Related papers: On the warping sum of knots
We show for an alternating knot the minimal boundary slope of an essential spanning surface is given by the signature plus twice the minimum degree of the Jones polynomial and the maximal boundary slope of an essential spanning surface is…
We study the band-unknotting number $u_{nb}(K)$ of a knot $K$, and how it behaves with respect to connect sums. We show that this sub-additive function is not additive under connected sums, by finding infinitely many examples of knots $K_1,…
A $\textit{knot}$ is a possibly wild simple closed curve in $S^3$. A knot $J$ is $\textit{semi-isotopic}$ to a knot $K$ if there is an annulus $A$ in $S^3\times[0,1]$ such that $A\cap(S^3\times\{0,1\})=\partial…
Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the…
A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the…
We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are…
We say that a knot $k_1$ in the $3$-sphere {\it $1$-dominates} another $k_2$ if there is a proper degree 1 map $E(k_1) \to E(k_2)$ between their exteriors, and write $k_1 \ge k_2$. When $k_1 \ge k_2$ but $k_1 \ne k_2$ we write $k_1 > k_2$.…
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…
A knot is a closed loop in space without self-intersection. Two knots are equivalent if there is a self homeomorphism of space bringing one onto the other. An arc presentation is an embedding of a knot in the union of finitely many half…
A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to…
In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given…
We study the connected sum of Hopf links in $S^3$. Particularly, we compute the entanglement entropy (EE) as a function of the number of link components. We find evidence of lower and upper bounds for the entanglement entropy. We show that…
We show that there exist knots K in S^3 with g(E(K))=2 and g(E(K#K#K))=6. Together with Theorem~1.5 of [1], this proves existence of counterexamples to Morimoto's Conjecture (Conjecture 1.5 of [2]). This is a special case of…
Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points.…
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot…
Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot…
We define the symmetric braid index $b_s(K)$ of a ribbon knot $K$ to be the smallest index of a braid whose closure yields a symmetric union diagram of $K$, and derive a Khovanov-homological characterisation of knots with $b_s(K)$ at most…
We consider the relationship between the crosscap number $\gamma$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ and $J$, we say $K \geq J$ if there exists an epimorphism…
In this paper, we introduce the concept of the warping degree for twisted knots, construct an invariant for them, and utilize it to establish a labeling scheme for these knots, known as ``warping labeling". We have identified that a warping…
The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the l1-norm. We show analogous results to those of Gromov, Pardon and…