can be linked in four dimension. And I have created a movie by cutting four dimensional space with ...
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### Thurston-Bennequin number vs. checkerboard coloring difference

For an alternating knot K, checkerboard-color the knot (if this is a lousy ASCII crossing: %, white goes to the left/right and black to top/bottom). Assume no surplus Reidemeister 1 kinks exist (K has ...
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### Can Reidemeister 3 be weakened?

If you take the diagram of the Reidemeister 3 move and "shortcircuit" two ends, you get (click http://imgur.com/kRvZa if Imgur hotlink doesn't work): ...
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### Does this knot invariant distinguish trefoil chiralities?

Let $C_N$ denote the labelled configuration of $N^{th}$ roots of unity with $p_J = e^{\frac{2\pi iJ}{N}}$ for $J = 1\ldots N$. As a corollary of something else I was playing around with, I recently ...
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### 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 ...
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### “Skein” equations sets that can reduce any graph

Consider for example the approach to the Kuperberg G2 or the Yamada polynomial. I don't know whether relations of graph (in contrast to knot) theory are also usually called "skein" but I simply carry ...
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### Can a closed trefoil appear as a space-time “cut” of an open trefoil?

An open trefoil is a trefoil tied in an infinitely long line. An open trefoil that is at rest in (flat) space, in space-time is a knotted hypersurface. Different observers in space-time have ...
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### How many lanes has a freeway? (Crossing free Kuperberg G2, that is.) EDITED

For referencing, I keep the original title and post and ask only about the simplest case (and forget the freeway with crossings for now). Consider a trivalent graph, e.g. the dodecahedron or cube ...
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### Simple terminology question about the Dubrovnik (Kauffman) polynomial

In my S matrix classification attempts I encounter a lot of Dubrovnik polynomials of the form D(z-1/z,z^n) and D(-z+1/z,z^n). [Second variable is for writhe, n is an integer; for the first I don't ...
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### On Birman-Wenzlyfying the B2 spider

Prelude: First of all, let "S matrix" denote "an abstract 4D tensor satisfying the usual isotypy rules (with no arrows!)". I'm busy trying to classify all possible S matrices (paper pending) - it's ...
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### Braid*Temperley-Lieb=?

I would be very astonished if this algebra isn't named. You simply have the braid AND the Temperley-Lieb generator in the algebra. Rules are the usual Reidemeister equivalents plus the kink and ...
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### Cubic skein relations

Hi, please note that this question deals with undirected knots/links! The most generic cubic skein relation for a knot polynome would be where w^3 is one positive writhe unit. The form is fairly ...
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### Discriminant locus in knot space

Consider the space $K$ of all immersions of $S^1$ into $\mathbb R^3$. The set of knots with self-intersection is a discriminant in $K$ and divide it into "chambers". Let $f$ be a knot with $n$ double ...
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I'm trying to fill a woeful gap in my topological knowledge and learn a little knot and link theory (I'll be recording my progress on the nLab, starting with a page on links). Not wishing to write ...
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### SO(3) knot polynomials

Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix ...
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### Is the pure braid group on three strands generated as a normal subgroup of the braid group by the six-crossing braid?

Artin's presentation of braid group on three strands is: $$B_3 = \langle l,r : lrl = rlr \rangle$$ where you should think of "$l$" as the positive crossing between the left and middle strands and ...
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### SU(2) representations of alternating knot groups

Suppose that $K$ is an $\textit{alternating}$ knot in $S^3$, and let $R_0$ be the space of homomorphisms from $\pi_1(S^3 - K)\to SU(2)$ which send meridians to trace free matrices. Denote the subset ...
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### What is Floer homology of a knot?

I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology ...