We define the knotting probability of a knot K by the probability for a random polygon or self-avoiding polygon (SAP) of N segments having the knot type K. We show fundamental and generic properties of the knotting probability particularly its dependence on the excluded volume. We investigate them for the SAP consisting of hard cylindrical segments of unit length and radius rex. For various prime and composite knots, we numerically show that a compact formula describes the knotting probabilities for the cylindrical SAP as a function of segment number N and radius rex. It connects the small-N to the large-N behavior and even to lattice knots in the case of large values of radius. As the excluded volume increases, the maximum of the knotting probability decreases for prime knots except for the trefoil knot. If it is large, the trefoil knot and its descendants are dominant among the nontrivial knots in the SAP. From the factorization property of the knotting probability, we derive a sum rule among the estimates of a fitting parameter for all prime knots, which suggests the local knot picture and the dominance of the trefoil knot in the case of large excluded volumes. Here we remark that the cylindrical SAP gives a model of circular DNA which is negatively charged and semiflexible, where radius rex corresponds to the screening length.

1.
H. A.
Kramers
,
J. Chem. Phys.
14
,
415
(
1946
).
2.
Cyclic Polymers
, edited by
J. A.
Semlyen
(
Elsevier Applied Science Publishers
,
New York
,
1986
); 2nd ed. (Kluwer Academic Publishers, Dordrecht, 2000).
3.
A. D.
Bates
and
A.
Maxwell
,
DNA Topology
(
Oxford University Press
,
2005
).
4.
J.
Vinograd
,
J.
Lebowitz
,
R.
Radloff
,
R.
Watson
, and
P.
Laipis
,
Proc. Natl. Acad. Sci. U. S. A.
53
,
1104
(
1965
).
5.
M. A.
Krasnow
,
A.
Stasiak
,
S. J.
Spengler
,
F.
Dean
,
T.
Koller
, and
N. R.
Cozzarelli
,
Nature
304
,
559
(
1983
).
6.
F. B.
Dean
,
A.
Stasiak
,
T.
Koller
, and
N. R.
Cozzarelli
,
J. Biol. Chem.
260
,
4975
(
1985
).
9.
J. J.
Danon
et al.,
Science
355
,
159
162
(
2017
).
10.
H.
Oike
,
H.
Imaizumi
,
T.
Mouri
,
Y.
Yoshioka
,
A.
Uchibori
, and
Y.
Tezuka
,
J. Am. Chem. Soc.
122
,
9592
(
2000
).
11.
Y.
Tezuka
and
H.
Oike
,
J. Am. Chem. Soc.
123
,
11570
(
2001
).
12.
C. W.
Bielawski
,
D.
Benitez
, and
R. H.
Grubbs
,
Science
297
,
2041
2044
(
2002
).
13.
D.
Cho
,
K.
Masuoka
,
K.
Koguchi
,
T.
Asari
,
D.
Kawaguchi
,
A.
Takano
, and
Y.
Matsushita
,
Polym. J.
37
,
506
511
(
2005
).
14.
A.
Takano
,
Y.
Kushida
,
K.
Aoki
,
K.
Masuoka
,
K.
Hayashida
,
D.
Cho
,
D.
Kawaguchi
, and
Y.
Matsushita
,
Macromolecules
40
,
679
681
(
2007
).
15.
B. A.
Laurent
and
S.
Grayson
,
J. Am. Chem. Soc.
128
,
4238
4239
(
2006
).
16.
N.
Sugai
,
H.
Heguri
,
K.
Ohta
,
Q.
Meng
,
T.
Yamamoto
, and
Y.
Tezuka
,
J. Am. Chem. Soc.
132
,
14790
14802
(
2010
).
17.
N.
Sugai
,
H.
Heguri
,
T.
Yamamoto
, and
Y.
Tezuka
,
J. Am. Chem. Soc.
133
,
19694
19697
(
2011
).
18.
Topological Polymer Chemistry: Progress in Cyclic Polymers in Syntheses, Properties, and Functions
, edited by
Y.
Tezuka
(
World Scientific Publishing
,
Singapore
,
2013
).
19.
A. V.
Vologodskii
,
A. V.
Lukashin
,
M. D.
Frank-Kamenetskii
, and
V. V.
Anshelevich
,
Sov. Phys. JETP
39
,
1059
(
1974
).
20.
J. P. J.
Michels
and
F. W.
Wiegel
,
Phys. Lett. A
90
,
381
(
1982
).
21.
E. J.
Janse van Rensburg
and
S. G.
Whittington
,
J. Phys. A: Math. Gen.
23
,
3573
(
1990
).
22.
K.
Koniaris
and
M.
