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We consider the question of how knots and their reverses are related in the concordance group \mathcal {C}. There are examples of knots for which K \ne K^r \in \mathcal {C}. This paper studies the cobordism distance, d(K, K^r). If K \ne K^r \in \mathcal {C}, then d(K, K^r) >0 and it is elementary to see that for all K, d(K, K^r) \le 2g_4(K), where g_4(K) denotes the four-genus. Here we present a proof that for non-slice knots satisfying g_3(K) = g_4(K), one has d(K,K^r) \le 2g_4(K) -1. This family includes all strongly quasi-positive knots and all non-slice genus one knots. We also construct knots K of arbitrary four-genus for which d(K,K^r) = g_4(K). Finding knots for which d(K,K^r) > g_4(K) remains an open problem. [ABSTRACT FROM AUTHOR]

In this paper we investigate the 0-concordance classes of 2-knots in S4, an equivalence relation that is related to understanding smooth structures on 4-manifolds. Using Rochlin's invariant, and invariants arising from Heegaard-Floer homology, we will prove that there are infinitely many 0-concordance classes of 2-knots. [ABSTRACT FROM AUTHOR]

In 1965, E. C. Zeeman proved that the (±1)-twist spin of any knotted sphere in (n−1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization of a knotted sphere and prove that a (±1)-twist spin of a knotted trivalent graph may be knotted. We then construct some families of knotted graphs for which the (±1)-twist spins are always unknotted. [ABSTRACT FROM AUTHOR]

We show that there are no distance one surgeries on non-null-homologous knots in M that yield -M (M with opposite orientation) if M is a 3-manifold obtained by a Dehn surgery on a knot K in S^{3}, such that the order of its first homology is divisible by 9 but is not divisible by 27. As an application, we show several knots, including the (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the (2,k) torus knot (k>1) has a chirally cosmetic banding if and only if k=5. [ABSTRACT FROM AUTHOR]

Let T be the group of smooth concordance classes of topologically slice knots, and let {0} ⊂ ··· ⊂ Tn+1 ⊂ Tn ⊂ ·· · ⊂ T0 ⊂ T be the bipolar filtration. Hedden, Kim, and Livingston showed that T contains a subgroup isomorphic to Z∞2 . In this paper, we show that a subset of the set of knots constructed by Hedden, Kim, and Livingston generates a subgroup isomorphic to Z∞2 in T0/T1. [ABSTRACT FROM AUTHOR]

This paper explores the relationship between the existence of an exact embedded Lagrangian filling for a Legendrian knot in the standard contact R3 and the hierarchy of positive, strongly quasi-positive, and quasipositive knots. On one hand, results of Eliashberg and especially Boileau and Orevkov show that every Legendrian knot with an exact, embedded Lagrangian filling is quasi-positive. On the other hand, we show that if a knot type is positive, then it has a Legendrian representative with an exact embedded Lagrangian filling. Further, we produce examples that show that strong quasi-positivity and fillability are independent conditions. [ABSTRACT FROM AUTHOR]

We address the primary decomposition of the knot concordance group in terms of the solvable filtration and higher order von Neumann ρ-invariants by Cochran, Orr, and Teichner. We show that for a non-negative integer n, if the connected sum of two n-solvable knots with coprime Alexander polynomials is slice, then each of the knots has vanishing von Neumann ρ-invariants of order n. This gives positive evidence for the conjecture that nonslice knots with coprime Alexander polynomials are not concordant. As an application, we show that if K is one of Cochran-Orr-Teichner's knots which are the first examples of nonslice knots with vanishing Casson-Gordon invariants, then K is not concordant to any knot with Alexander polynomial coprime to that of K. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a polynomial invariant fK(t) for virtual knots. We show that fK(t) can be used to distinguish some virtual knot from its inverse and mirror image. The behavior of fK(t) under a connected sum is also given. Finally, we discuss which kinds of polynomials can be realized as fK(t) for some virtual knot K. [ABSTRACT FROM AUTHOR]

