Articles of knot theory

Has knot theory led to the development of better knots?

Knot theory was likely originally motivated by the study of real-world knots such as these: Indeed, mathematical knot tables to this day look not too dissimilar from the familiar “age of sail”-style knot collections that decorate the walls of countless homes and restaurants around the world: So, has knot theory as a purely topological discipline […]

How does the topology of the graphs' Riemann surface relate to its knot representation?

Let me give a worked-out example: The following cubic planar non-simple graph $\hskip2.3in$ has the adjacency matrix $A=\pmatrix{0&3\\3&0}$. The graph has three faces, so the rank of $G$ is $\chi(G)=2$. The reciprocal of Ihara’s $\zeta$ function can be evaluated $$ \frac{1}{\zeta_G(u)}={(1-u^2)^{\chi(G)-1}\det(I – Au + 2u^2I)}\\ ={(1-u^2) (4u^4-5u^2+1)} $$ EDIT: Then $\zeta_G(u)=\frac{(1-u^2)^{-1} }{(4u^4-5u^2+1)}=\prod_p (1-u^{L(p)})$ with the […]

embedding projective plane in 4-space?

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?

Braid invariants resource

What are some braid invariants (analogous to the idea of knot invariants) or a resource where I can find them?

How is PL knot theory related to smooth knot theory?

I really want to like knot theory but the PL condition seems sort of ugly. I was hoping someone could give me a justification for secretly thinking about smooth knots as I read through a book like Rolfsen which works with PL knots. I’m really hoping for some grand unifying or almost-unifying theorem. Thank you […]

A generalization of the connected sum of links

A connected sum of two links $K$ and $L$ involves cutting a segment in each link and joining them up as illustrated in the top diagram, the connected sum of two trefoil knots. Is there any terminology for taking a link and doing the same process but to itself, as illustrated by the bottom diagram, […]

The Abelianization of $\langle x, a \mid a^2x=xa\rangle$

I wish to verify the following statement (which comes from Fox, “A Quick Trip Through Knot Theory”, although that is probably not important). “$\Gamma=\pi_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so the homology of $M$ is infinite cyclic.” So, I need to find the Abelianization of the fundamental group. Using the relations I get $$y_1:=[x,a]=x^{-1}ax,\qquad y_2:=[a,x]=x^{-1}a^{-1}x$$ […]

Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable connected surface with nonempty boundary and abelian fundamental group must be a disk or an annulus. Therefore $\pi_1(S^3 \setminus L)$ is abelian only […]

Are isomorphic the following two links?

Please consider the following links with four components My question is if such two links are isomorphic. The corresponding Jones polynomials are respectively It is observed that the ratio of the Jones polynomials is $q^{9⁄2}$. It is to say the only difference between the two Jones polynomials is a simple monomial. According with such fact […]

Are there Kirby diagrams for manifolds with boundaries?

There are Kirby diagrams for 3- and 4-manifolds which consist of framed links corresponding to 1- and 2-handles attached to a single 0-handle. Any such diagram will give a unique closed manifold since the 3- (and 4-) handles that close the 0-1-2-handlebody are unique. Is there a good notation for Kirby diagrams for manifolds with […]