Singquandle Shadows and Singular knot Invariants
aa r X i v : . [ m a t h . G T ] J a n SINGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS
JOSE CENICEROS, INDU R. CHURCHILL, AND MOHAMED ELHAMDADI
Abstract.
We introduce shadow structures for singular knot theory. Precisely, we define two invariants of singular knots and links. First, we introduce a notion of action of a singquandle on aset to define a shadow counting invariant of singular links which generalize the classical shadowcolorings of knots by quandles. We then define a shadow polynomial invariant for shadowstructures. Lastly, we enhance the shadow counting invariant by combining both the shadowcounting invariant and the shadow polynomial invariant. Explicit examples of computationsare given.
Contents
1. Introduction 12. Basics of Quandles 23. Oriented Singquandles and the Counting Invariant 34. Review of the Singquandle Polynomial 55. Singquandle Shadows 76. Enhanced Shadow Counting Invariant 137. Examples 138. Acknowledgments 15References 161.
Introduction
Quandles are non-associative algebraic structures that are modeled on the three Reidemestermoves in classical knot theory. Thus, they are appropriate algebraic structures for construct-ing invariants of knots and links in 3-space and knotted surfaces in 4-space. Quandles wereintroduced independently by Joyce [15] and Matveev [17] in the 1980s. Quandles have beeninvestigated in many areas of mathematics such as quasigroups and Moufang loops [10], theYang-Baxter equation [3, 6], representation theory [12], and ring theory [11]. For more infor-mation on quandles, the reader is advised to consult the book [13]. Knot theory has beenextended in several directions, for example, singular knot theory. In [2], connections betweenJones type invariants defined in [14] and Vassiliev invariants of singular knots defined in [21]were established. In [16], a variation of the Hecke algebra was used to construct a Jones-typeinvariant for singular knots. Combinatorial singular knot theory has the so-called generalizedReidemeister moves [16]. These generalized moves were used in [9] to extend the concept ofquandle to an algebraic structure called singquandle to provide invariants for singular knots.In [1], generating sets of the generalized Reidemeister moves for oriented singular links were
Mathematics Subject Classification.
Primary 57K12, 05C38; Secondary 05A15.
Key words and phrases.
Quandle polynomial, Singular Knots and Links, Singquandle polynomial.M.E. was partially supported by Simons Foundation collaboration grant 712462. introduced and used to distinguish singular knots and links. In [5], the quandle cocycle invari-ant [4] was extended to oriented singular knots and used to construct a state sum invariant forsingular links. Furthermore in [7], the quandle polynomial invariant was extended to the caseof singquandles. A singular link invariant was constructed from the singquandle polynomialand it was shown to generalize the singquandle counting invariant in [7].The article is organized as follows. In Section 2, the basics of quandle theory are reviewed,and some examples are given. Section 3 provides a review of the basic constructions of ori-ented singquandles as well as the singquandle counting invariant. In Section 4, we discuss thesingquandle polynomial and the subsingquandle polynomial, which we introduced in a previouspaper [7] and used to define a singular link invariant. Section 5 defines singquandle shadows which is used to generalize the shadow colorings of knot diagrams by quandles previously definedin [6]. Furthermore, the shadow singquandle polynomial and the singquandle shadow polynomialinvariant for a singular link L is defined. In Section 6, the shadow counting invariant is usedto define an enhanced shadow link invariant by combining the shadow singquandle countinginvariant and the shadow singquandle polynomial. Lastly, Section 7 examines the strength ofthe singquandle shadow polynomial invariant. Two examples are provided to illustrate that thesingquandle shadow polynomial is sensitive to differences in singular links not detected by thesingquandle coloring invariant and singquandle polynomial invariant.2. Basics of Quandles
In this paper we will consider only finite quandles and singquandles. We’ll review the basicsof quandles; more details on the topic can be found in [13, 15, 17].
Definition 2.1.
