OFFSET
0,2
COMMENTS
T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial <K>(A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.
REFERENCES
Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.
Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.
Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.
Franck Maminirina Ramaharo, Illustration of T(2,2)
Franck Maminirina Ramaharo, Note on sequence A321125 and related ones
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Eric Weisstein's World of Mathematics, Bracket Polynomial
Wikipedia, 2-bridge knot
Wikipedia, Bracket polynomial
FORMULA
EXAMPLE
Square array begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
2, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
3, 3, 3, 4, 5, 6, 7, 8, 9, 10, ...
4, 4, 4, 5, 6, 7, 8, 9, 10, 11, ...
5, 5, 5, 6, 7, 8, 9, 10, 11, 12, ...
6, 6, 6, 7, 8, 9, 10, 11, 12, 13, ...
7, 7, 7, 8, 9, 10, 11, 12, 13, 14, ...
8, 8, 8, 9, 10, 11, 12, 13, 14, 15, ...
9, 9, 9, 10, 11, 12, 13, 14, 15, 16, ...
10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
...
MATHEMATICA
Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten
PROG
(Maxima) create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);
CROSSREFS
T(n,1) = degree of the (n+1)-th row polynomial in A300453.
KEYWORD
AUTHOR
Franck Maminirina Ramaharo, Nov 21 2018
STATUS
approved