Quantum Enhancements and Biquandle Brackets

@article{Nelson2015QuantumEA,
  title={Quantum Enhancements and Biquandle Brackets},
  author={Sam Nelson and Michael E. Orrison and Veronica Rivera},
  journal={arXiv: Geometric Topology},
  year={2015}
}
We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are… 
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17 Citations
Highly Influential Citations
3
Background Citations
12
Methods Citations
7

17 Citations

Citation Type

Citation Type

Kaestner brackets
Biquandle Brackets and Knotoids
Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of…
Biquandle virtual brackets
We introduce an infinite family of quantum enhancements of the biquandle counting invariant which we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored…
The structure of biquandle brackets
In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. We…
Cocycle enhancements of psyquandle counting invariants
We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant…
Biquandle Bracket Quivers
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting…
Quantum Enhancements via Tribracket Brackets
We enhance the tribracket counting invariant with \textit{tribracket brackets}, skein invariants of tribracket-colored oriented knots and links analogously to biquandle brackets. This infinite family…
Categorifying Biquandle Brackets
In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links.…
A Survey of Quantum Enhancements
  • Sam Nelson
  • Mathematics
    Knots, Low-Dimensional Topology and Applications
  • 2019
In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants…
Trace Diagrams and Biquandle Brackets
We introduce a method of computing biquandle brackets of oriented knots and links using a type of decorated trivalent spatial graphs we call trace diagrams. We identify algebraic conditions on the…
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