Research Publications

 

 

 

On a nonorientable analogue of the Milnor Conjecture
Comparing nonorientable three genus and nonorientable four genus of torus knots

Joint with Slaven Jabuka, Journal of Knot Theory and Its Ramifications, 29 (2020), no. 3, 15 pages. (Click here.)

 

Abstract: We compare the values of the nonorientable three genus (or crosscap number) and the nonorientable four genus of torus knots. In particular, we show that the difference between these two invariants can be arbitrarily large. This contrasts with the orientable case: Seifert proved in 1935 that the orientable three genus of the torus knot T(p, q) is (p-1)(q-1)/2, and subsequently in 1993, Kronheimer and Mrowka proved that the orientable four genus of T(p, q) is also this value.

Joint with Slaven Jabuka, submitted. Posted here.

 

Abstract: The nonorientable 4-genus of a knot K is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot K. We study a conjecture proposed by Batson about the nonorientable 4-genus for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound formulated by Ozsvath, Stipsicz, and Szabo. As a side product we obtain new closed formulas for the signature of torus knots.

The four-genus of connected sums of torus knots

Joint with Charles Livingston, Mathematical Proceedings of the Cambridge Philosophical Society, (2017), 1 -- 20.

 

Abstract: We study the four-genus of linear combinations of torus knots: aT (p, q )# -􀀀bT (p',q'). Fixing positive p, q, p', and q', our focus is on the behavior of the four-genus as a function of positive a and b . Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsvath-Stipsicz-Szabo determines the four-genus for all a and b ; for the third, a surprising interplay between signatures and Upsilon appears.

An obstruction to slicing iterated Bing doubles

Journal of Knot Theory and its Ramifications22 (2013), no. 6, 10 pages

 

Abstract: Recent advances in understanding slicing properties of Bing doubles of knots have depended on properties of iterated covering links. We expand and refine this covering link calculus. Our main application here is a simplified proof of the following result of Cha and Kim: If the iterated Bing double of a knot K is slice, then K is algebraically slice. Further applications are included in joint work with Livingston studying 4-genera of Bing doubles. The techniques also appear in the work of Adam Levine studying mixed Bing-Whitehead doubles.

Ozsvath-Szabo and Rasmussen invariants of cable knots

Algebraic & Geometric Topology10 (2010), 825 -- 836.

 

Abstract: We study the behavior of the Ozsvath--Szabo and Rasmussen knot concordance invariants K(m,n), the (m,n)-cable of a knot K where m and n are relatively prime.  We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K(m,n) differ from their value on the torus knot T(m,n) by fixed constants for all but finitely many n>0.  Combining this result together with Hedden's extensive work on the behavior of the Ozsvath--Szabo invariant on (m,mr+1)-cables yields bounds on the value of the invariant on any (m,n)-cable of K.  In addition, several of Hedden's obstructions for cables bounding complex curves are extended.

 

Concordance of Bing doubles and boundary genus

Joint with Charles Livingston, Mathematical Proceedings of the Cambridge Philosophical Society151 (2011), 459--470.

 

Abstract: Cha and Kim proved that if a knot K is not algebraically slice, then no iterated Bing double of K is concordant to the unlink.  We prove that if K has nontrivial signature, then the nth-iterated Bing double of K is not concordant to any boundary link with boundary surfaces of genus less than 2^{n-1}*signature. The same result holds with signature replaced by twice the Ozsvath-Szabo knot concordance invariant.

Relationships between braid length and the number of braid strands

Algebraic & Geometric Topology7 (2007), 181 -- 196.

 

Abstract: For a knot K, let l(K,n) be the minimum length of an n-stranded braid representative of K.  Fixing a knot K, l(K,n) can be viewed as a function of n, which we denote by l_K(n).  Examples of knots exist for which l_K(n) is a non-increasing function.  We investigate the behavior of l_K(n), developing bounds on the function in terms of the genus of K.  The bounds lead to the conclusion that for any knot K the function l_K(n) is eventually stable.  We study the stable behavior of l_K(n), with stronger results for homogeneous knots.  For knots of nine or fewer crossings, we show that l_K(n) is stable on all of its domain and determine the function completely.