A Knot Model in Psychology

Akio KAWAUCHI

ABSTRACT

By evaluating the five-factors in Five-Factor Model of personality estab-lished by H. J. Eysenck-S. B. G. Eysenck and P. T. Costa-R. R. McCrae, weconstruct a knot model of a human mind in terms of 2-bridge knots so that

the knot type of a mind-knot gives the personality%We relate the classifica-tion of the self-releasability relations of mind-links to a certain classification of

links%The (normal) self-releasability relations for two mind-knots and threemind-knots are classified into 3 relations and 30 relations, respectively.

Mathematical Subject Classification 2000: 57M25, 91E45, 92-08

Keywords: Five-Factor Model, Personality, Human mind, Psychology, Knot, Link,Self-releasability relation

1. Introduction

In this paper, we propose to visualize a mind situation by constructing a knot modelof a human mind (briefly, a mind) as an application of knot theory. For severalyears ago (cf.[9]), the author considered that there is no contradiction by consideringminds as knots whose knot types are regarded as the personalities and whose crossingchanges are regarded as mind-changes, from the reason that several expressions onminds, such as a tame (straightforward) character:G>J-J (sunaona seikaku) inJapanese and a twisted character:RM/l?-J (hinekureta seikaku) in Japanese1,are represented in daily life by strings which are main research objects in knot theory.Besides, the author knew by R. Rucker’s book [12] that B. Stewart and P. G. Tait2

appears to say in their book [16] that the soul exists as a knotted vortex ring in theaether, althoughFF the aether hypothesisGGis known as a wrong story and the author

1Furthermore, W$Ne (+)(omoi no ito) meaning a string connecting human feelings, 33mNW~ (kokoro no kotosen) meaning one’s heartstring, MVX8NbDl (ningen kankei no motsure)meaning an emotional entanglement on human relations, 4Ko@+^k (kokoro ni wadakamaru)meaning being rooted in one’s mind.

2P. G. Tait is known as a pioneer of knot theory.

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does not clearly understand what they mean. For our purpose, the following pointsare some difficulty points which we must overcome to construct our model (cf. [3]):

(1) (Birth-Time Mind Situation) By a genetic character inherited from Parents,we cannot always assume that the mind in birth time is untwisted.

(2)(Estimation on Changes of Environments) There are many sources of mind-changes coming from changes of environments on

(i) Age(ii) History(iii) Non-standard event factors.

In our knot model of a mind, we count the conditions (1) and (2). Throughoutthis paper, we mean by knots unoriented loops embedded in the 3-space R3, and bylinks, disjoint unions of finitely many knots in R3. The types of knots and links areunderstood as the equivalence classes of them under the Reidemeister moves in Fig.1.For general terms of knot theory, we refer to [8].

Figure 1: Reidemeister moves

Concluding this introduction, we mention here some studies on applying topology topsychology. E. C. Zeeman’s work on the topology of the brain and visual perceptionin [20] and the topological psychology by K. Lewin [10] which is an application oftopological space are respectively pointed out by R. Fenn and by J. Simon withan implication by S. Kinoshita during the international conferenceH InternationalWorkshop on Knot Theory for Scientific ObjectsI. L. Rudolph reported to the authorhis works of several applications of topology to psychology which are published in[13, 14, 15] and his works (in preparation) on a rehabilitation of K. Lewin’s topological

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psychology. Incidentally, the author also knew J. Valsiner’s book [18] by Rudolph’sreport.

2. Constructing a knot model of a mind

To clarify our viewpoint of a mind, we set up the following mind hypothesis:

Mind Hypothesis 2.1.

(1) A mind is understood as a knot, so that an untwisted mind is a trivial knot anda twisted mind is a non-trivial knot.(2) A personality is understood as the type of a knot, so that the untwisted personalityis the knot type of a trivial knot and a twisted personality is the knot type of a non-trivial knot.(3) A mind change is understood as a crossing change of a knot(see Fig.2).

Figure 2: A crossing change

To construct a concrete knot model, we consider the basic factors of a mind by H.J. Eysenck [4, 5] and H. J. Eysenck-S. B. G. Eysenck[6, 7], which are described asfollows:

(1) Introversion-Extroversion.(2) Neuroticism.(3) Psychoticism.

