knots
Renzo L Ricca
1,2,4and Xin Liu
31
Department of Mathematics and Applications, U. Milano-Bicocca, Via Cozzi 55, I-20125 Milano, Italy
2
BDIC, Beijing U. Technology, 100 Pingleyuan, Beijing 100124, Peopleʼs Republic of China
3
BDIC & Inst. Theoretical Physics, Beijing U. Technology, 100 Pingleyuan, Beijing 100124, Peopleʼs Republic of China
E-mail: renzo.ricca@unimib.it and xin.liu@bjut.edu.cn Received 17 January 2017
Accepted for publication 13 March 2017 Published 22 November 2017
Communicated by Philip Boyland Abstract
In this paper we derive and compare numerical sequences obtained by adapted polynomials such as HOMFLYPT, Jones and Alexander-Conway for the topo- logical cascade of vortex torus knots and links that progressively untie by a single reconnection event at a time. Two cases are considered: the alternate sequence of knots and co-oriented links (with positive crossings) and the sequence of two- component links with oppositely oriented components (negative crossings). New recurrence equations are derived and sequences of numerical values are com- puted. In all cases the adapted HOMFLYPT polynomial proves to be the best quanti fier for the topological cascade of torus knots and links.
Keywords: vortex knots and links, torus knots, helicity, HOMFLYPT, structural complexity
(Some figures may appear in colour only in the online journal)
1. Adapted knot polynomials for vortex knots
In recent years the present authors have shown that adapted knot polynomials can be usefully employed to study and quantify topological complexity of fluid knots (vortex or magnetic)
4 Author to whom any correspondence should be addressed.
(Liu and Ricca 2012, 2015 ). These adapted polynomials are function of knot type and fluid conserved quantities (such as vortex strength), and represent new powerful invariants of ideal fluid mechanics (Ricca and Liu 2014 ). In 2016, stimulated by the laboratory experiments of Kleckner and Irvine ( 2013 ) on the production and decay of vortex knots in water, and by the work of Shimokawa et al ( 2013 ) on recombinant DNA plasmid reactions, we have applied the adapted the HOMFLYPT polynomial to study the alternate sequence of torus knots
+
( )
T 2, 2 n 1 and torus links T ( 2, 2 n ) (n positive integer) of co-oriented components, when they progressively untie under the assumption of decreasing topological complexity, due to a single reconnection event at a time (see figure 1 ). We found that under certain simplifying assumptions the alternate cascade from T ( 2, 2 n + 1 to ) T ( 2, 2 n ) (as n 0) can be detected by a unique, monotonically decreasing sequence of HOMFLYPT numerical values (Liu and Ricca 2016 ).
The derivation of the HOMFLYPT polynomial for fluid knots has allowed the inter- pretation of the two standard variables of this polynomial in terms of the average values of writhe and twist, and it has also helped to understand the physical implications of the analytical constraints that reduce the two HOMFLYPT variables to a single variable of other knot polynomials, such as Jones and Alexander-Conway. The scope of this paper is to re- examine the numerical sequence obtained by HOMFLYPT and compare it with the numerical sequences obtained by Jones and Alexander-Conway polynomials. In doing so we shall also consider the case of a cascade of torus links T
o( 2, 2 n ) , whose components are oppositely oriented and compare the results. This will give us useful indications as to which polynomial is best suited to detect and quantify topological cascade processes. The material is arranged as follows. In section 2 we brie fly recall the results obtained by HOMFLYPT for the sequence of figure 1. In section 3 we derive the expressions of Jones and Alexander-Conway polynomial for the sequence above and compute the numerical values. We then apply HOMFLYPT to the case of links with oppositely-oriented components (section 4 ), compute the results and compare these to the results obtained by deriving Jones and Alexander-Conway polynomials from HOMFLYPT (section 5 ). Conclusions are drawn in the final section.
2. HOMFLYPT polynomial for the alternate sequence of knots and co-oriented links
In (Liu and Ricca 2016 ) we considered the topological cascade sequence of torus knots and links shown in figure 1 under the following assumptions:
Figure 1.
Top: topological cascade of torus knots and co-oriented links by single
reconnection events. Bottom: monotonically decreasing sequence of corresponding
HOMFLYPT numerical values.
