arXiv subjects

Taylor Martin

Publications and source records attributed to Taylor Martin.

4 recordsLinked to original sources

arXiv2026

Gordian distance and clasper surgery for links

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, Taniyama, and Yamada proved that $C_k$-equivalence, and by extension finite type invariants, say very little about the unknotting number by showing that any knot is at most one crossing change away from being $C_k$-trivial for any $k\in \mathbb{N}$. The same is not true for links, since the pairwise linking number gives a lower bound on unlinking and is an invariant of $C_2$-equivalence. We prove that, aside from the linking number, the result of Ohyama, Taniyama, and Yamada extends to links: any $n$-component link with linking number zero can be reduced to a $C_k$-trivial link in at most $n^2$ crossing changes. As a consequence, Milnor's invariants carry only limited information about the unlinking number. To establish a lower bound, we produce a sequence of $n$-component links for which the crossing change distance to a $C_k$-trivial link grows quadratically in $n$. Notably, these bounds are independent of the choice of $k\in \mathbb{N}$. Finally, we determine the exact number of crossing changes to a $C_k$-trivial link for links with nonzero linking numbers and where no component is $C_k$-trivial.

math.GT↗

arXiv2024

How many crossing changes or Delta-moves does it take to get to a homotopy trivial link?

The homotopy trivializing number, $n_h(L)$, and the Delta homotopy trivializing number, $n_\Delta(L)$, are invariants of the link homotopy class of $L$ which count how many crossing changes or Delta moves are needed to reduce that link to a homotopy trivial link. In 2022, Davis, Orson, and Park proved that the homotopy trivializing number of $L$ is bounded above by the sum of the absolute values of the pairwise linking numbers and some quantity $C_n$ which depends only on $n$, the number of components. In this paper we improve on this result by using the classification of link homotopy due to Habegger-Lin to give a quadratic upper bound on $C_n$. We employ ideas from extremal graph theory to demonstrate that this bound is close to sharp, by exhibiting links with vanishing pairwise linking numbers and whose homotopy trivializing numbers grows quadratically. In the process, we determine the homotopy trivializing number of every 4-component link. We also prove a cubic upper bound on the difference between the Delta homotopy trivializing number of $L$ and the sum of the absolute values of the triple linking numbers of $L$.

math.GT↗

arXiv2024

Rectangular mosaics for virtual knots

Mosaic knots, first introduced in 2008 by Lomanoco and Kauffman, have become a useful tool for studying combinatorial invariants of knots and links. In 2020, by considering knot mosaics on $n \times n$ polygons with boundary edge identification, Ganzell and Henrich extended the study of mosaic knots to include virtual knots - knots embedded in thickened surfaces. They also provided a set of virtual mosaic moves preserving knot and link type. In this paper, we introduce rectangular mosaics for virtual knots, defined to be $m \times n$ arrays of classical knot mosaic tiles, along with an edge identification of the boundary of the mosaic, whose closures produce virtual knots. We modify Ganzell and Henrich's mosaic moves to the rectangular setting, provide several invariants of virtual rectangular mosaics, and give algorithms for computations of common virtual knot invariants.

math.GT↗

arXiv2021

Moves relating C-complexes: A correction to Cimasoni's "A geometric construction of the Conway potential function"

In groundbreaking work from 2004, Cimasoni gave a geometric computation of the multivariable Conway potential function in terms of a generalization of a Seifert surface for a link called a C-complex. Lemma 3 of that paper provides a family of moves which relates any two C-complexes for a fixed link. This allows for an approach to studying links from the point of view of C-complexes and in following papers it has been used to derive invariants. This lemma is false. We present counterexamples, a correction with detailed proof, and an analysis of the consequences of this error on subsequent works that rely on this lemma.

math.GT↗