A set X with binary operation ∗ is called a quandle if the following threeidentities are satisfied.(i) For all x ∈ X , x ∗ x = x .(ii) For all y, z ∈ X , there is a unique x ∈ X such that x ∗ y = z .(iii) For all x, y, z ∈ X , we have ( x ∗ y ) ∗ z = ( x ∗ z ) ∗ ( y ∗ z ) . From Axiom (ii) of Definition 2.1 we can write the element x as z ¯ ∗ y = x . Notice that thisoperation ¯ ∗ defines a quandle structure on X . The axioms of a quandle correspond respectivelyto the three Reidemeister moves of types I, II and III (see [13] for examples). In fact, one ofthe motivations of defining quandles came from knot diagrammatic.A quandle homomorphism between two quandles ( X, ∗ ) and ( Y, ⊲ ) is a map f : X → Y suchthat f ( x ∗ y ) = f ( x ) ⊲ f ( y ), where ∗ and ⊲ denote respectively the quandle operations of X and Y . Furthermore, if f is a bijection, then it is called a quandle isomorphism between X and Y .Some typical examples of quandles: • Any non-empty set X with the operation x ∗ y = x, for all x, y ∈ X, is a quandle calleda trivial quandle. • Any group X = G with conjugation x ∗ y = y − xy is a quandle. • Let G be an abelian group. For elements x, y ∈ G , define x ∗ y ≡ y − x . Then ∗ definesa quandle structure on G called Takasaki quandle. In case G = Z n (integers mod n )the quandle is called dihedral quandle . This quandle can be identified with the set ofreflections of a regular n -gon with conjugation as the quandle operation. • Any Λ = ( Z [ T ± ])-module M is a quandle with x ∗ y = T x + (1 − T ) y , x, y ∈ M , calledan Alexander quandle . INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 3 Oriented Singquandles and the Counting Invariant
We will provide a basic overview of an oriented singquandle as well as the singquandle count-ing invariant. For a detailed construction of oriented singquandle and the singquandle countinginvariant, see [1, 9]. We will be adopting the following conventions at classical and singularcrossings. x yx ∗ yy y xx ¯ ∗ y y yxR ( x, y ) R ( x, y )( a ) ( b ) ( c ) Figure 1. (a) Coloring of arcs at a positive crossing, (b) coloring of arcs at anegative crossing, (c) colorings of semi-arcs at a singular crossing.Generating sets of oriented singular Reidemeister moves were studied and were used to defineoriented singquandles in [1]. We will follow the naming convention for oriented singular Reide-meister moves used in [1]. Note that if we let ( S, ∗ ) be a quandle, we only need to consider thecolorings from singular Reidemeister moves in Figures 2, 3, and 4. a a ¯ ∗ bR ( a ¯ ∗ b, c ) ∗ b cb R ( a ¯ ∗ b, c ) a cb c ∗ bR ( a, c ∗ b ) R ( a, c ∗ b )¯ ∗ b Figure 2.
The Reidemeister move Ω4 a with colors. Definition 3.1.
Let ( X, ∗ ) be a quandle. Let R and R be two maps from X × X to X . Thetriple ( X, ∗ , R , R ) is called an oriented singquandle if the following axioms are satisfied for all a, b, c ∈ X : R ( a ¯ ∗ b, c ) ∗ b = R ( a, c ∗ b ) (3.1) R ( a ¯ ∗ b, c ) = R ( a, c ∗ b )¯ ∗ b (3.2)( b ¯ ∗ R ( a, c )) ∗ a = ( b ∗ R ( a, c ))¯ ∗ c (3.3) R ( a, b ) = R ( b, a ∗ b ) (3.4) R ( a, b ) ∗ R ( a, b ) = R ( b, a ∗ b ) . (3.5) INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 4 ab ¯ ∗ R ( a, c ) R ( a, c ) ( b ¯ ∗ R ( a, c )) ∗ a cb R ( a, c ) a cb b ∗ R ( a, c ) R ( a, c ) R ( a, c )( b ∗ R ( a, c ))¯ ∗ c Figure 3.
The Reidemeister move Ω4 e with colors. aR ( a, b ) R ( a, b ) bR ( a, b ) R ( a, b ) ∗ R ( a, b ) abR ( b, a ∗ b ) ba ∗ bR ( b, a ∗ b ) Figure 4.
