arXiv subjects

Holt Bodish

Publications and source records attributed to Holt Bodish.

4 recordsLinked to original sources

arXiv2024

Genus, Fiberedness, $τ$ and $ε$ of Satellite Knots with $n$-Twisted Generalized Mazur patterns

We study a family of $(1,1)$-pattern knots that generalize the Mazur pattern, and compute the concordance invariants $τ$ and $ε$ of $n$-twisted satellites formed from these patterns. We show that none of the $n$-twisted patterns from this family act surjectively on the smooth or rational concordance group. We also determine when the $n$-twisted generalized Mazur patterns are fibered in the solid torus, compute their genus in $S^1 \times D^2$, and show that $n$-twisted satellites with generalized Mazur patterns and non-trivial companions are not Floer thin.

math.GT↗

arXiv2023

Some Three and Four-Dimensional Invariants of Satellite Knots with (1,1)-Patterns

We use bordered Floer homology, specifically the immersed curve interpretation of the bordered pairing theorem, to compute various three- and four-dimensional invariants of satellite knots with arbitrary companions and patterns from a family of knots in the solid torus that have the knot type of the trefoil in $S^3$. We compute the three-genus, and bound the four-genus of these satellites. We show that all patterns in this family are fibered in the solid torus. This implies that satellites with fibered companions and patterns from this family are also fibered. We also show that satellites with thin fibered companions or companions $K$ with $τ(K)=\pm g(K)$ formed from these patterns have left or right veering monodromy. We then use this to show that satellites with thin fibered companion knots $K$ so that $|τ(K)|<g(K)$ formed from these patterns do not have thin knot Floer homology.

math.GT↗

arXiv2022

Non-Trivial Steenrod Squares on Prime, Hyperbolic and Satellite Knots

We show that there are prime knots so that the Steenrod operations of Lipshitz and Sarkar arXiv:1204.5776 are non trivial on their Khovanov homology. This answers a question posed by Lipshitz and Sarkar in their paper arXiv:1709.03602. We then go on to show that there are hyperbolic and satellite knots so that the Steenrod operations are non trivial on their Khovanov homology.

math.GT↗

arXiv2022

Obstructing Reducible Surgeries: Slice Genus and Thickness Bounds

In this paper, we study reducible surgeries on knots in $S^3$. We develop thickness bounds for L-space knots that admit reducible surgeries, and lower bounds on the slice genus for general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots, which was mostly worked out in \cite{DeY21b}. We also provide a new upper bound on reducing slopes for fibered, hyperbolic slice knots and on multiple reducing slopes for slice knots. Our techniques involve the $d$-invariants and mapping cone formula from Heegaard Floer homology.

math.GT↗