6
votes
1answer
143 views

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.
4
votes
2answers
222 views

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. ...
1
vote
2answers
105 views

Questions about knot (link) of surface in four dimension

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 ...
2
votes
1answer
96 views

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 ...
0
votes
1answer
176 views

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 ...
2
votes
1answer
323 views

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): ...
26
votes
1answer
1k views

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 ...
1
vote
0answers
214 views

subset embedding gives trefoil knot

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 ...
0
votes
1answer
230 views

“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 ...
2
votes
1answer
260 views

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 ...
2
votes
2answers
995 views

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 ...
2
votes
0answers
128 views

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 ...
0
votes
0answers
165 views

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 ...
1
vote
1answer
339 views

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 ...
4
votes
1answer
333 views

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 ...
3
votes
2answers
255 views

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 ...
2
votes
1answer
331 views

Is the Hopf link a Brunnian link?

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 ...
1
vote
3answers
731 views

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 ...
12
votes
5answers
1k views

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 ...
3
votes
2answers
408 views

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 ...
3
votes
1answer
1k views

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 ...
5
votes
2answers
908 views

Computations in Knot Homology Theories

1) Relative to one another, how computable are the various knot homology theories? For example, how many crossings can we allow a knot and still hope to compute its Khovanov homology, versus its knot ...
15
votes
4answers
815 views

HOMFLY and homology; also superalgebras

My understanding is that an analogy along the following lines is (roughly) true: "The Alexander polynomial is to knot Floer homology is to gl(1|1) as the Jones polynomial is to Khovanov homology is ...