Narrow Results

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K₁#K₂)≥Cr(K₁) or Cr(K₁#K₂)≥Cr(K₂) holds in general, here K₁#K₂ is the connected sum of K₁ and K₂ and Cr(K) stands for the crossing number of the link K. However, for alternating links K₁ and K₂, Cr(K₁#K₂)=Cr(K₁)+Cr(K₂) does hold. On the other hand, if K₁ is an alternating link and K₂ is any link, then we have Cr(K₁#K₂)≥Cr(K₁). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K₁ is a connected sum of any given number of links from this class and K₂ is a non-trivial knot, we prove that Cr(K₁#K₂)≥Cr(K₁)+3.

The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.