# (t,s)-racks and their link invariants

@article{Ceniceros2010tsracksAT, title={(t,s)-racks and their link invariants}, author={Jessica Ceniceros and Sam Nelson}, journal={arXiv: Geometric Topology}, year={2010} }

A (t,s)-rack is a rack structure defined on a module over the ring $\ddot\Lambda=\mathbb{Z}[t^{\pm 1},s]/(s^2-(1-t)s)$. We identify necessary and sufficient conditions for two $(t,s)$-racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t,s)-racks and give some computations and examples. As an application, we use these enhanced invariants to obtain obstructions to knot ordering.

## 7 Citations

Birack Dynamical Cocycles and Homomorphism Invariants

- Mathematics
- 2012

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented classical and virtual knots and links via algebraic structures…

Blackboard biracks and their counting invariants

- Mathematics
- 2010

A blackboard birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In…

Enhancements of rack counting invariants via dynamical cocycles

- Mathematics
- 2012

We introduce the notion of N-reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to…

Twisted Virtual Biracks

- Mathematics
- 2011

This thesis will take a look at a branch of topology called knot theory. We will first look at what started the study of this field, classical knot theory. Knot invariants such as the Bracket…

Bikei, Involutory Biracks and unoriented link invariants

- Mathematics
- 2011

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which…

Tribracket Modules

- MathematicsInternational Journal of Mathematics
- 2020

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link…

Enhancements of the rack counting invariant via N-reduced dynamical cocycles

- Mathematics
- 2011

We introduce the notion of N-reduced dynamical cocycles and use these objects to dene enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to…

## References

SHOWING 1-10 OF 12 REFERENCES

Rack Module Enhancements of Counting Invariants

- Mathematics
- 2010

We introduce a modified rack algebra Z[X] for racks X with finite rack rank N. We use representations of Z[X] into rings, known as rack modules, to define enhancements of the rack counting invariant…

RACKS AND LINKS IN CODIMENSION TWO

- Mathematics
- 1992

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a…

An isomorphism theorem for Alexander biquandles

- Mathematics
- 2006

We show that two Alexander biquandles M and M' are isomorphic iff there is an isomorphism of Z[s,1/s,t,1/t]-modules h:(1-st)M --> (1-st)M' and a bijection g:O_s(A) --> O_s(A') between the s-orbits of…

Link invariants from finite racks

- Mathematics
- 2008

We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant…

Classification of Finite Alexander Quandles

- Mathematics
- 2002

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander…

Two-bridge knots with common ORS covers

- Mathematics
- 2010

Given a 2-bridge knot K, Ohtsuki, Riley, and Sakuma show how to construct infinitely many other 2-bridge knots or links which are “greater than,” or “cover” K. In this paper we explore the question…

UPPER BOUNDS IN THE OHTSUKI–RILEY–SAKUMA PARTIAL ORDER ON 2-BRIDGE KNOTS

- Mathematics
- 2010

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions…

Epimorphisms between 2-bridge link groups

- Mathematics
- 2008

We give a systematic construction of epimorphisms between 2‐bridge link groups. Moreover, we show that 2‐bridge links having such an epimorphism between their link groups are related by a map between…