These basic factors are refined by P. T. Costa and R. R. McCrae [1] as Five-FactorModel (= Big-Five Model) described as follows:

(1) Introversion-Extroversion.(2) Neuroticism.(3.1) Openness to Experience.(3.2) Agreeableness(3.3) Conscientiousness

Our idea of constructing a knot model of a mind is to evaluate the degrees of thefive-factors of the personality of a mind at the n-year-old as follows(See [19] for aconcrete test):

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Definition 2.2.

(1) The introversion-extroversion degree IEn at n-year-old takes the value IEn =−1, 1, 0 according to whether the mind is introverted, extroverted, or insignificant.(2) The neuroticism degree Nn at n-year-old takes the value Nn = −1, 0 according towhether the mind is neurotic or not.(3.1) The openness to experience degree On at n-year-old takes the value On = 0,−1according to whether the mind is open to the experience or not.(3.2) The agreeableness degree An at n-year-old takes the value An = −1, 0 accordingto whether the mind is disagreeable or not.(3.3) The conscientiousness degree Cn at n-year-old takes the value Cn = −1, 0according to whether the mind is unconscientious or not.

We define the psychoticism degree at n-year-old from (3.1)-(3.3) as follows:

(3) The psychoticism degree OACn at n-year-old to be the product

OACn = On ·An ·Cn = −1 or 0.

We also need Father’s and Mother’s data IEF , NF , OACF and IEM , NM , OACM ontheir introversion-extroversion, neuroticism and psychoticism degrees at the baby’sbirth time, respectively. Namely, we have the following definition:

Definition 2.3.

(1) Father’s and Mother’s introversion-extroversion degrees IEF and IEM at thebaby’s birth time take the values IEF = −1, 1, 0 and IEM = −1, 1, 0 according towhether Father’s mind and Mother’s mind are introverted, extroverted, or insignifi-cant, respectively.(2) Father’s and Mother’s neuroticism degrees NF and NM at the baby’s birth timetake the values NF = −1, 0 and NM = −1, 0 according to whether Father’s mind andMother’s mind are neurotic or not, respectively.(3.1) Father’s and Mother’s openness to experience degrees OF and OM at the baby’sbirth time take the values OF = 0,−1 and OM = 0,−1 according to whether Father’smind and Mother’s mind are open to the experience or not, respectively.(3.2) Father’s and Mother’s agreeableness degrees AF and AM at the baby’s birthtime take the values AF = −1, 0 and AM = −1, 0 according to whether Father’s mindand Mother’s mind are disagreeable or not, respectively.(3.3) Father’s and Mother’s conscientiousness degrees CF and CM at the baby’s birthtime take the values CM = −1, 0 and CF = −1, 0 according to whether Father’s mindand Mother’s mind are unconscientious or not, respectively.

We define Father’s and Mother’s psychoticism degrees at the baby’s birth time from(3.1)-(3.3) as follows:

(3) Father’s and Mother’s psychoticism degrees OACF and OACM at the baby’s birthtime are the products

OACF = OF · AF · CF = −1 or 0 and OACM = OM · AM · CM = −1 or 0.

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We define Parents’ introversion-extroversion, neuroticism and psychoticism degreesIEP , NP and OACP to be

IEP = ℓF IEF + ℓM IEM , NP = mFNF + mMNM , OACP = nFOACF + nMOACM

for non-negative integral constants ℓF , ℓM , mF , mM , nF , nM . In our argument here,we take ℓF = ℓM = mF = mM = nF = nM = 1, but the values shuold be chosenby estimating carefully a genetic character inherited from Parents (cf. [3]). Thetotal introversion-extroversion, neuroticism and psychoticism degrees of a mind atn-year-old are defined as follows:

IE[n] = IEP +n∑

i=1

IEi,

N[n] = NP +n∑

i=1

Ni,

OAC[n] = OACP +

n∑

i=1

OACi.

Let a = IE[n] + N[n] and b = OAC[n]. We have

−2n − 4 ≦ a ≦ n + 2 and − n − 2 ≦ b ≦ 0.

