2014/10/21 by Rama Mishra, Mishra, Rama, Hitesh Raundal +1
Mathematics · #14P25 #57M25 #57M27 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:14P25 #msc:57M25 #msc:57M27
paper · pdf · doi:10.48550/arxiv.1410.5728
32 pages, 11 figures, 3 tables
arxiv created 2021/01/02 · arxiv updated 2021/01/05
We show that all knots up to 6 crossings can be represented by polynomial knots of degree at most 7, among which except for 52, 52^*, 61, 61^*, 62, 62^* and 63 all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any 5 crossing knot in degree 6? In this paper we try to partially answer this question. For an integer d≥2, we define a set \mathcalPd to be the set of all polynomial knots given by t↦(f(t),g(t),h(t)) such that deg(f)=d-2, deg(g)=d-1 and deg(h)=d. This set can be identified with a subset of ℝ3d and thus it is equipped with the natural topology which comes from the usual topology ℝ3d. In this paper we determine a lower bound on the number of path components of \mathcalPd for d≤ 7. We define a path equivalence for polynomial knots in the space \mathcalPd and show that it is stronger than the topological equivalence.