9 papers
A surprising generalization of the Möbius function
George E. Andrews, Louis H. Kauffman, Divyamaan Sahoo
We introduce a broad generalization of a recursive formula for the Möbius function due to George Spencer-Brown. By replaci…
A counterexample for the polar conjecture of Spencer-Brown
Scott Baldridge, Louis H. Kauffman, Ben McCarty
In 1976, George Spencer-Brown announced a proof of the four color theorem, using operations on Tait colorings for trivalent plane graphs. In subsequent work he formulated these ope…
Categorification of Chromatic, Dichromatic and Penrose Polynomials
Louis H Kauffman
This paper discusses ways to categorify chromatic, dichromatic and Penrose polynomials, including categorifications of integer evaluations of chromatic polynomials. We show that wi…
Primes Between Squares -- Commentary on Appendix 8 of Laws Of Form
J. M. Flagg, Louis H. Kauffman, Divyamaan Sahoo
This paper provides a commentary and guide to Appendix 8 of Laws Of Form, which is a chapter (appendix) on number theory in the book Laws of Form by Spencer-Brown. (Spencer-Brown,L…
The clock theorem for knotoids and linkoids
Neslihan Gügümcü, Louis H. Kauffman
In this paper, we generalize the \textit{Clock Theorem} of Formal Knot Theory to knotoids in . The clock theorem implies that clock states of a knotoid diagram form a lattice…
Fusion and Fission of Particle-like Chiral Nematic Vortex Knots
Darian Hall, Jung-Shen Benny Tai, Louis H. Kauffman +1
Vortex knots have been seen decaying in many physical systems. Here we describe topologically protected vortex knots, which remain stable and undergo fusion and fission while conse…
The Mock Alexander Polynomial for Knotoids and Linkoids
Joanna A. Ellis-Monaghan, Neslihan Gügümcü, Louis H. Kauffman +1
The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this p…
Mock Alexander Polynomials
Neslihan Gügümcü, Louis H. Kauffman
In this paper, we construct mock Alexander polynomials for starred links and linkoids in surfaces. These polynomials are defined as specific sums over states of link or linkoid dia…
A state sum for the total face color polynomial
Scott Baldridge, Louis H. Kauffman, Ben McCarty
The total face color polynomial is based upon the Poincaré polynomials of a family of filtered -color homologies. It counts the number of -face colorings of ribbon graphs for…