By applying the standard HOMFLYPT skein relations, given by
ð1Þ
ð2Þ in the independent variables a and z, we can compute the polynomial P
Kfor any given knot K.
Explicit computations of some basic examples are given in (Liu and Ricca 2015 ). By applying equations ( 1 )–( 2 ) to the ordered set { ( T 2, n )} one can prove the following general result (Liu and Ricca 2016 ).
Theorem (Liu and Ricca 2016). Let us consider the ordered set { ( T 2, n )} (n integer,
)
n 2 of torus knots and links. The HOMFLYPT polynomial P
T(2,n)of T ( 2, n ) ( n 4 is ) given by
= - -
+ + - -
+
- - -
- -
- - -
- -
( )
( )
( )
( ) ( )
( )
( )
( )
( )
( )
P k k
a k k P k k
a k k P , 3
T n
n n
n T
n n
n T
2,
2 2
3 1 2,3
3 3
2 1 2,2
where P
T 2,3( )= 2 a
-2+ a z
-2 2- a
-4and P
T 2,2( )= a z
-1+ ( a
-1- a
-3) z
-1.
Initial values are given by the HOMFLYPT computation of the 2-component unlink unknot T 2, 0 ( ) , unknot T 2, 1 ( ) , Hopf link T 2, 2 ( ) and trefoil knot T 2, 3 ( ) . In general, for any physical knot /link K of writhe Wr, twist Tw and unit strength (flux of vorticity), we have (Liu and Ricca 2015 )
=
lt=
lw( )
a e
Tw, k e
2 Wr, 4
with z = - k k
-1and uncertainty (probability) factors l l Î {
w,
t} ( 0, 1 . Consistently with ) the above assumptions, let l
w= l
t= 1 2 (equivalent to an equi-distribution of uncertainty of writhe and twist values ) and Wr = Tw = + 1 2 (admissible values of writhe and twist of standardly embedded positive torus knots (Oberti and Ricca 2016 ). Hence, the sequence of numerical values shown in figure 1 and reproduced in the table of figure 3 is obtained by taking
= = ( )
a e
1 4, k e
1 2. 5
Note that a different choice of l
wand l
tvalues would simply produce a vertical shift of the
numerical sequence in the plot.
3. Jones and Alexander-Conway polynomials for the alternate sequence of knots and co-oriented links
3.1. Jones polynomial V
T ð2;nÞThe HOMFLYPT polynomial P
Kdegenerates to the Jones polynomial V
Kwhen we take
t t t
=
-= -
-= -
-( )
a
1, z k k
1 12 12, 6
where τ is the new polynomial variable. The standard Jones polynomial is indeed a single- variable polynomial obtained by combining the two HOMFLYPT variables according to
=
ak
21 , i.e. a = k
-2. For physical knots this amounts to a framing prescription on the admissible values of writhe (through z) and twist (through a), given by
l l l l
= = - ( =
w t) ( )
ak
21 Tw 4 Wr . 7
If we take l = 1 , then Tw = - 4 Wr , that for Wr = 1 2 we have Tw = - 2 , that represents a rather peculiar constraint for fluid systems. In any case, by taking( 6 ) the skein relations of the Jones polynomial reduce to
ð8Þ
ð9Þ Explicit computations of some basic examples are given in (Liu and Ricca 2012 ). By substituting the positions ( 6 ) into ( 3 ), and after some straightforward algebra, we obtain the Jones polynomial V
T(2,n)for the ordered set { ( T 2, n )} ( n 4 : )
t t
t t
t t
t t
= + -
+ + + -
+
- - -
-
- - -
-
( ) ( )
( )
( ) ( )
( ) ( )
( )
V 1 V 1 V
, 10
T n
n n n
T
n n n
2, T
4 1 2
2,3
3 7 2 1
2,2
3 2
1 2 1
2 1 2
1 2
1 2 1
2 1 2
with V
T 2,3( )= + t t
3- t
4and V
T 2,2( )= - t - t
1
2 5
2
. 3.2. Numerical values for the sequence of V
T ð2;nÞAccording to Ricca and Liu ( 2014 ) (section 6, equation ( 16 )) it is reasonable to take
t = e .
-1( 11 )
Hence, by ( 10 ) and the position above we have
= + -
+ + + -
+
- + - - +
-
- - - - -
-
( ) ( )
( )
( ) ( )
( ) ( )
( )
V e e
e e V e e
e e V
1 1
, 12
T n
n n n
T
n n n
2, T
4 1 2
2,3
3 7 2 1
2,2
3 2
1 2 1
2 1 2
1 2
1 2 1
2 1 2
with
=
-+
--
-= -
--
-( )
( ) ( )
V
T 2,3e
1e
3e
4, V
Te e , 13
2,2
1 2
5 2
and (Ricca and Liu ( 2014 ), section 6, equation ( 17 ))
= - ( +
-) ( )
( )
V
T 2,0e
121 e
1. 14
The numerical values of V
T(2,10), ¼ , V
T(2,0)are shown in the table of figure 3. As we see, the
Jones sequence is, in absolute values, a monotonically growing sequence as n 0. In terms
of signed numbers we actually have two distinct sequences, one of positive values for knots
and one of negative values for two-component links, both growing with decreasing values
of n.
ð16Þ
ð17Þ As above, by substituting the positions ( 15 ) into ( 3 ), and after some straightforward algebra, we obtain the Alexander-Conway polynomial D
T(2,n)for the ordered set { ( T 2, n )} ( n 4 : )
D = + -
+ D + + -
+ D
- - -
- - -
- - - -
( ) ( )
( )
( ) ( ) ( )
t t
t t
t t
t t
1 1
, 18
T n
n
T
n 2, T
1
2,3
2
2,2
n 2 n n n
2 2
2 1
2 1 2
3
2 3
2 1
2 1 2
where D
T 2,3( )= t + t
-1- 1 and D
T 2,2( )= t
12- t
-12. 3.4. Numerical values for the sequence of Δ
T ð2;nÞWe take
=
-( )
t e .
119
Hence, by ( 18 ) and the position above, we have
D = + -
+ D + + -
+ D
- -
-
- -
-
- - - -
( ) ( )
( )
( ) ( ) ( )
e e
e e
e e
e e
1 1
, 20
T n
n
T
n 2, T
1
2,3
2
2,2
n 2 n n n
2 2
2 1
2 1
2
3
2 3
2 1
2 1
2
with
D
T(2,3)= e
-1+ - e 1, D
T( )= e
-- e , D
T( )= 0. ( 21 )
2,2 2,0
1
2 1
2
The numerical values of D
T(2,10), ¼ D ,
T(2,0)are shown in the table of figure 3. As we see the Alexander-Conway sequence is, in absolute values, a monotonically decreasing sequence as
n 0. Similarly to Jones, in terms of signed numbers we actually have two distinct sequences, one of positive values for knots and one of negative values for two-component links, both decreasing with decreasing values of n.
4. HOMFLYPT polynomial for the sequence of links with oppositely oriented components
In the case of torus links with oppositely oriented components the family { ( T 2, n )} reduces to the ordered set { ( T
o2, 2 n )} . By direct application of the HOMFLYPT skein relations ( 1 )–( 2 ) we can see (figure 2 ) that at every reduction step we have the production of a torus link of lower topological complexity, and of an unknot. Hence, we can prove the following result in full generality.
