Related papers: Crosscap numbers of alternating links via state co…
This is the first in a series of four papers wherein we enumerate all prime alternating knots and links. In this first paper, we introduce four operators on knots and show that, when used according to very simple rules on the prime…
For any given number of crossings $c$, there exists a formula to determine the number of 2-bridge knots of $c$ crossings, and indeed it is a simple matter to actually construct presentations of these knots. However, the determination of…
This paper, to be regularly updated, lists those prime knots with the fewest possible number of crossings for which values of basic knot invariants, such as the unknotting number or the smooth 4-genus, are unknown. This list is being…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
Determining unknotting numbers is a large and widely studied problem. We consider the more general question of the unknotting number of a spatial graph. We show the unknotting number of spatial graphs is subadditive. Let $g$ be an embedding…
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.
Using exhaustive techniques and results from Lackenby and many others, we compute the tunnel number of all 1655 alternating 11 and 12 crossing knots and of 881 non-alternating 11 and 12 crossing knots. We also find all 5525 Montesinos knots…
We compute the smooth 4-genera of the prime knots with 12 crossings whose values, as reported on the KnotInfo website, were unknown. This completes the calculation of the smooth 4-genus for all prime knots with 12 or fewer crossings.
We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…
We study the integrable crosscap states of the integrable quantum spin chains and we classify them for the $\mathfrak{gl}(N)$ symmetric models. We also give a derivation for the exact overlaps between the integrable crosscap states and the…
We give three algorithms to determine the crosscap number of a knot in the 3-sphere using $0$-efficient triangulations and normal surface theory. Our algorithms are shown to be correct for a larger class of complements of knots in closed…
We introduced concept of meander knots, 2-component meander links and multi-component meander links and derived different families of meander knots and links from open meanders with at most 16 crossings. We also defined semi-meander knots…
A conjecture proposed by J. Tripp in 2002 states that the crossing number of any knot coincides with the canonical genus of its Whitehead double. In the meantime, it has been established that this conjecture is true for a large class of…
A quadruple crossing is a crossing in a projection of a knot or link that has four strands of the knot passing straight through it. A quadruple crossing projection is a projection such that all of the crossings are quadruple crossings. In a…
Given a knot in the 3-sphere, the non-orientable 4-genus or 4-dimensional crosscap number of a knot is the minimal first Betti number of non-orientable surfaces, smoothly and properly embedded in the 4-ball, with boundary the knot. In this…
There are 46,972 prime knots with crossing number 14. Among them 19,536 are alternating and have arc index 16. Among the non-alternating knots, 17, 477, and 3,180 have arc index 10, 11, and 12, respectively. The remaining 23,762 have arc…
This is the second of a part series devoted to enumerating prime alternating knots and links. In Part I, we introduced four operators on knots and showed that if these operators are applied to the set of all prime alternating knots of n…
Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…
An equilateral stick number $s_{=}(K)$ of a knot $K$ is defined to be the minimal number of sticks required to construct a polygonal knot of $K$ which consists of equal length sticks. Rawdon and Scharein [12] found upper bounds for the…
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples…