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In [B. Kwon and S. Kang, Rectangle conditions and families of 3-bridge prime knots, Topol. Appl. 291 (2021) 107453], using the set E A T k of all essential alternating rational 3-tangles for positive integer k , the authors showed that all knot diagrams in the numerator closure set C N (E A T 2 l + 1) and the denominator closure set C D (E A T 2 l + 2) with l > 0 are 3-bridge prime knot diagrams. In this paper, for n > 4 we construct a set A A T 4 n of additions of alternating rational tangles in E A T 4 . The set A A T 4 n generalizes E A T k and contains it as a subset for some k. We show that any closure set C (A A T 4 n) on A A T 4 n so that the resulting diagrams are reduced and alternating knot diagrams represent alternating 3-bridge prime knot diagrams. Since a tangle diagram in A A T 4 n + 1 is constructed inductively from a tangle diagram in A A T 4 n by adding only one crossing positively, the result of this paper supports the conjecture that 3-bridge property is preserved under one-crossing alternating addition positively to alternating 3-bridge knots in 3-bridge representations. [ABSTRACT FROM AUTHOR]

A slope p q is called a characterizing slope for a given knot K 0 ⊂ S 3 if whenever the p q -surgery on a knot K ⊂ S 3 is homeomorphic to the p q -surgery on K 0 via an orientation preserving homeomorphism, then K = K 0 . In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot T 5 , 2 . More explicitly besides all negative integer slopes there are 2 4 7 slopes which were unknown to be characterizing for T 5 , 2 , including 8 9 nontrivial L -space slopes. Applying recent work of Baldwin–Hu–Sivek, we improve our result by showing that a nontrivial slope p q is a characterizing slope for T 5 , 2 if p q > − 1 and p q ∉ { 0 , 1 , ± 1 2 , ± 1 3 }. In particular every nontrivial L -space slope of T 5 , 2 is characterizing for T 5 , 2 . More explicitly this work yields 1 2 1 new characterizing slopes for T 5 , 2 . Another interesting consequence of this work is that if a nontrivial p q -surgery on a non-torus knot in S 3 yields a manifold of finite fundamental group, then | p | > 9. [ABSTRACT FROM AUTHOR]

A knot shadow is a diagram without over/ under information at all crossings. Many knot types are obtained from a knot shadow by assigning over/ under information to each crossing. The purpose of the paper is to observe numbers of knot types obtained from closed 3-braid knot shadows. As a result, we discover that given an arbitary odd integer n ≥ 3 , there exists a closed 3-braid knot shadow S such that the number of T (m , 2) (3 ≤ m ≤ n) torus knots is more than that of trivial knots in the knots obtained from S by assigning over/ under information to each crossing. [ABSTRACT FROM AUTHOR]

A nontrivial knot is called minimal if its knot group does not surject onto the knot groups of other nontrivial knots. In this paper, we determine the minimality of the rational knots C (2 n + 1 , 2 m , 2) in the Conway notation, where m ≠ 0 and n ≠ 0 , − 1 are integers. When | m | ≥ 2 , we show that the nonabelian SL 2 (ℂ) -character variety of C (2 n + 1 , 2 m , 2) is irreducible and therefore C (2 n + 1 , 2 m , 2) is a minimal knot. The proof of this result is an interesting application of Eisenstein's irreducibility criterion for polynomials over integral domains. [ABSTRACT FROM AUTHOR]

Taking the Levine–Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017, Cimasoni and Conway generalized this formula to the multivariable signature of the closure of colored tangles. In this paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor ℱ ω . We also show that in the case of colored braids this defect can be rewritten in terms of the Meyer cocycle and the colored Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys. [ABSTRACT FROM AUTHOR]

A petal diagram of a knot is a projection with a single multi-crossing such that there are no nested loops. The petal number p (K) of a knot K is the minimum number of loops among all petal diagrams of K. Let T n , s denote the (n , s) -torus knot for relatively prime integers 2 ≤ n < s. Recently, Kim et al. proved that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 whenever s ≡ ± 1 mod n. They conjectured that the inequality holds without the assumption s ≡ ± 1 mod n. They also showed that p (T n , s) = 2 s − 1 whenever 2 ≤ n < s < 2 n and n ≡ 1 mod s − n. Their proofs construct petal grid diagrams for those torus knots. In this paper, we prove the conjecture that p (T n , s) ≤ 2 s − 2 ⌊ s n ⌋ + 1 holds for any 2 ≤ n < s. We also show that p (T n , s) = 2 s − 1 holds for any 2 ≤ n < s < 2 n. Our proofs construct petal grid diagrams for any torus knots. [ABSTRACT FROM AUTHOR]