Our knot model of a mind is the knot M(n; a, b) which is represented by a knotdiagram with 2|a| + 2|b| crossings illustrated in Fig.3 and called the nth mind-knot.In the figure, we illustrate the case that a is negative. The picture for the case thata is positive is given by changing all the the crossings in this 2|a|-tangle part intothe crossings with the opposite sign. When the knot diagrams M(n − 1, a′, b′) andM(n; a, b) are distinct, we consider that some mind-changes are made during (n−1)-year-old and n-year-old. Thus, in our model, the total picture of a mind-knot fromthe birth time to the n-year-old is considered as a cylinder immersed properly inthe 4-dimensional space R3 × [0, n]. If we evaluate the degrees of the five-factorsfiner, then a finer mind-knot model can be constructed. See § 5 later for the othermind-knot models.

The following proposition is a consequence of the well-known classification of 2-bridgeknots in knot theory (cf. [8, 17]).

Proposition 2.4. A mind-knot M(n; a, b) is untwisted if and only if a = 0 or b = 0.Two twisted mind-knots M(n; a, b), M(n′; a′, b′) have the same personality if and onlyif (a′, b′) = (a, b) or (−b,−a).

Proof. The knot M(n; a, b) is a 2-bridge knot whose type is calculated as follows:

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2a +1

2b

=2b

4ab + 1.

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Figure 3: A mind-knot M(n; a, b)

Thus, M(n; a, b) is a trivial knot if and only if 4ab + 1 = ±1 that is ab = 0. LetM(n; a, b) and M(n′; a′, b′) be non-trivial knots. If they belong to the same knottype, then we have 4ab + 1 = ±(4a′b′ + 1), so that ab = a′b′ 6= 0. Further, we have2b ≡ 2b′ (mod 4ab+1) or 4bb′ ≡ 1 (mod 4ab+1). If the first case occurs, then there isan integer k such that 2(b− b′) = k(4ab+1). The integer k must be even, i.e., k = 2k′

and we have b− b′−k′ = 4abk′. Suppose k′ 6= 0. Then for N = max{|b|, |b′|, |k′|} > 0,we have 3N ≧ 4N , a contradiction. Hence we have k′ = 0 and b = b′, a = a′. If thesecond case occurs, then we have 2b ≡ −2a′ (mod 4ab + 1). In fact,

2b′(2b + 2a′) ≡ 4bb′ + 4a′b′ = 4bb′ + 4ab ≡ 1 − 1 = 0 (mod 4ab + 1)

and, since 2b′ is coprime with 4a′b′ +1 = 4ab+1, we have 2b+2a′ ≡ 0 (mod 4ab+1).Using that M(n′; a′, b′) and M(n′;−b′,−a′) belong to the same knot type, we see fromthe argument of the first case that −b′ = a and −a′ = b.

For example, M(n; 1,−1) and M(n;−1, 1) are trefoil knots which are each other’smirror images, and M(n; 1, 1) = M(n;−1,−1) is the figure-eight knot.

3. Self-releasability relations on mind-links

When we consider a mind as a knot, it is an interesting problem to consider a mind-link, that is, a link of mind-knots. We consider that a mind-link is generated by acrossing change (called also a mind-change) between a pair of mind-knot componentsand a mind-change on a mind-knot component. We shall introduce here a concept ofself-releasability relation on links of n(≧ 2) mind-knots.

Definition 3.1. Two non-splittable mind-links L and L′ have the similar self-releasability relation if there is a bijection from the mind-knots Ai (i = 1, 2, . . . , n)

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of L onto the mind-knots A′

i (i = 1, 2, . . . , n) of L′ such that for every mind-knotcomponent Ai and every sublink Li of L\Ai

(1) the splittability between Ai and Li coincides with the splittability between thecorresponding A′

i and L′

i, and(2) the splittability between Ai and Li up to finitely many mind-changes of Ai co-incides with the splittability of the corresponding A′

i and L′

i up to finitely manymind-changes of A′

i.

