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Results tagged with knot-theory
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user 428
Knot theory is dealing with embedding of curves in manifolds of dimension 3. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. Many knot invariants are known and can be used to distinguish knots.
18
votes
Is Murasugi's conjecture still open?
It's actually a conjecture of Fox, sometimes known as the trapezoidal conjecture: the absolute values of the coefficients of $\Delta_K(t)$ are nonincreasing if K is an alternating knot. I think the o …
- 6,589
16
votes
Accepted
What is Floer homology of a knot?
I can say something about this for Heegaard Floer homology. Given a 3-manifold Y, you can take a Heegaard splitting, i.e. a decomposition of Y into two genus g handlebodies joined along their boundar …
- 6,589
13
votes
Accepted
Knot theory question: bridge number vs. min generators of fundamental group of complement
The (p,q) torus knot has a presentation with two generators, namely $\langle x,y \mid x^p = y^q\rangle$, but if $p,q>2$ then it's non-alternating and so it must have bridge index greater than 2.
- 6,589
13
votes
Accepted
$0$-surgeries on trefoil and figure-eight
If you're happy bringing in heavy machinery then you could compute some sort of Floer homology, like the 'hat' version of Heegaard Floer homology: this has rank 2 for $S^3_0(3_1)$ and rank 4 for $S^3_ …
- 6,589
12
votes
Accepted
Knot groups with big number of generators
If $\pi_1(S^3\setminus K)$ has a presentation with $n$ generators then its representation variety $\mathrm{Hom}(\pi_1(S^3\setminus K),SL_2(\mathbb{C}))$ is a subvariety of $(SL_2(\mathbb{C}))^n$, whic …
- 6,589
11
votes
Accepted
Is every quasipositive knot strongly quasipositive?
As pointed out by Hedden, Livingston showed that strongly quasipositive knots have $g(K) = g_4(K) = \tau(K)$, where $g_4$ is the smooth slice genus and $\tau$ is the Ozsváth-Szabó concordance invarian …
- 6,589
10
votes
Accepted
Counting knots with fixed number of crossings
There are some known exponential bounds on the number. For example, if kn is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that
…
- 6,589
9
votes
Prime decomposition for knots in manifolds
This was addressed in Miyazaki, Conjugation and the prime decomposition of knots in closed, oriented 3-manifolds, using the definition of connected sum suggested by Ryan Budney in the comments.
The m …
- 6,589
9
votes
Accepted
Which knot complements are double branched covers?
The complements of strongly invertible knots can always be realized as branched double covers of tangles. Torus knots are strongly invertible, so their complements actually do arise in this way.
For …
- 6,589
8
votes
Accepted
Are there spaces in which there are no fibered knots?
The answer for knots is still "no", because if you have an open book decomposition with disconnected binding then you can stabilize it (see section 2 of Etnyre's lecture notes) by attaching a handle t …
- 6,589
8
votes
Property P and R for general 3-manifolds
The generalized Property R conjecture stated above is known for nullhomologous knots $K$ in a rational homology 3-sphere $Y$. The only surgery that can produce $Y \# (S^1\times S^2)$ is the zero-surg …
- 6,589
6
votes
Accepted
Surgery along an arc connecting the components of a $2$-component link gives the unknot
One source of restrictions is the Montesinos trick: if you take the branched double cover of $L$, then a small neighborhood of the framed arc lifts to a solid torus because it intersects $L$ in two sm …
- 6,589
6
votes
Sliceness of knots
Both $6_1$ and $3_1 \# m(3_1)$ are smoothly slice (as is the unknot), and I claim that all other knots of at most seven crossings are not integrally slice. This will follow from two claims: first, if …
- 6,589
5
votes
Accepted
Distinct knots with same $A$-polynomial
The torus knots $T_{7,15}$ and $T_{3,35}$ have the same A-polynomials. In general, if $p,q>1$ are coprime and odd then $T_{p,q}$ has A-polynomial $(L-1)(LM^{pq}+1)(LM^{pq}-1)$, which only depends on t …
- 6,589
5
votes
Covering of a knot complement
Gonzalez-Acuña and Whitten answered this question for coverings by knot exteriors, as opposed to link exteriors more generally, in chapter 3 of "Imbeddings of three-manifold groups". They prove that …
- 6,589