BPS spectra of complex knots

Marino’s Conjecture remains underexplored within the framework of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>S</a:mi><a:mi>O</a:mi><a:mo stretchy="false">(</a:mo><a:mi>N</a:mi><a:mo stretchy="false">)</a:mo></a:math> string dualities. In this article, we investigated the reformulated invariants of one-parameter families of knots <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mrow><e:mo stretchy="false">[</e:mo><e:mi mathvariant="script">K</e:mi><e:mo stretchy="false">]</e:mo></e:mrow><e:mi>p</e:mi></e:msub></e:math> derived from tangle surgery on Manolescu’s quasialternating knot diagrams [C. Manolescu, ]. Within topological string dualities, we have verified Marino’s integrality conjecture for these families of knots up to the Young diagram representation <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi mathvariant="bold">R</j:mi></j:math>, with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mrow><m:mo stretchy="false">|</m:mo><m:mi mathvariant="bold">R</m:mi><m:mo stretchy="false">|</m:mo></m:mrow><m:mo>≤</m:mo><m:mn>2</m:mn></m:math>. Furthermore, through our analysis, we have conjectured a closed structure for the extremal refined Bogomol’nyi–Prasad–Sommerfeld (BPS) integers for the torus knots <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mrow><r:mo stretchy="false">[</r:mo><r:msub><r:mn mathvariant="bold">3</r:mn><r:mn mathvariant="bold">1</r:mn></r:msub><r:mo stretchy="false">]</r:mo></r:mrow><r:mrow><r:mn>2</r:mn><r:mi>p</r:mi><r:mo>+</r:mo><r:mn>1</r:mn></r:mrow></r:msub></r:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msub><x:mrow><x:mo stretchy="false">[</x:mo><x:msub><x:mn mathvariant="bold">8</x:mn><x:mn mathvariant="bold">20</x:mn></x:msub><x:mo stretchy="false">]</x:mo></x:mrow><x:mrow><x:mn>2</x:mn><x:mi>p</x:mi><x:mo>+</x:mo><x:mn>1</x:mn></x:mrow></x:msub></x:math>, <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"><db:mi>p</db:mi><db:mo>∈</db:mo><db:msub><db:mi mathvariant="double-struck">Z</db:mi><db:mrow><db:mo>≥</db:mo><db:mn>0</db:mn></db:mrow></db:msub></db:math>. As the parameter <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:mi>p</gb:mi></gb:math> of the knot diagram increases, the total crossing number of a knot exceeds 16, which we describe as a complex knot. Interestingly, we discovered the maximum number of gaps in the BPS spectra associated with complex knot families. Moreover, our observations indicated that as <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"><ib:mi>p</ib:mi></ib:math> increases, the size of these gaps also expands.