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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]

In the paper, we give an equivalent description of the lens space L (p , q) with p prime in terms of any corresponding Heegaard diagrams as follows: Let M be a closed orientable 3-manifold with π 1 (M) ≠ 1 , and U ∪ F V a Heegaard splitting of genus n for M with an associated Heegaard diagram (U ; J 1 , ... , J n). Assume p is a prime integer. Then M is homeomorphic to the lens space L (p , q) if and only if there exists an embedding f : U ↪ L (p , q) such that L = { f (J 1) , ... , f (J n) } bounds a complete system of surfaces for W = L (p , q) \ f (U) ¯. [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]

For coprime integers p(> 0) and q, the (p, q)-cable Γ-polynomial of a knot is the Γ- polynomial of the (p, q)-cable knot of the knot, where the Γ-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. In this paper, we show that there exist infinitely many knots with the trivial (2, 1)-cable Γ- polynomial, that is, the (2, 1)-cable Γ-polynomial of the trivial knot. Moreover, we see that the knots have the trivial Γ-polynomial, the trivial first coefficient HOMFLYPT and Kauffman polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we first investigate the minimal sufficient sets of colors for and . For odd prime and any -colorable link with , we give alternative proofs of for and for . We also elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo , and only one such class of five colors modulo 13. [ABSTRACT FROM AUTHOR]

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on 'Quantum knots and mosaics' to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot -mosaic is an matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot -mosaics for any positive integers and , toroidal knot -mosaics for co-prime integers and , and furthermore toroidal knot -mosaics for a prime number . We also analyze the asymptotics of the growth rates of their cardinality. [ABSTRACT FROM AUTHOR]

In [Simple ribbon fusions and genera for links, J. Math. Soc. Japan 68 (2016) 1033-1045], we introduced special types of fusions, so-called simple-ribbon fusions on links, and showed that they never decrease the genera of links. In this paper, we also gave a geometric condition for a simple-ribbon fusion to preserve a link type, and it turns out that a 'complicated' simple-ribbon fusion may preserve a link type if the link is split. In this paper, we show that only 'clearly trivial' simple-ribbon fusion preserves a link type if the link is non-split. This enables us to give a lower bound for the difference between genera of knots related by a simple-ribbon fusion. [ABSTRACT FROM AUTHOR]

For any classical knot k 1 , we can construct a ribbon 2 -knot spun (k 1) by spinning an arc removed a small segment from k 1 about R 2 in R 4 . A ribbon 2 -knot is an embedded 2 -sphere in R 4 . If k 1 has an n -crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun (k 1). Thus, we can define naturally a notion on ribbon 2 -knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer n there exists an amphicheiral ribbon 2 -knot with the ribbon crossing number n. [ABSTRACT FROM AUTHOR]

For a virtual knot K and an integer r ≥ 0 , the r -covering K (r) is defined by using the indices of chords on a Gauss diagram of K. In this paper, we prove that for any finite set of virtual knots J 0 , J 2 , J 3 , ... , J m , there is a virtual knot K such that K (r) = J r (r = 0  and  2 ≤ r ≤ m) , K (1) = K , and otherwise K (r) = J 0 . [ABSTRACT FROM AUTHOR]

In this paper, we describe the equivalence classes of simple arcs between the two punctures on a 2-punctured torus Σ 1 , 2 up to isotopy by using the given four generators g 1 , g 2 , g 3 and g 4 . Actually, we show that a class of simple arcs is represented by an ordered sequence of four integers. Also, we introduce an algorithm to check whether or not an ordered sequence of four integers represents a class of simple arcs in Σ 1 , 2. We want to point out that this result classifies the (1 , 1) -positions. [ABSTRACT FROM AUTHOR]

It is proved by Ozawa that every knot in the 3-sphere has a unique essential tangle decomposition if it admits an essential free 2-tangle sphere. In this paper, we show that for every integer with , there is a knot in the 3-sphere which admits an essential free -tangle decomposition and also admits at least distinct essential tangle spheres. To this end, we give examples of nested essential free -tangles for every integer with . [ABSTRACT FROM AUTHOR]

