State summation invariants for knots, links, knotoids, and linkoids

<p> In this paper, we construct state summation invariants for knots, links, knotoids and linkoids. These invariants are defined as specific sums over states of link or linkoid diagrams that satisfy <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f equals n"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">f=n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the number of regions and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the number of crossings of diagrams. The invariants considered here can distinguish planar knotoids <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K factorial Superscript star Baseline"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:msup> <mml:mo>!</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> ⋆ </mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">K!^{\star }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where the latter denotes a double mirror image of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> obtained by reflecting in the diagram plane and in a plane perpendicular to the diagram plane, and they can detect invertibility of knotoids and linkoids. </p>