Knot Theory Timeline (Summer 2010-Present) Summer 2010 – AMRE, David Freund (2013) and Louisa Catalano (2011) - We know that T(p,q) and T(q,p) are equivalent knots and if (p,q)=1 then T(p,q) is a knot (has one component) - Theorem: K(p,0) is the unlink having p components - Theorem: K(p,1) is the unknot Fall 2010 – Sophomore Research, David Freund (2013) and Rutendo Ruzvidzo (2013) November 2010 – MCURCSM 2010 publication (“A Preliminary Study of Klein Knots”) at Wittenberg University, David Freund (2013), Louisa Catalano (2011), Rutendo Ruzvidzo (2013), Jennifer Bowen, John Ramsay - Theorem: K(p,0) is the unlink having p components - Theorem: K(p,1) is the unknot Spring 2011 – Sophomore Research, David Freund (2013) and Rutendo Ruzvidzo (2013) Summer 2011 – AMRE, David Freund (2013), Sarah Smith-Polderman (2013) Fall 2011 – Sophomore Research, Danielle Shepherd (2014) and Sarah Smith-Polderman (2013) October 2011 – Faculty at Large presentation (“Knots, Links, and Bottles: Pure Math with a Twist”), The College of Wooster, Jennifer Bowen and John Ramsay Spring 2012 – Sophomore Research, Danielle Shepherd (2014) and Joseph Smith (2015) Summer 2012 – AMRE, Danielle Shepherd (2014), Joseph Smith (2015), and Sarah Smith-Polderman (2013) July 2012 – UnKnot Conference presentation (“A Recursive Algorithm for Classifying Klein Links”) at Denison University, Danielle Shepherd (2014), Joseph Smith (2015), and Sarah Smith-Polderman (2013) Fall 2012 – Sophomore Research, Danielle Shepherd (2014) and Joseph Smith (2015) November 2012 – MCURCSM 2012 publication (“The Classification of a Subset of Klein Links”) at Ohio Wesleyan University, Joseph Smith (2015), Sarah Smith-Polderman (2013), Danielle Shepherd (2014), Jennifer Bowen, John Ramsay - Reduction result from results during Summer 2012 - If m<n then K(m,n) is the disjoint union of K(m,m) and K(n-m,n-m)* - Generalized braid word for Klein Links - Crossing number calculations for K(0,n) - Number of components calculations for K(0,n) Spring 2013 – Sophomore Research, Michael Bush (2016) and Joseph Smith (2015) Spring 2013 – Rose-Hulman Undergraduate Mathematics Journal publication (“Klein Links and Braids”), David Freund(2013) and Sarah Smith-Polderman (2013) - Publication based on results from Summer 2011 AMRE work - K(m,2)=T(m-1,2) - If n>=2 then K(1,n) is a disjoint union of an unknot and K(0,n-1) - K(0,n) and K(n,n) are mirror images of the same link - K(n,n) and K(n,n-1) are equivalent links Summer 2013 – AMRE, Michael Bush (2016), Katelyn French (2016), and Joseph Smith (2015) - Linking Number work - P-Colorability work Fall 2013 – Sophomore Research, Michael Bush (2016) and Joseph Smith (2015) - Submit article (“Total Linking Number of Torus Links and Klein Links”) to Rose-Hulman Undergraduate Mathematics Journal, under review in Fall 2013 - Alternate proof of linking number for T(m,n) (already known in literature) - Linking number for Klein Links, K(m,n) - Reworking paper submission on crossing number of Klein Links Spring 2014 – Sophomore Research, Michael Bush (2016) and Kiera Dobbs (2016) - Reworking information on hole placement construction for Klein Links - P-Colorability work, specifically determinants of coloring matrices for Klein Links - Submit article (“Braid Computations for the Crossing Number of Klein Links”) to Involve: A Journal of Mathematics Spring 2014 - Rose-Hulman Undergraduate Mathematics Journal publication (“Total Linking Number of Torus Links and Klein Links”), Michael Bush (2016), Katelyn French (2016), and Joseph Smith (2015) Summer 2014 – AMRE, Kiera Dobbs (2016) and Brian Foley (2017) - Finalizing information for hole placement construction for Klein links - Computer algorithm for the Alexander polynomial - Preliminary work on the Unknotting Problem computer algorithm Fall 2014 – Sophomore Research, Kiera Dobbs (2016) and Brian Foley (2017) - Finalizing information for hole placement construction for Klein links - Continuing work on the Unknotting Problem computer algorithm November 2014 – MCURCSM 2014 publication (“Automated Reidemeister Moves: A Numerical Approach to the Unknotting Problem”) at The College of Wooster, Brian Foley (2017) Spring 2015 – Sophomore Research, Kiera Dobbs (2016) and Brian Foley (2017) - Exploring other options for hole placement construction for Klein links - Continuing work on the Unknotting Problem computer algorithm February 2015 – Ohio Celebration of Women in Computing (OCWiC) Winning Poster (“Alexander Polynomial Program”) at Sawmill Creek Resort, Sandusky, OH, Kiera Dobbs (2016) Spring 2015 – Involve: A Journal of Mathematics publication (“Braid Computations for the Crossing Number of Klein Links”), Michael Bush (2016), Danielle Shepherd (2014), Joseph Smith (2015), Sarah Smith-Polderman (2013), Jennifer Bowen, John Ramsay Summer 2015 – AMRE, Michael Bush (2016) and Brian Foley (2017) - Continuing work on the Unknotting Problem computer algorithm June 2015 – Submit “Klein Link Multiplicity and Recursion” for review .