We show that all knots up to six crossings can be represented by polynomial knots of degree at most , among which except for and all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any -crossing knot in degree ? In this paper we try to partially answer this question. For an integer , we define a set to be the set of all polynomial knots given by such that and . This set can be identified with a subset of and thus it is equipped with the natural topology which comes from the usual topology . In this paper we determine a lower bound on the number of path components of for . We define a path equivalence for polynomial knots in the space and show that it is stronger than the topological equivalence. [ABSTRACT FROM AUTHOR]

Satoh and Taniguchi introduced the n -writhe J n for each non-zero integer n , which is an invariant for virtual knots. The n -writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The values of n -writhes changed by some local moves are calculated. However for the virtualization, it is unknown. In this paper, we show that for any given non-zero integer n and any given integer N , there exists a virtual knot whose unknotting number by the virtualization is one and the value of the n -writhe equals N. Namely, the virtualization of a real crossing changes the value of n -writhe by any given integer N. [ABSTRACT FROM AUTHOR]

An (m , n) -branched twist spin is a fibered 2 -knot in S 4 which is determined by a 1 -knot K and coprime integers m and n. For a 1 -knot, Nagasato proved that the number of conjugacy classes of irreducible S L (2 , ℂ) -metabelian representations of the knot group of a 1 -knot is determined by the knot determinant of the 1 -knot. In this paper, we prove that the number of irreducible S L (2 , ℂ) -metabelian representations of the knot group of an (m , n) -branched twist spin is determined up to conjugation by the determinant of the associated 1 -knot in the orbit space by comparing a presentation of the knot group of the branched twist spin with the Lin's presentation of the knot group of the 1 -knot. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that a complete graph with n vertices has an optimal diagram, i.e. a diagram whose crossing number equals the value of Guy's formula, with a free maximal linear tree and without free Hamiltonian cycles for any odd integer n ≥ 7. [ABSTRACT FROM AUTHOR]

The ordinary unknotting number of 1-dimensional knots, which is defined by using the crossing-change, is a very basic and important invariant. Let n be a positive integer. It is very natural to consider the "unknotting number" associated with other local moves on n -dimensional knots. In this paper, we prove the following. For the ribbon-move on 2-knots, which is a local move on knots, we have the following: There is a 2-knot which is changed into the unknot by two times of the ribbon-move not by one time. The "unknotting number" associated with the ribbon-move is unbounded. For the pass-move on 1-knots, which is a local move on knots, we have the following: There is a 1-knot such that it is changed into the unknot by two times of the pass-move not by one time and such that the ordinary unknotting number is 4. For any positive integer n , there is a 1-knot whose "unknotting number" associated with the pass-move is > n and whose ordinary unknotting number is 4 n. Let p and q be positive integers. For the (p , q) -move on (p + q − 1) -knots, which is a local move on knots, we have the following: Let k be a non-negative integer. There is a (4 k + 2) -knot which is changed into the unknot by two times of the (2 k + 1 , 2 k + 2) -move not by one time. The "unknotting number" associated with the (2 k + 1 , 2 k + 2) -move is unbounded. There is a (4 k + 1) -knot which is changed into the unknot by two times of the (2 k + 1 , 2 k + 1) -move not by one. The "unknotting number" associated with the (2 k + 1 , 2 k + 1) -move is unbounded. We prove the following: For any positive integer n and any positive integer m < n , there is a (4 k + 1) -knot which is changed into the unknot by n times of the twist-move not by m times. [ABSTRACT FROM AUTHOR]

Knot mosaics are used to model physical quantum states. The mosaic number of a knot is the smallest integer m such that the knot can be represented as a knot m -mosaic. In this paper, we establish an upper bound for the crossing number of a knot in terms of the mosaic number. Given an m -mosaic and any knot K that is represented on the mosaic, its crossing number c is bounded above by (m − 2) 2 − 2 if m is odd, and by (m − 2) 2 − (m − 3) if m is even. In the process, we develop a useful new tool called the mosaic complement. [ABSTRACT FROM AUTHOR]

