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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.
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
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 …
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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 …
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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 …
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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
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_ …
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3
votes
Accepted
Does tangle closure determine the triviality of the tangle?
The branched double covers $\Sigma(NS(T))$ and $\Sigma(D_+(T))$ are both $S^3$ by assumption, and they're each built by gluing $\Sigma(T)$ to the branched double cover of a different rational tangle ( …
- 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
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 …
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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 …
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2
votes
{0,1} Maslov potentials on Legendrian knots
I don't have anything close to a complete answer, but for many $tb$-maximizing knots we can get some obstructions from Legendrian contact homology. The generators for the DGA $A(K)$ associated to a g …
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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
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.
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4
votes
Accepted
Why Tristram-Levine signature jumps at the zeros of alexander polynomial?
If $A$ is a Seifert matrix for $K$ and $\omega \in \mathbb{C}$ has norm 1, then the Tristram-Levine signature $\sigma_\omega(K)$ is the signature of the matrix
$(1-\omega)A + (1-\bar{\omega})A^T = ( …
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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 …
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