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Results tagged with knot-theory
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user 27111
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.
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One question about the quandle
Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston i …
- 63.9k
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Is there a known invariant for knotted surfaces defined by skein relations?
It is known that a knotted surface can be presented by a marked graph diagram, which is just a knot diagram while some crossing points are equipped with markers. On the other hand, two marked graph di …
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Jones polynomial of 2-knots
Question: is it possible to define the Jones polynomial for knotted surfaces (or $S^2$ for simplicity) in $R^4$?
Jones polynomial has several definitions (see How many definitions are there of the Jo …
- 6,881
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Braid group and knot group
It is well known that braid groups and knot groups share many common properties. For example, they have the same $H_1$ and they are both residually finite and (hence) Hopfian. On the other hand, we kn …
- 96.4k
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Invariance of Khovanov homology under first Reidemester move
I do not understand your example of Hopf link, since one cannot remove a crossing by the first Reidemeister move on this diagram. As a simple exercise to check the result, you can write down the chain …
- 517
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Knot invariants with skein relation of order 3 or 4
The following two papers discuss the link invariants which satisfy "higher order" skein relations. Similar to the Jones polynomial, here only one crossing point is resolved, but all possible smoothing …
- 517
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Distance between two knots
If you want to define some distance on knots, you should have a (local) transformation/move on knots or or knot diagrams such that any two knots can be connected by finitely many of them.
As mentione …
- 517
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1
answer
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Unknotting number and crossing number
It is well known that if $c(K)=2n+1$, then $u(K)$ is less than $n+1$. It can not be sharper because of the trefoil knot. On the other hand, if $c(K)=2n$, then similarly we have $u(K)$ is less than $n+ …
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answer
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Quandle colorings under Reidemeister moves
Let $D$ be a knot diagram and $Q$ a quandle. We use $c$ to denote a fixed coloring of $D$ with $Q$. If $D'$ is another knot diagram of the same knot, and $R_1$ is a sequence of Reidemeister moves conn …
- 22.1k
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Reference on representations of knot groups
F. Gonzalez-Acuna. Homomorphs of knot groups. Ann. of Math. (2) 102 (1975), 373-377
In this paper the author studied the homomorphic images of knot groups. It was proved that a finite group is the h …
- 517
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Links with same Jones polynomial
It is well-known that the Jones polynomial of the connected sum of $L_1$ and $L_2$ is exactly the product of the Jones polynomial of $L_1$ and $L_2$. However for a pair of links $L_1$ and $L_2$, the c …
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