Related papers: Surface distance on knots
The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) =…
This paper has been withdrawn by the authors. Significantly revised versions of the results of this paper are now available in arXiv:0707.0487v2 and arXiv:0808.3169v1.
We define a pants distance for knotted surfaces in 4-manifolds which generalizes the complexity studied by Blair-Campisi-Taylor-Tomova for surfaces in the 4-sphere. We determine that if the distance computed on a given diagram does not…
The fundamental quandle is an invariant for distinguishing surface knots, yet computable presentations have traditionally been limited to surfaces embedded in the $4$-sphere. Building on the framework of banded unlink diagrams introduced by…
Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.
This paper has been withdrawn by the author due to its main result being included in cond-mat/0403309 by the same author.
This paper has been withdrawn by the author.
This paper has been withdrawn by the authors.
This paper has been withdrawn by the author.
This paper has been withdrawn.
This paper has been withdrawn by the author due to time-consuming revision.
This paper has been withdrawn by the author. It will be replaced, substantially modified, by sections of the author's completed PhD thesis.
This paper has been withdrawn by the authors due to a gap in the proof of the main result (in 5.3).
This paper has been withdrawn by the author.
Following Goussarov's paper `Interdependent Modifications of Links and Invariants of Finite Degree' [Topology 37 (1998) 595--602] we describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory…
This paper has been withdrawn by the author due to the incomplete results.
In 2000, Habiro introduced the notion of $C_k$-equivalence of knots and links. This geometric filtration is closely connected to finite type invariants, a class of invariants including Milnor's invariants. Shortly thereafter, Ohyama,…
This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling.
This paper has been withdrawn by the authors due to a likely hole in the proof.
This paper has been withdrawn by the author due to pending experimental investigation to avoid certain potential experimental uncertainty.