In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in S3 is true for knots K1,K2 in S3 in which each Ki is either a tunnel number one or m-small, i = 1, 2. This extends two known results by Morimoto. [ABSTRACT FROM AUTHOR]

We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots. [ABSTRACT FROM AUTHOR]

We prove the cosmetic crossing conjecture for genus one knots with non-trivial Alexander polynomial. We also prove the conjecture for genus one knots with trivial Alexander polynomial, under some additional assumptions. [ABSTRACT FROM AUTHOR]

Negami found an upper bound on the stick number $ s(K)$ in terms of the minimal crossing number $ c(K)$ $ s(K) \leq 2 c(K)$ $ s(K) \leq c(K) 3$-bridge knot or link, except in the cases of the unlink and the Hopf link. In this paper we construct any $ 2$ of at least six crossings by using only $ c(K)$-bridge knots and links in terms of crossing numbers. [ABSTRACT FROM AUTHOR]

Any surface-knot $F$ in 4-space can be projected into 3-space with a finite number of triple points, and its triple point number, ${\rm t}(F)$, is defined similarly to the crossing number of a classical knot. By definition, we have ${\rm t}(F_1# F_2)\leq {\rm t}(F_1)+{\rm t}(F_2)$ for the connected sum. In this paper, we give infinitely many pairs of surface-knots for which this equality does not hold. [ABSTRACT FROM AUTHOR]

For a knotted surface in $4$-space, its generic projection into $3$-space has branch points as its singularities, and its successive projection into $2$-space has fold points and cusps as its singularities. In this paper, we show that for non-orientable knotted surfaces, the numbers of branch points and cusps can be minimized by isotopy. [ABSTRACT FROM AUTHOR]

In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation. [ABSTRACT FROM AUTHOR]

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. Let $L$ denote the fundamental group of the link consisting of a chain of $4$ circles. It is shown that $L$ is not subgroup separable. Furthermore, it is shown that $L$ is a subgroup of every known non-subgroup separable compact 3-manifold group. It is asked whether all such examples contain $L$. [ABSTRACT FROM AUTHOR]

Let $K_1$ and $K_2$ be two knots in $S^3$ and $t(K_1)$, $t(K_2)$ the tunnel numbers of them. In this paper, we show that if both $K_1$ and $K_2$ are small, then $t(K_1 # K_2) \ge t(K_1) + t(K_2)$. Moreover we show that $t(K_1 # K_2 # \cdots # K_n) \ge t(K_1) + t(K_2) + \cdots + t(K_n)$ for any small knots $K_1, K_2, \cdots , K_n$. [ABSTRACT FROM AUTHOR]

In knot theory, Vassiliev's 1-knot invariants are defined in a combinatorial way as finite type invariants. By a natural generalization of the combinatorial definition, one has a certain family of 2-knot invariants, which should be called finite type 2-knot invariants. They form a subspace of the whole space of ``Vassiliev 2-knot invariants''. In this paper we prove that it is 1-dimensional. [ABSTRACT FROM AUTHOR]

For some families of two-bridge knots, including double twist knots with genus at least four, we determine precisely the set of integers n > 1 such that the fundamental group of the n-fold cyclic branched cover of the 3-sphere along these knots is left-orderable. There are knots, including the figure-eight knot, for which this set is empty. We give the first class of hyperbolic knots, not of this type, for which these integers can be completely determined. [ABSTRACT FROM AUTHOR]

It is known that connected sums of positive torus knots are not concordant to L-space knots. Here we consider differences of torus knots. The main result states that the subgroup of the concordance group generated by two positive torus knots contains no nontrivial L-space knots other than the torus knots themselves. Generalizations to subgroups generated by more than two torus knots are also considered. [ABSTRACT FROM AUTHOR]

Hom gives an example of a knot with vanishing Upsilon invariant but nonzero epsilon invariant. We build more such knots that are linearly independent in the smooth concordance group. [ABSTRACT FROM AUTHOR]

