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We present a short proof of a theorem of Tanaka that if a composite ribbon knot admits a symmetric union presentation with one twist region, then it has a non-trivial knot and its mirror image as connected summands. [ABSTRACT FROM AUTHOR]

The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal z -degree part of the HOMFLY polynomial of a naturally associated link. Note that the latter can be any oriented link. This result fits into a program aimed at deriving the HOMFLY polynomial from Floer homology. We also establish some other, more basic properties of the signed interior polynomial. For example, the HOMFLY polynomial of the mirror image of L is given by P L (− v − 1 , z). This implies a mirroring formula for the signed interior polynomial in the planar case. We prove that the same property holds for any bipartite graph and the same graph with all signs reversed. The proof relies on Ehrhart reciprocity applied to the so-called root polytope. We also establish formulas for the signed interior polynomial inspired by the knot theoretical notions of flyping and mutation. This leads to new identities for the original unsigned interior polynomial. [ABSTRACT FROM AUTHOR]

We call a link (knot) L to be strongly Jones (respectively, Homfly) undetectable, if there are infinitely many links which are not isotopic to L but share the same Jones (respectively, Homfly) polynomial as L. We reconstruct Kanenobu's knot [Kanenobu, Infinitely many knots with the same polynomial invariant, Proc. Amer. Math. Soc. 97(1) (1986), 158–162] and give two new constructions. Using these constructions, we give some examples of strongly Jones undetectable: 8 8 , 8 9 , 1 0 2 2 , 1 0 3 5 , 1 0 1 5 5 , 4 1 # 4 1 , 5 2 # 5 2 ! ( 5 2 ! is the mirror image of 5 2 ) and etc. For some special cases, these constructions will be shown to be strongly Jones undetectable and strongly Homfly undetectable. [ABSTRACT FROM AUTHOR]

A symmetric union in the 3-space ℝ 3 is obtained from a knot in ℝ 3 and its mirror image, which is symmetric with respect to a 2-plane in ℝ 3 , by connecting them with some vertical 2-string tangles with twists and a horizontal 2-string tangle with no twists along the plane. We introduce the minimal twisting number of a symmetric union and show that if a knot K is a composite symmetric union with minimal twisting number one, then K has a non-trivial knot and its mirror image as connected summands. As an application, we study the minimal twisting number of a symmetric union and show that there exist infinitely many symmetric unions with minimal twisting number two. [ABSTRACT FROM AUTHOR]

Let u(K) and g(K) denote the unknotting number and the genus of a knot K, respectively. For a 3-braid knot K, we show that u(K) ≤ g(K) holds, and that if u(K) = g(K) then K is either a 2-braid knot, a connected sum of two 2-braid knots, the figure-eight knot, a strongly quasipositive knot or its mirror image. [ABSTRACT FROM AUTHOR]

An oriented and ordered n-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring orientations and ordering of the components. For an oriented and ordered n-component link L, let λL be the product of linking numbers of all 2-component sublinks in L. For n = 4m + 3, where m is a non-negative integer, we show that if L is achiral then λL = 0. And for n ≠ 4m + 3, we show that there exists an n-component achiral link L with λL ≠ 0 by using achiral embeddings of complete graphs with n vertices Kn. [ABSTRACT FROM AUTHOR]

L. H. Kauffman defined the binary bracket polynomial of a virtual link by introducing binary labelings into the states of a virtual link diagram. We use the invariant by a slight modification, and call it the modified b-polynomial. We prove that if a virtual link K has a period pl for a prime p and a positive integer l, then the modified b-polynomial InvK (A) of K is congruent to InvK* (A) modulo p and A4pl-1 where K* is the mirror image of K. We exhibit examples of virtual links whose periods are completely determined by the invariant. [ABSTRACT FROM AUTHOR]

A symmetric union of a knot is a generalization of the operation of the connected sum of a knot and its mirror image. In this paper, we show that every ribbon 2-knot of 1-fusion has a ribbon presentation associated with a symmetric union which has either the unknot or a 2-bridge knot as a partial knot. [ABSTRACT FROM AUTHOR]

The image of a polygonal knot K under a spherical inversion of ℝ3 ∪ ∞ is a simple closed curve made of arcs of circles, perhaps some line segments, having the same knot type as the mirror image of K. But suppose we reconnect the vertices of the inverted polygon with straight lines, making a new polygon $\widehat{K}$. This may be a different knot type. For example, a certain 7-segment figure-eight knot can be transformed to a figure-eight knot, a trefoil, or an unknot, by selecting different inverting spheres. Which knot types can be obtained from a given original polygon K under this process? We show that for large n, most n-segment knot types cannot be reached from one initial n-segment polygon, using a single inversion or even the whole Möbius group. The number of knot types is bounded by the number of complementary domains of a certain system of round 2-spheres in ℝ3. We show the number of domains is at most polynomial in the number of spheres, and the number of spheres is itself a polynomial function of the number of edges of the original polygon. In the analysis, we obtain an exact formula for the number of complementary domains of any collection of round 2-spheres in ℝ3. On the other hand, the number of knot types that can be represented by n-segment polygons is exponential in n. Our construction can be interpreted as a particular instance of building polygonal knots in non-Euclidean metrics. In particular, start with a list of n vertices in ℝ3 and connect them with arcs of circles instead of line segments: Which knots can be obtained? Our polygonal inversion construction is equivalent to picking one fixed point p ∈ ℝ3 and replacing each edge of K by an arc of the circle determined by p and the endpoints of the edge. [ABSTRACT FROM AUTHOR]