In Definition 3.1, we say that a mind-knot component Ai is self-releasable from Li ifAi is made split from Li by finitely many mind-changes of Ai, which is equivalent tosaying, in terms of topology, that Ai is null-homotopic in R3\Li. The similar self-releasability relation is an equivalence relation and we call the equivalence class of anon-splittable mind-link under the similar self-releasability relation a self-releasabilityrelation. To consider the classification of self-releasability relations, we consider arestrictive mind-link which satisfies the following hypothesis:

Normal Relation Hypothesis(NRH). For every i, Ai is self-releasable from Li ifand only if Ai is self-releasable from Aj for every mind-knot component Aj of Li.

Figure 4: An anomalous self-releasability relation for three mind-knots

We call a mind-link satisfying NRH a normal mind-link. The other mind-link iscalled an anomalous mind-link. A self-releasability relation on normal mind-links iscalled a normal self-releasability relation, and the other self-releasability relation ananomalous self-releasability relation. A self-releasability relation for 2 mind-knots isalways normal. However, for three mind-knots, there are anomalous self-releasabilityrelations. A typical example is the Borromean rings in Fig. 4 (We denote a split linkof two mind-knots A and B by A · ·B in the schematic diagram.) The anomalyfollows from a classification of link-homotopy (see J. Milnor[11]). The classificationof self-releasability relations for 2 mind-knots is given as follows:

Proposition 3.3. The following cases (1)-(3) give the complete list of the self-releasability relations for 2 mind-knots A and B.

(1) Neither mind-knot component is self-releasable from the other. This relation isdenoted by A−B.

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Figure 5: Self-releasability relations for two mind-knots

(2) Both mind-knots are self-releasable from each other. This relation is denoted byA↔B.(3) One mind-knot component, say A is self-releasable from the other mind-knot-component B, but B is not self-releasable from A. This relation is denoted by A→B.In this case, A is necessarily a twisted mind.

Proof. Let A ∪ B be a 2-component non-split link. For example, if the linkingnumber Link(A, B) 6= 0, then it satisfies (1), because A is not null-homotopic in thespace R3\B and B is not null-homotopic in the space R3\A. If A and B are untwistedmind-knots and the linking number Link(A, B) = 0, then A and B are respectivelynull-homologous in the spaces R3\B and R3\A which are homotopy equivalent toS1, so that A and B are null-homotopic in R3\B and R3\A, respectively. Thus, thislink in this case satisfies (2). If Link(A, B) = 0 and B is an untwisted mind andnot null-homotopic in R3\A (in this case, A must be a twisted mind-knot), then A isself-releasable from B, but B is not self-releasable from A, satisfying (3). The typicalexamples of the cases (1)-(3) are given in Fig. 5. In (3) of Fig. 5, we note that Ais a trefoil knot and B is a trivial knot and represents the element x−1y in a grouppresentation (x, y|xyx = yxy) of the fundamental group π1(R

3\A) (see [2]). Sincethis group is non-abelian, we have x−1y 6= 1 in (x, y|xyx = yxy) and we see that B isnot null-homotopic in R3\A.

4. Classification of normal self-releasability relations for three mind-knots

We give here the classification of normal self-releasability relations for three mind-knots.

Proposition 4.1. The list of 30 links in Fig. 6 and Fig. 7 is the complete list ofnormal self-releasability relations for three mind-knots.

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Figure 6: Normal self-releasability relations for three mind-knots

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Figure 7: Normal self-releasability relations for three mind-knots(continued)

Proof. This construction is done by listing first all the distinct triangle relationswith virtices A, B, C up to permutations of A, B, C and then realizing them bylinks. The non-splittability and normalness of the links listed here are easily checkedif an argument similar to the proof of Proposition 3.3 is used.

