DOI: 10.46810/tdfd.1862200 ISSN: 2149-6366

A Note on the Relationship Between Knots and Dichromatic Polynomial

Abdulgani Şahin, Hatice Nevra Başcı
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Knot theory deals with how a circle or the disjoint union of circles can be embedded in R^3. Knots (or links) can be thought of topologically as circles embedded in space or geometrically as simple closed curves in space, however they can also be defined in a combinatorial sense. In other words, we can define the knots as equivalence classes of knot diagrams under an equivalence relation determined by certain diagrammatic movements. In this situation, it becomes much easier to manipulate (deform) the regular diagrams of the knots and their crossings. Based on these deformations, it becomes possible to define polynomials that are matched with specific coefficients. One of these special polynomials is the dichromatic polynomial. The definition of this polynomial has led to connections between knots and graph theory, as well as between nodes and fields such as physics and biology. This study examines information regarding these relationships. The situations mentioned are examined in detail, both structurally and through calculations based on a specific example.

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