A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The crosscap number of a knot in the 3-sphere is the minimal genus of non-orientable surface bounded by the knot. We determine the crosscap numbers of torus knots.
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).
We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twe…
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm is known. We present three methods for computing crosscap number that offer varyi…
Study knots with immersed crosscap number 1 and find nonorientable representatives of homology classes.
problem Understanding knots and their nonorientable representations in 3-manifolds.
method Analyzing knots with immersed crosscap number 1, studying differences between immersed and embedded crosscap numbers, and constructing specific 3-manifolds.
result Knots with immersed crosscap number 1 are specific types of torus or cable knots. Immersed crosscap number can differ from embedded crosscap number by large amounts and is not bounded.
For a torus knot K, we bound the crosscap number c(K) in terms of the genus g(K) and crossing number n(K): c(K) \leq [(g(K)+9)/6] and c(K) \leq [(n(K) + 16)/12]. The (6n-2,3) torus knots show that these bounds are sharp.
Study compares nonorientable genus values of torus knots.
problem Comparing nonorientable genus values of torus knots.
method Examined torus knots T(p,q) with p even, q odd, and calculated differences in nonorientable three and four genus values.
result The difference between nonorientable three and four genus values on torus knots T(p,q) grows arbitrarily large for any fixed odd q, as p ranges over values of a fixed congruence class modulo q.
Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a clos…
M. Scharlemann has recently proved that any genus one tunnel number one knot is either a satellite or 2-bridge knot, as conjectured by H. Goda and M. Teragaito; all such knots admit a (1,1) decomposition. In this paper we give a classification of the family of (1,1) knots in S3 with crosscap number two (i.e., boundi…
Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class…
A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus, orientability, and aggregate slope that a surface spanning that link can have. To thi…
We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus g≥3. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.
We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap …
We define the crosscap number of a 2-component link as the minimum of the first Betti numbers of connected, non-orientable surfaces bounding the link. We discuss some properties of the crosscap numbers of 2-component links.
The crosscap number of a knot in the 3-sphere is defined as the minimal first Betti number of non-orientable subsurfaces bounded by the knot. In this paper, we determine the crosscap numbers of pretzel knots. The key ingredient to obtain the result is the algorithm of enumerating all essential surfaces for Montesinos k…
A knot in S^3 is said to have crosscap number two if it bounds a once-punctured Klein bottle but not a Moebius band. In this paper we give a method of constructing crosscap number two hyperbolic (1,2)-knots with tunnel number one which are neither 2-bridge nor (1,1)-knots. An explicit infinite family of such knots is d…
For a knot K, the concordance crosscap number, c(K), is the minimum crosscap number among all knots concordant to K. Building on work of G. Zhang, which studied the determinants of knots with c(K) < 2, we apply the Alexander polynomial to construct new algebraic obstructions to c(K) < 2. With the exception of low cross…
We consider non-orientable closed surfaces of minimum crosscap number in the (p,q)-lens space L(p,q)≅V1∪∂V2, where V1 and V2 are solid tori. Bredon and Wood gave a formula for calculating the minimum crosscap number. Rubinstein showed that L(p,q) with p even has only one isotopy cla…