A state of a virtual knot diagram D is a collection of circles obtained from D by splicing all the real crossings. For each integer n ≥ 1 , we denote by s n ( D ) the number of states of D with n circles. The n -state number s n ( K ) of a virtual knot K is the minimum number of s n ( D ) for D of K. Let 𝒦 n ( i ) be the set of virtual knots K with s n ( K ) = i for an integer i ≥ 0. In this paper, we study the finiteness of 𝒦 n ( i ). We determine the finiteness of 𝒦 n ( 1 ) for any n ≥ 1 and 𝒦 1 ( i ) for any i ≥ 0. [ABSTRACT FROM AUTHOR]

For coprime integers and , the -cable -polynomial of a knot is the -polynomial of the -cable knot of the knot, where the -polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the -polynomial is computable in polynomial time, the -cable -polynomial is also computable in polynomial time. In this paper, we show that the -cable -polynomial completely classifies the unoriented knots up to ten crossings including the chirality information. [ABSTRACT FROM AUTHOR]

In this paper, we describe a method of making an invariant of virtual knots that is defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots using the invariant above. Moreover, we derive a relation of two invariants. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]

In this paper, the -Gordian complex of knots is defined. The authors show that for any knot and any positive integer , there is a finite set of knots of size containing , such that the -Gordian distance between any two knots in this set is 1 for all . [ABSTRACT FROM AUTHOR]

A framed knot with an integral coefficient determines a simply-connected 4-manifold by 2-handle attachment. Its boundary is a 3-manifold obtained by Dehn surgery along the framed knot. For a pair of such Dehn surgeries along distinct knots whose results are homeomorphic, it is a natural problem: Determine the closed 4-manifold obtained by pasting the 4-manifolds along their boundaries. We study pairs of lens space surgeries along distinct knots whose lens spaces (i.e. the resulting lens spaces of the surgeries) are orientation-preservingly or -reversingly homeomorphic. In the authors' previous work, we treated with the case both knots are torus knots. In this paper, we focus on the case where one is a torus knot and the other is a Berge's knot Type VII or VIII, in a genus one fiber surface. We determine the complete list (set) of such pairs of lens space surgeries and study the closed 4-manifolds constructed as above. The list consists of six sequences. All framed links and handle calculus are indexed by integers. [ABSTRACT FROM AUTHOR]

The -Turk's head link is presented by the alternating diagram which is obtained from the standard diagram of the -torus link by crossing changes. In this paper, we show that for any integers and , the coefficients of in the Conway polynomials of the - and -Turk's head links coincide for and differ by sign for . We conjecture that this property holds for any . [ABSTRACT FROM AUTHOR]

Quasi-alternating links are a generalization of alternating links. They are homologically thin for both Khovanov homology and knot Floer homology. Recent work of Greene and joint work of the first author with Kofman resulted in the classification of quasi-alternating pretzel links in terms of their integer tassel parameters. Replacing tassels by rational tangles generalizes pretzel links to Montesinos links. In this paper we establish conditions on the rational parameters of a Montesinos link to be quasi-alternating. Using recent results on left-orderable groups and Heegaard Floer L-spaces, we also establish conditions on the rational parameters of a Montesinos link to be non-quasi-alternating. We discuss examples which are not covered by the above results. [ABSTRACT FROM AUTHOR]

In this paper, we describe a method of making a polynomial invariant of flat virtual knots in terms of an integer labeling of the flat virtual knot diagram and an invariant of virtual links. We show that the polynomial is sometimes useful to detect non-invertibility and also to determine the virtual crossing number of a given flat virtual knot. [ABSTRACT FROM AUTHOR]

We show that for any positive integers a,b,m, and n, the Alexander polynomial of the (am,bn)-Turk’s head link is divisible by that of the (m,n)-Turk’s head link. [ABSTRACT FROM AUTHOR]

We compute the bridge spectra of cables of 2-bridge knots. We also give some results about bridge spectra and distance of Montesinos knots. [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]

