4-move kills Alexander polynomial
problem Whether the 4-move is an unknotting operation
method showing every knot can be reduced via 4-moves and isotopies
result every knot can be reduced to one with a trivial Alexander polynomial
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9 results for “4-moves”
Paper provides a condition to distinguish links up to 4-moves.
problem Distinguishing links up to 4-moves is generally difficult.
method Defined a quotient of the fundamental group invariant preserved by 4-moves.
result Provides a necessary condition for triviality up to 4-moves.
We study the 4-move invariant \crl\ for links in the 3-sphere developed by Dabkowski and Sahi, which is defined as a quotient of the fundamental group of the link complement. We develop techniques for computing this invariant and show that for several classes of knots it is equal to the invariant for the unknot; theref…
A -move is a local operation for links consisting in replacing two parallel arcs by four half twists. At the present time, it is not known if this induces an unkotting operation for knots. Studying the Dabkowski-Sahi invariant, we prove that any invariant of knots based on the fundamental group …
We show that every alternating link of 2-components and 12 crossings can be reduced by 4-moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan.
In this paper we describe three geometric applications of quandle homology. We show that it gives obstructions to tangle embeddings, provides the lower bound for the 4-move distance between links, and can be used in determining periodicity of links.
On C_n-moves for linksmath.GT
A C_n-move is a local move on links defined by Habiro and Goussarov, which can be regarded as a `higher order crossing change'. We use Milnor invariants with repeating indices to provide several classification results for links up to C_n-moves, under certain restrictions. Namely, we give a classification up to C_4-move…
Knots of genus twomath.GT
We classify all knot diagrams of genus two and three, and give applications to positive, alternating and homogeneous knots, including a classification of achiral genus 2 alternating knots, slice or achiral 2-almost positive knots, a proof of the 3- and 4-move conjectures, and the calculation of the maximal hyperbolic v…
This paper is based on my talks (`Skein modules with a cubic skein relation: properties and speculations' and `Symplectic structure on colorings, Lagrangian tangles and its applications') given in Kyoto (RIMS), September 11 and September 18 respectively, 2001. The first three sections closely follow the talks: starting…