Related papers: Pretzel links, mutation, and the slice-ribbon conj…

It is well-known that all 2-knots are slice. Are all 2-links slice? This is an outstanding open question. In this paper we prove the following: For any 2-component 2-link (J,K)in the 4-sphere which bounds the 5-ball B^5, there is an…

In this paper we use Heegaard Floer link homology to determine the dual Thurston polytope for pretzel links of the form P(-2r_1-1, 2q_1, -2q_2, 2r_2+1) where r_i and q_i are positive integers. We apply this result to determine the Thurston…

We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we…

In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots.…

Twisted torus links $T(p,q;r,s)$ generalize torus links by introducing $s$ additional twists on $r$ adjacent strands of the torus link $T(p,q)$. It is well known that the number of components of a torus link $T(p, q)$ is given by the…

We study Kauffman's 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…

We prove that an odd pretzel knot is doubly slice if it has $2n+1$ twist parameters consisting of $n+1$ copies of $a$ and $n$ copies of $-a$ for some odd integer $a$. Combined with the work of Issa and McCoy, it follows that these are the…

The colored $\mathfrak{sl}_{3}$ Jones polynomial $J_{(n_{1}, n_{2})}^{\mathfrak{sl}_{3}}(L;q)$ are given by a link and an $(n_{1}, n_{2})$-irreducible representation of $\mathfrak{sl}_{3}$. In general, it is hard to calculate $J_{(n_{1},…

There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S,…

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating…

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen…

In this paper, we introduce \textit{graph-pretzel links}, a generalization of classical pretzel links based on spatial graph projections. As our main result, we investigate a subfamily associated with the complete graph on four vertices to…

In this paper, we study the self delta-equivalence of pretzel links. If the number of components is 2, then we know the complete invariants in terms of the Conway polynomial for classification. We calculate the values. For pretzel links…

We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always…

In the 60's Levine proved that if $R$ is a slice knot, then on any genus $g$ Seifert surface for $R$ there is a $g$ component link $J$, called a derivative of $R$, on which the Seifert form vanishes. Many subsequent obstructions to $R$…

For n >1, if the Seifert form of a knotted 2n-1 sphere K in S^{2n+1} has a metabolizer, then the knot is slice. Casson and Gordon proved that this is false in dimension three (n = 1). However, in the three dimensional case it is true that…

We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it…

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…

The $T$-genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots…

This paper compares notions of double sliceness for links. The main result is to show that a large family of 2-component Montesinos links are not strongly doubly slice despite being weakly doubly slice and having doubly slice components.…