Lemma. Let us consider the ordered set of oppositely oriented torus links { ( T
o2, 2 n )} (n
integer, n ) 1 . The HOMFLYPT polynomial P
To(2,2n)is given by
= -
+ -
- ( )
( )
P a
az a a
a az
1 1
1 . 22
T n n
n 2,2
2
2 2
2
o
Proof. Let us apply the skein relations ( 1 )–( 2 ) to the top diagram of figure 2. We have
-
- += ( )
( ) ( )
aP
T 2,2na P
1 T nz, 23
2,2 2
o o
that is
= -
+
( )
( ) ( )
P
T 2,2n 2a P
2 T naz. 24
o o2,2
By applying the same relation recursively, we have
=
-- ( )
( ) ( ( ))
a P
2 T na P
T na z, 25
2,2 4
2,2 1 3
o o
= -
- -
( )
( ( )) ( ( ))
a P a P a z,
26
T n T n
4 2,2 1 6
2,2 2 5
o o
= -
- + - + - +
( )
( )
( ) ( )
( ) ( )
a
nP
Ta
nP a z. 27
T n
2 1 2
2,2 2 1 4
2,0 2 1 3
o o
By substituting the l.h.s. term of equation ( 25 ) into the r.h.s. of ( 24 ) and by doing the same for the subsequent terms in the equations above, we obtain
= - + + + +
= - -
-
+ - +
+ +
( )
( ) ( )
( ) ( )
( )
( )
( )
P a P az a a a
a P az a
a n
1 1
1 1 . 28
T n n
T n
n T
n
2,2 2 2 1 4
2,0 2 4 2
2 2 2,0
2 1
2
o o
o
( )
P
T 2,0odenotes the polynomial of a disjoint union of two unknots given by Liu and Ricca ( 2015 ) section 4.1, equation (4.2):
d
= -
-
=
( )
( )
P a a
z . 29
T 2,0
1
o
Figure 2.
Application of the skein relation (2) to a portion of torus knot/link diagram.
The over-crossing (top diagram, encircled region) is converted to the under-crossing
(left diagrams) and the non-crossing (right diagrams). Double arrows denote
topological equivalence.
Thus, we have
d d
= + -
( )
( )
P a 1 a
, 30
T n n
n
2,2 2 2
o
or, equivalently
= -
+ -
( )
-
P a
az a a
a az
1 1
1 ,
T n n
n 2,2
2
2 2
2
o
that proves the Lemma. +
4.1. Numerical values for the sequence of P
Toð2;nÞWithout loss of generality we can take l
w= l
t= 1 2 and Wr = Tw = - 1 2 (admissible values of writhe and twist of standardly embedded negative torus links (Oberti and Ricca 2016 ). Hence, we have
=
-=
-( )
k e
1 2, a e
1 4, 31
and according to ( 28 ), we have numerically:
d
= = = - = -
-= ( )
a k z a a
0.78, 0.61, 1.04, z 0.48. 32
1
Hence, for HOMFLYPT we have
d d
= + -
= ´ + -
( )
( )
P a 1 a
0.78 0.48 1 0.78
0.48 . 33
T n n
n
n
n
2,2 2 2
2 2
o
The numerical values of P
To(2,10), ¼ , P
To(2,0)are shown in the table of figure 3. For oppositely oriented components (negative crossings) the sequence decreases with decreasing values of n.
Figure 3.
Numerical values of HOMFLYPT, Jones and Alexander-Conway adapted
polynomials for the sequence T ( 2, 10 ) - T ( 2, 0 and ) T
o( 2, 10 ) - T
o( 2, 0 ) . Note that
Wr = 1 2 for co-oriented link components (with positive crossings) and Wr = - 1 2
for oppositely oriented link components (with negative crossings).
This behavior is consistent with that obtained for the alternate sequence of torus knots and co- oriented links.
5. Jones and Alexander-Conway polynomials for the sequence of links with oppositely-oriented components
5.1. Jones polynomial V
Toð2;nÞAs mentioned in section 3.1 HOMFLYPT degenerates to Jones by taking equations ( 6 ). Thus, equation ( 22 ) reduces to
t t t t t t
= - + - - - t
-
- - - - -
( ) ( )
-( ) ( )
( )
V 1 n
1 1 . 34
T n n
n
2,2 2 2
2
o
1
2 1
2 1
2 3
2
5.2. Numerical values for the sequence of V
Toð2;nÞAs above, we can take the same uncertainty value for Jones and the average writhe
= -
Wr 1 2 for standardly embedded negative torus links; then, we have t = e . By sub- stituting this value into ( 34 ), we have
= - + - - -
-
- - - - -
( ) ( )
-( ) ( )
( )
V e e e e e e
e n
1
1 1 , 35
T n n n
2,2 2 2
2
o
1 2
1 2
1 2
3 2
with (Ricca and Liu ( 2014 ), equation (6))
= -
-- ( )
( )
V
T 2,0oe e . 36
1
2 1
2