Has the knot complement conjecture been proven for knots in solid tori?
2026-03-27 04:56:42.1774587402
Knot complement conjecture in solid tori
312 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in KNOT-THEORY
- Is unknot a composite knot?
- Can we modify one component of a link and keep the others unchanged
- Can we split a splittable link by applying Reidemeister moves to non-self crossings only
- Involution of the 3 and 4-holed torus and its effects on some knots and links
- Equivalence polygonal knots with smooth knots
- Can a knot diagram be recovered from this data?
- Does Seifert's algorithm produce Seifert surfaces with minimal genus?
- Equivalence of links in $R^3$ or $S^3$
- Homotopy type of knot complements
- The complement of a knot is aspherical
Related Questions in LOW-DIMENSIONAL-TOPOLOGY
- Getting a self-homeomorphism of the cylinder from a self-homeomorphism of the circle
- Does $S^2\times[-1,1]$ decompose as $B^3\#B^3$
- Homologically zero circles in smooth manifolds
- Can we really move disks around a compact surface like this?
- Why is this not a valid proof of the Poincare Conjecture?
- Regarding Surgery and Orientation
- Can a one-dimensional shape have volume?
- The inside of a closed compact surface $\sum_g$
- How do you prove that this set is open?
- Understanding cobordisms constructed from a Heegaard triple
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
If the conjecture is what I think it is (i.e. knots are determined by their compliments), then not, at least when knots are considered up to isotopy. A solid torus is an exterior of the unknot, so take a Whitehead link and twist around one of the components. The exterior of this link stays the same, and it is the exterior of the second component in the solid torus. But the second component changes (because you are performing meridional twists in the solid torus), and in fact the second component becomes a different knot (more or less) each time you twist, so these different knots cannot be isotopic in the solid torus (as they are not isotopic in $S^3$). I don't know the answer if you consider knots up to homeomorphism. But surely the answer factors through the corresponding question for two-component links with one component an unknot.