arXiv research

Proves cosmetic crossing conjecture for certain knots.

Alexander polynomial condition blocks crossing changes in some knots.

Formula for Casson-Walker invariant; applies to knot complements and cosmetic crossing conjecture.

Proves special alternating knots can't have cosmetic crossings.

The cosmetic crossing conjecture (also known as the "nugatory crossing conjecture") asserts that the only crossing changes that preserve the oriented isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery characterization of the unknot to prove this conjecture for knots in integer homology sphere…

Study knots with genus one, finds Gordian distance and cosmetic crossing constraints.

We study cosmetic crossings in knots of genus one and obtain obstructions to such crossings in terms of knot invariants determined by Seifert matrices. In particular, we prove that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic …

Let K' be a knot that admits no cosmetic crossing changes and let C be a non-trivial, prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K' admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-…

Proves cosmetic surgery conjecture for strongly invertible knots.

We show that for genus one knots the Alexander polynomial and the homology of the double cover branching over the knot provide obstructions to cosmetic crossings. As an application we prove the nugatory crossing conjecture for the negatively twisted, positive Whitehead doubles of all knots. We also verify the conjectur…

A symmetric union of two knots is a classical construction in knot theory which generalizes connected sum, introduced by Kinoshita and Terasaka in the 1950s. We study this construction for the purpose of finding an infinite family of hyperbolic non-fibered three-bridge knots of constant determinant which satisfy the we…

We show that two Dehn surgeries on a knot $K$ never yield manifolds that are homeomorphic as oriented manifolds if $V_K''(1)\neq 0$ or $V_K'''(1)\neq 0$. As an application, we verify the cosmetic surgery conjecture for all knots with no more than $11$ crossings except for three $10$-crossing knots and five $11$-crossin…

The paper verifies no cosmetic surgeries on knots and 3-manifolds using hyperbolic geometry.

Knots from a specific band sum have similar homologies but are distinct.

Study shows alternating knots can't have similar crossings.

Two knot families meet cosmetic surgery conjecture.

We prove the nugatory crossing conjecture for fibered knots. We also show that if a knot $K$ is $n$-adjacent to a fibered knot $K'$, for some $n>1$, then either the genus of $K$ is larger than that of $K'$ or $K$ is isotopic to $K'$.

If a knot $K$ in $S^3$ admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either $\pm 2$ or $\pm 1/q$ for some value of $q$ that is explicitly determined by the knot Floer homology of $K$. Moreover, in the former case the genus of $K$ must be two, and in the latter case there is bound relati…

New quantum obstructions prevent purely cosmetic surgeries on knots.

We prove that the property of admitting no cosmetic crossing changes is preserved under the operation of forming certain satellites of winding number zero. We also define strongly cosmetic crossing changes and we discuss their behavior under the operation of inserting full twists in the strings of closed braids.

Paper proves knots satisfy a conjecture using Jones polynomial.

Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.

We show that if K is a satellite knot which admits a generalized cosmetic crossing change of order q with |q| \geq 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot which admits a pattern knot that is fibered ca…

Study finds new knot distances and chirally cosmetic bands using grid diagrams.

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert $\mathbb{Z}/2\mathbb{Z}$-homology sphere or a toroidal irreducible non-Seifert surgery then t…

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.

Pseudo links have two crossing types: classical crossings and indeterminate crossings. They were first introduced by Ryo Hanaki as a possible tool for analyzing images produced by electron microscopy of DNA. A normalized bracket polynomial is defined for pseudo links and then used to construct and obstruction to cosmet…

Study shows most knots up to 10 crossings can't be chirally cosmetic.

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in $S^3$ do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for connected sums of knots by analysing the JSJ-structures.

Cosmetic surgeries on pretzel knots are unique.

In this paper, we generalize the Cosmetic Surgery Conjecture to an $n$-cusped hyperbolic $3$-manifold and prove it under the assumption of another well-known conjecture in number theory, so called the Zilber-Pink Conjecture. For $n=1$ and $2$, we show them without the assumption.

Study tangle equations linking enzyme actions to knot theory.

Update: The Cosmetic Surgery Conjecture modulo finitely many Dehn-filling coefficients has been a well-known classical result, so the first main result of this paper is not new. (But the author was initially unaware of this fact, and the tools and techniques used here are very different from all the classically known m…

The study confirms that most positive 2-bridge knots up to 31 crossings do not have chirally cosmetic surgeries.

Study confirms contact cosmetic surgery for most knots, with exceptions.

New method proves cosmetic surgery conjecture for certain knots.

This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in $S^3$ must be $\pm 1$ and the surgery must be toroidal but not Seifert fibred. As consequence we…

We present various examples of cosmetic bandings on knots and links, that is, bandings on knots and links leaving their types unchanged. As a byproduct, we give a hyperbolic knot which admits exotic chirally cosmetic surgeries yielding hyperbolic manifolds. This gives a counterexample to a conjecture raised by Bleiler,…

Special knots with many twists have no certain type of surgery.

The study shows a limit on cosmetic surgeries for certain knots.

The study proves large alternating Montesinos knots cannot have purely cosmetic surgeries.

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. It is seen that all the two-bridge knots at most 9 crossings other than $9_{27} = S(49,19)=C[2,2,-2,2,2,-2]$ admits no purely cosmetic surgery pairs. Then we show that any two-bridge knot of the Conway form $[2x,2,-2x,2x,2,-2x]$ with $x \ge …

Study extends contact cosmetic surgeries to non-trivial Legendrian knots in L-spaces.

Let $K$ be a knot in a rational homology sphere $M$. This paper investigates the question of when modifying $K$ by adding $m>0$ half-twists to two oppositely-oriented strands, while keeping the rest of $K$ fixed, produces a knot isotopic to $K$. Such a two-strand twist of order $m$, as we define it, is a generalized cr…

This paper gives an alternate definition of the Affine Index Polynomial (called the Wriggle Polynomial) using virtual linking numbers and explores applications of this polynomial. In particular, it proves the Cosmetic Crossing Change Conjecture for odd virtual knots and pure virtual knots. It also demonstrates that the…

Study satellite knots using bordered Floer theory, proving non-thinness and calculating genus.

New framework distinguishes knots via neighborhood invariants.

Let K be a knot in S^3, and M and M' be distinct Dehn surgeries along K. We investigate when M covers M'. When K is a torus knot, we provide a complete classification of such covers. When K is a hyperbolic knot, we provide partial results in the direction of the conjecture that M never covers M'.