Related papers: On Fox's trapezoidal conjecture
X.S. Lin's original definition of twisted Alexander knot polynomial is generalized for arbitrary finitely presented groups. J. Cha's fibering obstruction theorem is generalized. The group of a nontrivial virtual knot shown by L. Kauffman to…
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques…
For arborescent links, we present an efficient method of computing their Alexander polynomials. Applying this method, we express the Alexander polynomials of Montesinos links in terms of certain functions associated to rational tangles…
The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $\Delta_L(t)$ and has been shown to share some of its most geometric features in several…
A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of…
The Tait conjecture states that reduced alternating diagrams of links in S^3 have the minimal number of crossings. It has been proved in 1987 by M. Thistlethwaite, L.H. Kauffman and K. Murasugi studying the Jones polynomial. In this paper…
An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge…
We use Reidemeister torsion to study a twisted Alexander polynomial, as defined by Turaev, for links in the projective space. Using sign-refined torsion we derive a skein relation for a normalized form of this polynomial.
We give a complete proof of results announced by Hirasawa and Sakuma describing explicitly the Kakimizu complex of a non-split, prime, special, alternating link.
We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from…
In this work, we find a closed form formula for the braid index of an $n$-bridge braid, a class of positive braid knots which simultaneously generalizes torus knots, 1-bridge braids, and twisted torus knots. Our proof is elementary,…
The connected sum of two flat virtual knots depends on the choice of diagrams and basepoints. We show that any minimal crossing diagram of a composite flat virtual knot is a connected sum diagram. We also show the crossing number of flat…
Alternative proof is given for an earlier presented result that if a link in 3-space bounds a compact oriented proper surface (without closed component) in the upper half 4-space, then the link bounds a ribbon surface in the upper half…
Morifuji computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and the Reidemeister torsion of genus one two-bridge knots, a class of knots which…
By the Fox's re-embedding theorem, any compact submanifold of the 3-sphere can be re-embedded in the 3-sphere so that it is unknotted. It is unknown whether the Fox's re-embedding can be replaced with twistings. In this paper, we will show…
Checkerboard framings are an extension of checkerboard colorings for virtual links. According to checkerboard framings, in 2017, Dye obtained an independent invariant of virtual links: the cut point number. Checkerboard framings and cut…
The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. Generalizing work of Balm, Friedl, Kalfagianni and Powell, and of Lidman and Moore, we give an Alexander…
We present an accurate detailed exposition of the proof of existence of the Alexander-Conway polynomial (of links in 3-dimensional space). Other proofs were given by J. Alexander, J. Conway, V. Mantourov and L. Kauffman.
The Alexander polynomial of a knot has been generalized in three different ways to give twisted invariants. The resulting invariants are usually referred to as twisted Alexander polynomials, higher-order Alexander polynomials and…
We use twisted Alexander polynomials to show that certain algebraically slice 2-bridge knots are not topologically slice, even though all prime power Casson-Gordon signatures vanish. We also provide some computations indicating the efficacy…