For a given knot, we study the minimal number of positive eigenvalues of the double branched cover over spanning surfaces for the knot. The value gives a lower bound for various genera, the dealternating number and the alternation number of knots, and we prove that Batson's bound for the non-orientable 4-genus gives an estimate of the value. In addition, we use the value to give a necessary condition for being quasi-alternating. [ABSTRACT FROM AUTHOR]

Let ν be either the Ozsváth–Szabó τ–invariant or the Rasmussen s–invariant, suitably normalized. For a knot K, Livingston and Naik defined the invariant tν(K) to be the minimum of k for which ν of the k–twisted positive Whitehead double of K vanishes. They proved that tν(K) is bounded above by −TB(−K), where TB is the maximal Thurston–Bennequin number. We use a blowing-up process to find a crossing change formula and a new upper bound for tν in terms of the unknotting number. As an application, we present infinitely many knots K such that the difference between Livingston–Naik’s upper bound −TB(−K) and tν(K) can be arbitrarily large. [ABSTRACT FROM AUTHOR]

We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmüller space of the torus. A similar result holds for tunnel number n manifolds. As a consequence, for fixed n, there are infinitely many hyperbolic tunnel number n manifolds with at most one exceptional Dehn filling. This is in contrast to large volume Berge knots, which are tunnel number one manifolds, but with cusp shapes converging to a single point in Teichmüller space. [ABSTRACT FROM AUTHOR]

We present the strongest known knot invariant that can be computed effectively (in polynomial time). [ABSTRACT FROM AUTHOR]

We classify link diagrams with Turaev genus one and two in terms of an alternating tangle structure of the link diagram. The proof involves surgery along simple loops on the Turaev surface, called cutting loops, which have corresponding cutting arcs that are visible on the planar link diagram. These also provide new obstructions for a link diagram on a surface to come from the Turaev surface algorithm. We also show that inadequate Turaev genus one links are almost-alternating. [ABSTRACT FROM AUTHOR]

We study concordance of virtual knots. Our main result is that a classical knot K is virtually slice if and only if it is classically slice. From this we deduce that the concordance group of classical knots embeds into the concordance group of long virtual knots. [ABSTRACT FROM AUTHOR]

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite). [ABSTRACT FROM AUTHOR]

We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links. [ABSTRACT FROM AUTHOR]

We give infinitely many examples of 2-bridge knots for which the topological and smooth slice genera differ. The smallest of these is the 12-crossing knot 12a255. These also provide the first known examples of alternating knots for which the smooth and topological genera differ. [ABSTRACT FROM AUTHOR]

We present the complete classification of the subgroup of the classical knot concordance group generated by prime knots with eight or fewer crossings. Proofs are presented in summary. We also describe extensions of this work to the case of nine crossing knots. [ABSTRACT FROM AUTHOR]

We show that if K is a satellite knot in the 3-sphere S³ which admits a generalized cosmetic crossing change of order q with |q| ≥ 6, then K admits a pattern knot with a generalized cosmetic crossing change of the same order. As a consequence of this, we find that any prime satellite knot in S³ which admits a torus knot as a pattern cannot admit a generalized cosmetic crossing change of order q with |q| ≥ 6. We also show that if there is any knot in S³ admitting a generalized cosmetic crossing change of order q with |q| ≥ 6, then there must be such a knot which is hyperbolic. [ABSTRACT FROM AUTHOR]

We show that if a link L with a non-zero Alexander polynomial admits a locally flat cobordism to a 'weakly m-split link', then the cobordism must have genus at least ｢m/2｣. This generalises a recent result of J. Pardon. [ABSTRACT FROM AUTHOR]

It is shown that if the exterior of a link L in the three sphere admits a genus 2 Heegaard splitting, then L has Generalized Property R. [ABSTRACT FROM AUTHOR]

We give an infinite sequence of mutually non-conjugate surface braids with same degree representing the trivial surface-link with at least two components and a pair of non-conjugate surface braids with same degree representing a spun (2, t)-torus knot for t ≥ 3. To give these examples, we introduce new invariants of conjugacy classes of surface braids via colorings by Alexander quandles or core quandles of groups. [ABSTRACT FROM AUTHOR]