Narrow Results

We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call “Jones rational coincidences”. We provide moves on the continued fraction expansion of the associated rational which we prove do not change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant $<900$) that these moves are sufficient to generate all Jones rational coincidences. In the process we give a new formula for the Jones polynomial of a rational knot based on a continued fraction expansion of the associated rational, which has significantly fewer terms than other formulae known to us. The paper is based on the second author’s Ph.D. thesis and gives an essentially self-contained account.

Given a properly embedded graph Γ in a ball B and a punctured sphere Σ properly embedded in B-Γ we examine the conditions on Γ that are necessary to assure that Σ is boundary parallel.

Generalized Reidemeister moves provide an extended set of moves to work with virtual knots and links. We introduce virtual tangle moves, generalization of classical rational tangle moves and show that such generalizations are essential to develop new invariants of virtual knots and links. We show that every 2-algebraic virtual link is a virtual 4-move equivalent to a trivial link or Hopf link. The properties of virtual tangle move are analyzed on few existing invariants associated with virtual knots and links.

In this paper, a method is given to calculate the Jones polynomial of the 6-plat presentations of knots by using a representation of the braid group 𝔹₆ into a group of 5 × 5 matrices. We also can calculate the Jones polynomial of the 2n-plat presentations of knots by generalizing the method for the 6-plat presentations of knots. Also, it helps us to detect 3-bridge knots in 3-plat presentations.

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results.

We use Kauffman’s bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.

We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots, singular knots, and rigid vertex spatial graphs, Contemp. Math.689 (2017) 1–10] by inserting rational tangles at the labeled singularities to extend usual knot invariants to our class of singular knots. We show that we can use invariants of labeled singular knots to serve usual singular knots. Labeled framed pseudoknots are also introduced and discussed.

We extend the classical coloring rule to compute the tangle (fraction) number of a virtual rational tangle. We use the contemporary argument used by Kauffman [Virtual knot theory, Eur. J. Comb. 20(7) (1999) 662–690] and show that tangle (fraction) number is a complete invariant. The main purpose of this paper is to provide an alternate yet elementary proof of John Conway’s classification theorem for the virtual rational tangles.

In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set $EA{T}^k$ of all essential alternating rational 3-tangles for positive integer $k$, the authors showed that all knot diagrams in the numerator closure set $C_N(EA{T}^{2l+1})$ and the denominator closure set $C_D(EA{T}^{2l+2})$ with $l>0$ are 3-bridge prime knot diagrams. In this paper, for $n>4$ we construct a set $AA{T}_4^n$ of additions of alternating rational tangles in $EA{T}^4$. The set $AA{T}_4^n$ generalizes $EA{T}^k$ and contains it as a subset for some $k$. We show that any closure set $C(AA{T}_4^n)$ on $AA{T}_4^n$ so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in $AA{T}_4^{n+1}$ is constructed inductively from a tangle diagram in $AA{T}_4^n$ by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations.

We extend the classical notion of unknotting operation to include operations on rational tangles. We recall the “classical” conditions (on the signature, linking form etc.) for a knot to have integral (respectively rational) unknotting number one. We show that the generalised Casson invariant of the twofold branched cover of the knot gives a further necessary condition. We apply these results to some Montesinos knots and to knots with less than nine crossings.

A subset of legendrian 2-string tangles are defined to be minimal if the strands realize the minimum absolute value of the Bennequin invariant. The restrictiveness of this condition is examined by studying which topological rational tangles have minimal representatives. It is shown that a rational of finite parity has a minimal representative if and only if its standard continued fraction expansion contains only non-negative entries, is of odd length, and has every horizontal entry even. This is proved by applying recent results of Fuchs and Tabachnikov or Chmutov and Goryunov that give an upperbound for the Bennequin invariant of links in terms of the minimal exponent of the framing variable of the Kauffman polynomial and Yokota's precise formula for this topological invariant. A second geometric stratum of rational tangles with infinite parity is defined and it is then shown a positive rational has a minimal represenative if and only if its continued fraction expansion is a particular extension of a minimal of finite parity and a negative rational of infinite parity has a minimal representative if and only if it has even length with every vertical entry even.