Let M reg be the topological moduli space of long knots up to regular isotopy, and for any natural number n > 1 let M n reg be the moduli space of all n -cables of framed long knots which are twisted by a string link to a knot in the solid torus V 3 . We upgrade the Vassiliev invariant v 2 of a knot to an integer valued combinatorial 1-cocycle for M n reg by a very simple formula. This 1-cocycle depends on a natural number a ∈ ℤ ≅ H 1 (V 3 ; ℤ) with 0 < a < n as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on H 0 (M n reg ) × H 0 (M reg ) already for n = 2. [ABSTRACT FROM AUTHOR]

We show that, for any integers, g ≥ 3 and n ≥ 2 , there exists a link in S 3 such that its complement has a genus g Heegaard splitting with distance n. [ABSTRACT FROM AUTHOR]

In this study, we describe a method of making an invariant of virtual knots defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots and virtual doodles modifying the previous invariant. [ABSTRACT FROM AUTHOR]

A formula for the Alexander polynomial of a 2-bridge knot or link given by Hartley and also by Minkus has a beautiful interpretation as a walk on the integers. We extend this to the 2-variable Alexander polynomial of a 2-component, 2-bridge link, obtaining a formula that corresponds to a walk on the 2-dimensional integer lattice. [ABSTRACT FROM AUTHOR]

We construct an infinite tower of covering spaces over the configuration space of n − 1 distinct nonzero points in the complex plane. This results in an action of the braid group 𝔹 n on the set of n -adic integers ℤ n for all natural numbers n ≥ 2. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid B an infinite sequence of braids, whose braid types are invariants of B. We present computations for the cases of n = 2 and n = 3 and use these to show an infinite family of braids close to real algebraic links, i.e. links of isolated singularities of real polynomials ℝ 4 → ℝ 2 . [ABSTRACT FROM AUTHOR]

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist t c about a nonseparating circle c in the mapping class group ℳ (N g) of a nonorientable surface N g of genus g. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer n > 1 , then for each sufficiently large g , t c has a root of degree n in ℳ (N g). Moreover, for any possible degree n , we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in N g . [ABSTRACT FROM AUTHOR]

We introduce two sequences of two-variable polynomials { L K n (t , ℓ) } n = 1 ∞ and { F K n (t , ℓ) } n = 1 ∞ , expressed in terms of index value of a crossing and n -dwrithe value of a virtual knot K , where t and ℓ are variables. Basing on the fact that n -dwrithe is a flat virtual knot invariant, we prove that L K n and F K n are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using L K n we give sufficient conditions when virtual knot does not admit cosmetic crossing change. [ABSTRACT FROM AUTHOR]

Given a link L , a representation π 1 ( S 3 − L ) → SL ( 2 , ℂ ) is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when L is a Montesinos link. [ABSTRACT FROM AUTHOR]

A cord is a simple curve on a punctured plane. We introduce diagrams which represent isotopy classes of cords. Using such diagrams, we prove that there is a one-to-one correspondence from the set of the isotopy classes of cords to a set of symmetric matrices whose components are non-negative integers, and we give necessary conditions for matrices to represent the isotopy classes of cords. [ABSTRACT FROM AUTHOR]

We introduce the Schubert form for a -bridge link diagram, as a generalization of the Schubert normal form of a -bridge link. It consists of a set of six positive integers, written as , with some conditions and it is based on the concept of -butterfly. Using the Schubert normal form of a -bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers . The construction is a generalization of the form the link group presentation of the -bridge link depends on the integers and . [ABSTRACT FROM AUTHOR]

We discuss the higher order stabilization of the coefficients of the colored Jones polynomial. In particular, we find an expression for the second stable sequence of the colored Jones polynomial of a certain class of knots. We also determine which knots have the same higher order stability. [ABSTRACT FROM AUTHOR]

For any positive integer , we exhibit a prime knot with crossings whose Jones polynomial is equal to modulo . Our construction rests on a certain -crossing prime tangle which is undetectable by the Kauffman bracket polynomial pair mod . [ABSTRACT FROM AUTHOR]