2022/10/14 by Ana Wright, Wright, Ana
Computer Science · Mathematics · #57K10 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2210.07728
openalex publication_date 2022/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We call a knot K a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from K. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases. Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo's unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one which proves the same result. However, we show that Lickorish's obstruction does not subsume the obstruction coming from the condition on determinants.