A366190 - OEIS

%I #58 Nov 07 2023 04:13:45

%S 4,24,30,34,36,40,42,44,46,48,50,52,54,56,58,60,62,64

%N Minimal lengths of prime knots formed by orthogonal unit line segments of the cubic lattice.

%C The same term may correspond to more than one knot. For the initial terms, the minimum lengths were found within three layers of the lattice and it is conceivable that the tightest representation of larger knots expand in all three axes.

%C Length 24: 3_1.

%C Length 30: 4_1.

%C Length 34: 5_1.

%C Length 36: 5_2.

%C Length 40: 6_1, 6_2, 6_3.

%C Length 42: 8_19.

%C Length 44: 8_20.

%C Length 46: 7_2, 7_5, 7_6, 8_21.

%C Length 48: 8_3, 8_7, 9_42, 10_124.

%C Length 50: 8_1, 8_2, 8_4, 8_5, 8_6, 8_8, 8_9, 8_10, 8_11, 8_13, 8_14, 8_16, 9_43, 9_44, 9_46, 9_47, 10_139.

%C Length 52: 8_12, 8_15, 8_17, 8_18, 9_45, 9_48, 9_49, 10_132.

%C Length 54: 9_1, 9_3-5, 9_14, 9_19, 9_26, 9_31, 9_40, 9_41, ... .

%C Conjecture: All even numbers >= 40 will appear in this sequence.

%H Andrew Rechnitzer, <a href="https://personal.math.ubc.ca/~andrewr/knots/minimal_knots.html">List of knot data for different cubic lattices</a>.

%H Rob Scharein, Kai Ishihara, Javier Arsuaga, Yuanan Diao, Koya Shimokawa and Mariel Vazquez, <a href="https://iopscience.iop.org/article/10.1088/1751-8113/42/47/475006">Bounds for the minimum step number of knots in the simple cubic lattice</a>, J. Phys. A: Math. Theor. 42 475006 (2009).

%H Thomas Scheuerle, <a href="/A366190/a366190_2.pdf">Examples for a(2) - a(6) with up to 6 crossings</a>.

%H <a href="/index/K#knots">Index entries for sequences related to knots</a>

%e a(1) = 4 because the unknot is represented by four joined unit line segments, forming a closed loop, in the lattice.

%e a(2) = 24 because the second simplest knot, the trefoil knot, 3_1, can be described by 24 joined unit line segments, forming a self-avoiding closed loop in the lattice.

%Y Cf. A122059, A076770.

%K nonn,more

%O 1,1

%A _Tamas Sandor Nagy_ and _Thomas Scheuerle_, Oct 12 2023