The Reidemeister move Ω5 a with colors. Remark . We will only consider oriented singquandles in this paper. Therefore, we will simplyrefer to oriented singquandle as singquandles. For a description of unoriented singquandles, see[9].In [5], the following family of singquandles was introduced.
Example 3.3.
Let n be a positive integer, let a be an invertible element in Z n and let b, c ∈ Z n ,then the binary operations x ∗ y = ax + (1 − a ) y , R ( x, y ) = bx + cy and R ( x, y ) = acx + [ b + c (1 − a )] y make the quadruple ( Z n , ∗ , R , R ) into an oriented singquandle.The singquandles used in this paper are obtained from Example 3.3 with specific values of a, b, and c . Definition 3.4.
Let ( X, ∗ , R , R ) be a singquandle. A subset M ⊂ X is called a subsingquan-dle if ( M, ∗ , R , R ) is itself a singquandle. In particular, M is closed under the operations ∗ , R and R .We can define the notion of a homomorphism and isomorphism of oriented singquandles. INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 5
Definition 3.5.
A map f : X → Y is called a homomorphism of oriented singquandles( X, ∗ , R , R ) and ( Y, ⊲, R ′ , R ′ ) if the following conditions are satisfied for all x, y ∈ Xf ( x ∗ y ) = f ( x ) ⊲ f ( y ) (3.6) f ( R ( x, y )) = R ′ ( f ( x ) , f ( y )) (3.7) f ( R ( x, y )) = R ′ ( f ( x ) , f ( y )) . (3.8)An oriented singquandle isomorphism is a bijective oriented singquandle homomorphism. Wesay two oriented singquandles are isomorphic if there exists an oriented singquandle isomor-phism between them.The authors of this paper introduced the idea of a fundamental singquandle associated to asingular link L , denoted by SQ ( L ), in [7]. Therefore, for any oriented singular link L and an ori-ented singquandle ( S, ∗ , R ′ , R ′ ), the set of singquandle homomorphism from ( SQ ( L ) , ⊲, R , R )to ( S, ∗ , R ′ , R ′ ) is defined by:Hom( SQ ( L ) , S ) = { f : SQ ( L ) → S | f ( x ⊲ y ) = f ( x ) ∗ f ( y ) ,f ( R ( x, y )) = R ′ ( f ( x ) , f ( y )) , f ( R ( x, y )) = R ′ ( f ( x ) , f ( y )) } . The set defined above was shown to be an invariant of oriented singular links in [1]. Furthermore,this set can be used to define computable invariants of oriented singular links. For example, bytaking the cardinality of Hom( SQ ( L ) , S ), we obtain the following invariant of oriented singularlinks. Definition 3.6.
Let L be an oriented singular link and ( S, ∗ , R , R ) be an oriented singquan-dles. The singquandle counting invariant is S ( L ) = | Hom( SQ ( L ) , S ) | . Remark . We also note that the image, Im( f ), for each f ∈ Hom( SQ ( L ) , S ) is a subs-ingquandle of S as shown in [7].4. Review of the Singquandle Polynomial
The quandle polynomial was introduced in [19] and generalized in [18]. In [7], the authors ofthis paper defined the singquandle polynomial, the subsingquandle polynomial and a polynomialinvariant of singular links. In this section, we will give an overview of the construction ofthe singquandle polynomial, the subsingquandle polynomial, and the polynomial invariant ofsingular links. We will follow the construction and notation introduced in [7].
Definition 4.1.
Let ( X, ∗ , R , R ) be a finite singquandle. For every x ∈ X , define C ( x ) = { y ∈ X | y ∗ x = y } and R ( x ) = { y ∈ X | x ∗ y = x } ,C ( x ) = { y ∈ X | R ( y, x ) = y } and R ( x ) = { y ∈ X | R ( x, y ) = x } ,C ( x ) = { y ∈ X | R ( y, x ) = y } and R ( x ) = { y ∈ X | R ( x, y ) = x } . Let c i ( x ) = | C i ( x ) | and r i ( x ) = | R i ( x ) | for i = 1 , ,
3. Then the singquandle polynomial of X is sqp ( X ) = X x ∈ X s r ( x )1 t c ( x )1 s r ( x )2 t c ( x )2 s r ( x )3 t c ( x )3 . INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 6
We note that the value r i ( x ) is the number of elements in X that act trivially on x , while c i ( x ) is the number of elements of X on which x acts trivially via ∗ , R and R . Furthermore,if Y ⊂ X is a subsingquandle we can define the following singquandle polynomial for Y as asubsignquandle of X Definition 4.2.
