Narrow Results

In this paper, we study on knots and closed incompressible surfaces in the 3-sphere via Morse functions. We show that both knots and closed incompressible surfaces can be isotoped into a "related Morse position" simultaneously. As an application, we have the following results.

• Smallness of Montesinos tangles with length two and Kinoshita's theta curve.

• Classification of closed incompressible and meridionally incompressible surfaces in 2-bridge theta-curve and handcuff graph complements and the complements of links which admit Hopf tangle decompositions.

It has been conjectured that for knots K and K′ in S³, w(K # K′) = w(K) + w(K′) - 2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots for which they conjectured that where Bⁿ is a bridge number n knot. We show that for n > 2 none of the knots in produces such counterexamples.

We give the rectangle condition for strong irreducibility of Heegaard splittings of 3-manifolds with non-empty boundary. We apply this to a generalized Heegaard splitting of 2-fold covering of S³ branched along a link. The condition implies that any thin meridional level surface in the link complement is incompressible. We also show that the additivity of width holds for a composite knot satisfying the condition.

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 $\mathrm{\text{tr}}(K_1\#K_2)=max\{\mathrm{\text{tr}}(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any width-minimizing embedding of a knot $K$ also minimizes the trunk of $K$. We produce several families of probable counterexamples to this conjecture.

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer $m$ such that the knot can be represented as a knot $m$-mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an $m$-mosaic and any knot $K$ that is represented on the mosaic, its crossing number $c$ is bounded above by ${(m-2)}^2-2$ if $m$ is odd, and by ${(m-2)}^2-(m-3)$ if $m$ is even. In the process, we develop a useful new tool called the mosaic complement.