5. Further discussions

By taking aP = IEP + NP , bP = OACP , a = a − aP , and b = b − bP , we have adifferent knot model M(n; aP , bP , a, b) of a mind, illustrated in Fig. 8. If bP = 0,

then we have M(n; aP , b,a, b) = M(n; 0, 0, a, b) where Proposition 2.4 can be usedfor the classification. However, if bP 6= 0, then M(n; aP , bP , a, b) is on a differentbehavior from M(n; a, b) in general. We have also a further refined knot model by

decomposing a and b into the numbers ai = IEi + Ni and bi = OACi (i = 1, 2, . . . , n),

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Figure 8: A mind-knot M(n; aP , bP , a, b)

respectively, and by considering the knot M(n; aP , bP , a1, b1, . . . , an, bn) illustrated inFig. 9. In our construction, we used the full-twist tangles of the numbers a, aP , a, ai

and b, bP , b, bi. If we use the half-twist tangles of these numbers instead of the full-twists, then our knot model changes into a link model which is a mind-knot or a linkof just two untwisted mind-knots. Also, we note that there is a finer concept of theself-releasability relations for three or more mind-knots described as follows: Twonon-splittable mind-links L and L′ have the same global self-releasability relation ifthere is a bijection τ from the mind-knots of L onto the mind-knots of L′ such thatfor every mind-sublink L1 of L and every mind-sublink L2 of L\L1, we have

(1) the splittability on (L1, L2) coincides with the splittability of (τ (L1), τ (L2)), and(2) the splittability on (L1, L2) up to finitely many mind-changes on the components ofL1 coincides with the splittability of (τ (L1), τ (L2)) up to finitely many mind-changeson the components of τ (L1).

Figure 9: A mind-knot M(n; aP , bP , a1, b1, . . . , an, bn)

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The same global self-releasability relation is an equivalence relation. The equivalenceclass of a non-splittable mind-link under the same self-releasability relation is called aglobal self-releasability relation. Then we set the following problem, meaningful evenin knot theory when we replace n mind-knots by general links of n components, whichwill be also discussed elsewhere.

Problem 5.1. Classify the global self-releasability relations for n mind-knots.

Finally, the author does not yet succeed in constructing any process model of gener-ating a mind-link from mind-knots.

References

[1] P. T. Costa and R. R. McCrae, From catalog to classification: Murray’s needsand the five-factor model, Journal of Personality and Social Psychology, 55(1988),258-265.

[2] R. H. Crowell and R. H. Fox, Introduction to knot theory, Ginn and Co. (1963).

[3] H. Enomoto and T. Kuwabara, Personality psychology (in Japanese), A textbookof the University of the Air(2004).

[4] H. J. Eysenck, Dimensions of personality, Transaction Publishers(1997).

[5] H. J. Eysenck, The biological basis of personality, Transaction Publishers (2006).

[6] H. J. Eysenck and S. B. G. Eysenck, Personality structure and measurement,London: Routledge(1969).

[7] H. J. Eysenck and S. B. G. Eysenck, Psychoticism as a dimension of personality.London: Hodder and Stoughton (1976).

[8] A. Kawauchi, A survey of knot theory, Birkhauser (1996).

[9] A. Kawauchi, Physical-chemical systems and knots (in Japanese), in: Have funwith mathematics, 5(1998), 72-80, Nihon-hyoron-sha.

[10] K. Lewin, Principles of topological psychology, McGraw-Hill (1936).

[11] J. W. Milnor, Link groups, Ann. of Math., 59(1954), 177-195.

[12] R. Rucker, The fourth dimension, A guided tour of the higher universes,Houghton Mifflin Company, Boston(1984).

[13] L. Rudolph, The fullness of time, Culture and Psychology, 12(2006), 157-186.

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[14] L. Rudolph, Mathematics, models and metaphors, Culture and Psychology,12(2006), 245-265.

[15] L. Rudolph, Spaces of ambivalence: Qualitative mathematics in the modeling ofcomplex fluid phenomena, Estudios de Psicologia, 27(2006), 67-83.

[16] B. Stewart and P. G. Tait, The unseen universe or physical speculations on afuture state, MacMillan and Co., London and New York (1894).

[17] H. Schubert, Knoten mit zwei Brucken, Math. Z., 65(1956), 133-170.

[18] J. Valsiner, The guided mind: a sociogenetic approach to personality, HarvardUniversity Press(1998).

[19] http://en.wikipedia.org/wiki/Big five personality traits.

[20] E. C. Zeeman, The topology of the brain and visual perception, Topology of3-manifolds and related topics, pp. 240-256, Prentice-Hall(1962).

Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japankawauchi@sci.osaka-cu.ac.jp

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