Virtual Mosaic Knots



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The study of knots and links is a main focus of the mathematical branch of topology. Classical knot theory studies knots embedded in 3-dimensional real space and has been a primary field of study since the 1960’s. Virtual knot theory, first introduced by Kauffman in 1999, studies knots embedded in thickened surfaces. Lomanoco and Kauffman introduced mosaic diagrams in order to build a quantum knot system in 2008. In 2009, Garduño extended these mosaic diagrams to include virtual knots. In order to represent knots on surfaces, Ganzell and Henrich introduced virtual mosaic knot theory in 2020 by placing knots onto 𝑛 × 𝑛 polygonal representations of surfaces. We extend the idea of virtual mosaic knot theory to include virtual rectangular mosaics, a placement of virtual knots onto 𝑛 × 𝑚 polygonal representations of surfaces, as well as row mosaics, a placement of virtual knots onto 1 × 𝑚 polygonal representations of surfaces. In this thesis, we introduce virtual rectangular mosaics and give two rectangular mosaic invariants called the tile number and row number. Included as an appendix, we give a complete row mosaic tabulation of knots with 8 or fewer crossings and virtual knots up to 4 crossings.