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot K such that a (K) = 2 and g (K) = n , and that there exists a knot K' such that a (K ′) ≥ n and g (K ′) = 1 for any positive integer n. We also give an upper bound of the ascending number for a 2-bridge knot. [ABSTRACT FROM AUTHOR]

We say that a set of pairs of disjoint cycles Λ (G) of a graph G is linked if for any spatial embedding f of G there exists an element λ of Λ (G) such that the 2 -component link f (λ) is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, (1) we show that the set of all pairs of disjoint cycles of G is minimally linked if and only if G is essentially same as a graph in the Petersen family, and (2) for any two integers p , q ≥ 3 , we exhibit a minimally linked set of Hamiltonian (p , q) -pairs of cycles of the complete graph K p + q with at most 18 elements. [ABSTRACT FROM AUTHOR]

A group of matrices with entries in a number field K is integral if it has a finite index subgroup of matrices whose entries are algebraic integers. In this paper, we show that the fundamental groups, G (p / q , (m , n)) , of the orbifolds (p / q , (m , n) are integral as subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , for all the rational knots and links p / q and all the isotropies with m > 2 , n > 2. We obtain the same result for the fundamental group G (m , m , m) of the orbifold B (m , m , m) , m > 2 , where B are the Borromean rings. The only groups G (m , n , p) with m , n , p ∈ 3 , 4 , 5 , 6 , ∞ , which are integral subgroups of both S L (2 , ℂ) and of S L (4 , ℝ) , are for the following (m , n , p) : { (3 , 3 , 3) , (3 , 3 , ∞) , (3 , 4 , 4) , (3 , 4 , ∞) , (3 , 6 , 6) , (3 , ∞ , ∞) , (4 , 4 , 4) , (4 , 4 , ∞) , (4 , ∞ , ∞) , (6 , 6 , 6) , (6 , 6 , ∞) , (∞ , ∞ , ∞) , (3 , 3 , 5) , (3 , 5 , 5) , (3 , 5 , ∞) , (4 , 4 , 5) , (4 , 5 , ∞) , (5 , 5 , 5) , (5 , 5 , ∞) , (5 , 6 , 6) , (5 , ∞ , ∞) }. [ABSTRACT FROM AUTHOR]

Let r be an odd integer, r ≥ 3. Then the petal number of the torus knot of type (r , r + 2) is equal to 2 r + 3. [ABSTRACT FROM AUTHOR]

We give a necessary condition for a 4-tangle T to be embedded in a link by using a common divisor of two integers obtained from the Kauffman bracket polynomials of the numerator N(T) and the denominator D(T) of the tangle T. The Kauffman bracket polynomial 〈 L 〉 of a link L is a regular isotopy invariant of links taking values in the Laurent polynomial ring ℤ [ A , A − 1 ]. For each integer i, we associate an integer B L (i) , which is obtained from the Kauffman bracket polynomial 〈 L 〉 of L. Let d be the greatest common divisor of B N (T) (i) and B D (T) (i) for a 4-tangle T. We will show that there exists a natural number n such that d divides i n B L (i) if the tangle T is embedded in a link L. [ABSTRACT FROM AUTHOR]

We prove that every ℤ 2 H-thin link has no 2 k -torsion for k > 1 in its Khovanov homology. Together with previous results by Lee [The support of the Khovanov's invariants for alternating knots, preprint (2002), arXiv:math.GT/0201105; An endomorphism of the Khovanov invariant, Adv. Math.197(2) (2005) 554–586, arXiv:math.GT/0210213] and the author [Torsion of Khovanov homology, Fund. Math.225 (2014) 343–364, arXiv:math.GT/0405474], this implies that integer Khovanov homology of non-split alternating links is completely determined by the Jones polynomial and signature. Our proof is based on establishing an algebraic relation between Bockstein and Turner differentials on Khovanov homology over ℤ 2 . We conjecture that a similar relation exists between the corresponding spectral sequences. [ABSTRACT FROM AUTHOR]