Let ( X, ∗ , R , R ) be a finite singquandle and S ⊂ X a subsingquandle. Thenthe subsingquandle polynomial is Ssqp ( S ⊂ X ) = X x ∈ S s r ( x )1 t c ( x )1 s r ( x )2 t c ( x )2 s r ( x )3 t c ( x )3 . The subsingquandle polynomials can be thought of as the contributions to the singquandlepolynomial coming from the subsingquandles we are considering. Using the subsingquandlespolynomial we can now define the following polynomial invariant of singular links.
Definition 4.3.
Let L be a singular link, ( X, ∗ , R , R ) a finite singquandle. Then the multisetΦ Ssqp ( L, X ) = { Ssqp ( Im ( f ) ⊂ X ) | f ∈ Hom( SQ ( L ) , X } is the subsingquandle polynomial invariant of L with respect to X . We can also representthis invariant in the following polynomial-style form by converting the multiset elements toexponents of a formal variable u and converting their multiplicities to coefficients: φ Ssqp ( L, X ) = X f ∈ Hom( SQ ( L ) ,X ) u Ssqp ( Im ( f ) ⊂ X ) . Example 4.4.
Consider the L listed as 1 l in [20]. Let ( S, ∗ , R , R )be the singquandle with S = Z and operations x ∗ y = 3 x − y = x ¯ ∗ y , R ( x, y ) = 2 x + 3 y , and R ( x, y ) = x . We can identify each coloring of 1 l by S with the triple ( f ( x ) , f ( y ) , f ( z )). Using zx y Figure 5.
Diagram of 1 l .the fact that z = R ( x, y ) , x = R ( x, y ) and z ∗ x = y, by a straightforward computation weobtain the following colorings:Hom( SQ (1 l ) , S ) = { (1 , , , (1 , , , (1 , , , (1 , , , (2 , , , (2 , , , (2 , , , (2 , , , (3 , , , (3 , , , (3 , , , (3 , , , (0 , , , (0 , , , (0 , , , (0 , , } . Therefore, S (1 l ) = 16. We compute r i and c i for i = 1 , , INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 7 x r ( x ) c ( x )1 2 22 2 23 2 20 2 2 x r ( x ) c ( x )1 1 12 1 13 1 10 1 1 x r ( x ) c ( x )1 4 42 4 43 4 40 4 4 . In order to compute the subsingquandle polynomial invariant of 1 l we consider the correspond-ing subsingquandle for each coloring. Therefore, we obtain φ Ssqp (1 l , S ) = 4 u s t s t s t + 4 u s t s t s t + 8 u s t s t s t . Singquandle Shadows
In this section, we define singquandle shadows which can be used to generalize the shadowcolorings of knot diagrams by quandles previously defined in [6].
Definition 5.1.
Let ( S, ∗ , R , R ) be a singquandle. An S-set is a set X and a map · : X × S → X satisfying the following conditions:(i) For all s ∈ S , · s : X → X mapping x to x · s is a bijection.(ii) For all s , s ∈ S and x ∈ X ,( x · s ) · s = ( x · s ) · ( s ∗ s ) (5.1)( x · s ) · s = ( x · R ( s , s )) · R ( s , s ) . (5.2)The meaning of these two equations will become clear from Figure 7. Definition 5.2. A singquandle shadow or S -shadow is the pair of an oriented sinquandle( S, ∗ , R , R ) and a S -set ( X, · ), denoted by ( S, X, ∗ , R , R , · ) or simply by ( S, X ). Let S ′ be asubsingquandle of S . A subset Y of X closed under the action of S ′ is an subshadow of ( S, X ),which we will denote by ( S ′ , Y ) ⊂ ( S, X ).The following definition will allow us to present a shadow operation in an alternate form thatwill be useful in later sections.
Definition 5.3.