Muthukumar
,
Phys. Rev. Lett.
66
,
2211
(
1991
).
23.
T.
Deguchi
and
K.
Tsurusaki
,
J. Phys. Soc. Jpn.
62
,
1411
(
1993
).
24.
T.
Deguchi
and
K.
Tsurusaki
,
J. Knot Theory Ramif.
3
,
321
(
1994
).
25.
K.
Tsurusaki
and
T.
Deguchi
,
J. Phys. Soc. Jpn.
64
,
1506
(
1995
).
26.
T.
Deguchi
and
K.
Tsurusaki
,
Phys. Rev. E
55
,
6245
(
1997
).
27.
E.
Orlandini
,
M. C.
Tesi
,
E. J.
Janse van Rensburg
, and
S. G.
Whittington
,
J. Phys. A: Math. Gen.
29
,
L299
(
1996
).
28.
E.
Orlandini
,
M. C.
Tesi
,
E. J.
Janse van Rensburg
, and
S. G.
Whittington
,
J. Phys. A: Math. Gen.
31
,
5953
(
1998
).
29.
M. K.
Shimamura
and
T.
Deguchi
,
Phys. Lett. A
274
,
184
(
2000
).
30.
V.
Katritch
,
W. K.
Olsen
,
A.
Vologodskii
,
J.
Dubochet
, and
A.
Stasiak
,
Phys. Rev. E
61
,
5545
(
2000
).
31.
A.
Yao
,
H.
Matsuda
,
H.
Tsukahara
,
M. K.
Shimamura
, and
T.
Deguchi
,
J. Phys. A: Math. Gen.
34
,
7563
(
2001
).
32.
B.
Marcone
,
E.
Orlandini
,
A. L.
Stella
, and
F.
Zonta
,
J. Phys. A: Math. Gen.
38
,
L15
L21
(
2005
).
33.
M.
Baiesi
,
E.
Orlandini
, and
A. L.
Stella
,
J. Stat. Mech.: Theory Exp.
2010
,
P06012
.
34.
E. J.
Janse van Rensburg
and
A.
Rechnitzer
,
J. Phys. A: Math. Theor.
44
,
162002
(
2011
).
35.
L.
Tubiana
,
Phys. Rev. E
89
,
052602
(
2014
).
36.
E.
Uehara
and
T.
Deguchi
,
J. Phys.: Condens. Matter
27
,
354104
(
2015
).
37.
D. W.
Sumners
and
S.
Whittington
,
J. Phys. A: Math. Gen.
21
,
1689
(
1988
).
38.
N.
Pippenger
,
Discrete Appl. Math.
25
,
273
(
1989
).
39.
V. V.
Rybenkov
,
N. R.
Cozzarelli
, and
A. V.
Vologodskii
,
Proc. Natl. Acad. Sci. U. S. A.
90
,
5307
(
1993
).
40.
S. Y.
Shaw
and
J. C.
Wang
,
Science
260
,
533
(
1993
).
41.
C.
Plesa
,
D.
Verschueren
,
S.
Pud
,
J.
van der Torre
,
J. W.
Ruitenberg
,
M. J.
Witteveen
,
M. P.
Johnsson
,
A. Y.
Grosberg
,
Y.
Rabin
, and
C.
Dekker
,
Nat. Nanotechnol.
11
,
1093
(
2016
).
42.
J. A.
Schellman
and
D.
Stigter
,
Biopolymers
16
,
1415
(
1977
).
44.
M.
Le Bret
and
B. H.
Zimm
,
Biopolymers
23
,
287
(
1984
).
45.
A.
Yu Grosberg
and
A. R.
Khokhlov
,
Statistical Physics of Macromolecules
(
AIP Press
,
New York
,
1994
).
46.
K. C.
Millett
,
J. Knot Theory Ramif.
3
,
263
(
1994
).
47.
M.
Kapovich
and
J. J.
Milson
,
J. Differ. Geom.
44
,
479
(
1996
).
48.
S.
Alvarado
,
J. A.
Calvo
, and
K. C.
Millett
,
J. Stat. Phys.
143
,
102
(
2011
).
49.
T.
Deguchi
and
K.
Tsurusaki
,
Phys. Lett. A
174
,
29
37
(
1993
).
50.
M.
Polyak
and
O.
Viro
,
Int. Math. Res. Not.
1994
(
11
),
445
.
51.
K.
Murasugi
and
B.
Kurpita
,
Knot Theory and its Applications
(
Birkhäuser
,
Boston
,
1996
).
52.
A.
Dobay
,
P.-E.
Sottas
,
J.
Dubochet
, and
A.
Stasiak
,
Lett. Math. Phys.
55
,
239
(
2001
).
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