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The quandle homology theory is generalized to the case when the coefficient groups admit the structure of Alexander quandles, by including an action of the infinite cyclic group in the boundary operator. Theories of Alexander extensions of quandles in relation to low dimensional cocycles are developed in parallel to group extension theories for group… (More)

Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Graña. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second,… (More)

- J. SCOTT CARTER, DANIEL JELSOVSKY, SEIICHI KAMADA, LAUREL LANGFORD, MASAHICO SAITO, Walter Neumann
- 1999

State-sum invariants for classical knots and knotted surfaces in 4-space are developed via the cohomology theory of quandles. Cohomology groups of quandles are computed to evaluate the invariants. Some twist spun torus knots are shown to be noninvertible using the invariants.

- Natasa Jonoska, Masahico Saito
- DNA
- 2001

- W Edwin Clark, Mohamed Elhamdadi, Masahico Saito, Timothy Yeatman
- Journal of knot theory and its ramifications
- 2014

We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the… (More)

- Angela Angeleska, Natasa Jonoska, Masahico Saito, Laura F Landweber
- Journal of theoretical biology
- 2007

We propose molecular models for homologous DNA recombination events that are guided by either double-stranded RNA (dsRNA) or single-stranded RNA (ssRNA) templates. The models are applied to explain DNA rearrangements in some groups of ciliates, such as Stylonychia or Oxytricha, where extensive gene rearrangement occurs during differentiation of a somatic… (More)

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The selfdistributive operations of these structures provide solutions of the Yang–Baxter… (More)

The state-sum invariants for knots and knotted surfaces defined from quandle cocycles are described using the Kronecker product between cycles represented by colored knot diagrams and a cocycle of a finite quandle used to color the diagram. Such an interpretation is applied to evaluating the invariants. Algebraic interpretations of quandle cocycles as… (More)

- Angela Angeleska, Natasa Jonoska, Masahico Saito
- Discrete Applied Mathematics
- 2009

Motivated by DNA rearrangements and DNA homologous recombination modeled in [A. Angeleska, N. Jonoska, M. Saito, L.F. Landweber, RNA-guided DNA assembly, Journal of Theoretical Biology, 248(4) (2007), 706–720], we investigate smoothings on graphs that consist of only 4-valent and 1-valent rigid vertices, called assembly graphs. An assembly graph can be seen… (More)

- Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, Masahico Saito
- Discrete Applied Mathematics
- 2013

Genome rearrangement and homological recombination processes have been modeled by 4regular spacial graphs with rigid vertices, called assembly graphs [1]. These graphs can also be represented by double occurrence words called assembly words. The rearranged DNA segments are modeled by certain types of paths in the assembly graphs called polygonal paths. The… (More)