Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAlexander polynomial of ribbon knots
137
0
0.0
(
0
)
Authors
Sheng Bai
Ask ChatGPT about the research
No Arabic abstract
Conway-normalized Alexander polynomial of ribbon knots depend only on their ribbon diagrams. Here ribbon diagram means a ribbon spanning the ribbon knot marked with the information of singularities. We further give an algorithm to calculate Alexander polynomials of ribbon knots from their ribbon diagrams.
rate research
Read More
We define a family of virtual knots generalizing the classical twist knots. We develop a recursive formula for the Alexander polynomial $Delta_0$ (as defined by Silver and Williams) of these virtual twist knots. These results are applied to provide evidence for a conjecture that the odd writhe of a virtual knot can be obtained from $Delta_0$.
We show that if a link J in the 3-sphere is homotopy ribbon concordant to a link L then the Alexander polynomial of L divides the Alexander polynomial of J.
We study Kauffmans model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,p)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in crossing number. We give a new way to fold $(p,q)$ torus knots, and show that their folded ribbonlength is bounded above by $p+q$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 5. We then show that any $(p,q)$ torus knot $K$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $ccdot Cr(K)^{1/2}$, providing an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.
We give a new interpretation of the Alexander polynomial $Delta_0$ for virtual knots due to Sawollek and Silver and Williams, and use it to show that, for any virtual knot, $Delta_0$ determines the writhe polynomial of Cheng and Gao (equivalently, Kauffmans affine index polynomial). We also use it to define a second-order writhe polynomial, and give some applications.
The fusion number of a ribbon knot is the minimal number of 1-handles needed to construct a ribbon disk. The strong homotopy fusion number of a ribbon knot is the minimal number of 2-handles in a handle decomposition of a ribbon disk complement. We demonstrate that these invariants behave completely differently under cabling by showing that the (p,1)-cable of any ribbon knot with fusion number one has strong homotopy fusion number one and fusion number p. Our main tools are Juhasz-Miller-Zemkes bound on fusion number coming from the torsion order of knot Floer homology and Hanselman-Watsons cabling formula for immersed curves.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