When (
X, S ) is a singquandle shadow with X and S finite, the shadow matrix of the singquandle shadow ( X = { x , . . . , x m } , S = { s , . . . , s n } ) is the m × n matrix whose( i, j ) entry is k where x k = x i · s j .Let ( S, ∗ , R , R ) and ( S ′ , ⊲, R ′ , R ′ ) be singquandles. Furthermore, let ( X, · ) be an S -set and( X ′ , • ) be an S ′ -set. A map f : S → S ′ makes X ′ inherit a natural action of S via the map f by x ′ · s := x ′ • f ( s ). Definition 5.4. A homomorphism of sinquandle shadows between ( S, X, ∗ , R , R , · ) and( S ′ , Y, ⊲, R ′ , R ′ , • ) is a pair of maps φ : ( X, · ) → ( Y, • ) and f : ( S, ∗ , R , R ) → ( S ′ , ⊲, R ′ , R ′ ),such that f is a singquandle homomorphism, that is the identities (3.6), (3.7) and (3.8) aresatisfied and for all x ∈ X and s ∈ S , we have φ ( x · s ) = φ ( x ) • f ( s ) . (5.3)Furthermore, if φ and f are bijections then we have a singquandle shadow isomorphism .From this definition it is straightforward to obtain the following lemma. Lemma 5.5. ( Im ( f ) , Im ( φ ) , ⊲, R ′ , R ′ , • ) is a subshadow of ( S ′ , Y, ⊲, R ′ , R ′ , • ). INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 8
Example 5.6.
Let ( S, ∗ , R , R ) be an oriented singquandle with S = Z = { , , , , , , , } , x ∗ y = 5 x − y = x ¯ ∗ y , R ( x, y ) = 3 x + 4 y , and R ( x, y ) = 4 x + 3 y . Then the four element set X = Z = { , , , } with map · s : Z → Z for each s ∈ S defined by x · s = x + 2 s + s is asingquandle shadow. Note that · has the following operation table · . Furthermore, by Definition 5.3 the shadow operation · can be presented by the shadow matrix, . Example 5.7.
Let ( S, ∗ , R , R ) be any oriented singquandle and let X = S . Then X is asingquandle shadow under the shadow operation x · s = x ∗ s for all x, y ∈ S , since we have( x · s ) · s = ( x ∗ s ) ∗ s = ( x ∗ s ) ∗ ( s ∗ s )= ( x · s ) · ( s · s ) , (5.4)and ( x · s ) · s = ( x ∗ s ) ∗ s = ( x ∗ R ( s , s )) ∗ R ( s , s )= ( x · R ( s , s )) · R ( s , s ) . (5.5) Remark . Equation (5.4) is satisfied by the self-distributitive property of ∗ . On the otherhand, equation (5.5) is satisfied by applying a combination of singquandle properties. Considerequation (3.3) in the definition of a singquandle, let b = c . Therefore, we obtain( c ¯ ∗ R ( a, c )) ∗ a = ( c ∗ R ( a, c ))¯ ∗ c, which can be written as (( c ¯ ∗ R ( a, c )) ∗ a ) ∗ c = c ∗ R ( a, c ) . Now, we let w = c ¯ ∗ R ( a, c ) ⇐⇒ w ∗ R ( a, c ) = c . Next, we make the appropriate substitutionto obtain ( w ∗ a ) ∗ c = ( w ∗ R ( a, c )) ∗ R ( a, c ) . Lastly, let w = x , a = s , and b = s to obtain( x ∗ s ) ∗ s = ( x ∗ R ( s , s )) ∗ R ( s , s ) . Let D be a diagram of an oriented singular link L . We will denoted the set of arcs of D by A ( D ) and the connected regions of R \ D by R ( D ). Using the notion of a singquandle homo-morphism given in Definition 3.5, we have the following notion of colorings by singquandles. INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 9
Definition 5.9.
Let ( S, ∗ , R , R ) be an oriented singquandle. An S -coloring of D is a map f : A ( D ) → S such that at a crossing with u , u , o ∈ A ( D ) and at a singular crossing with a , a , a , a ∈ A ( D ) the following conditions are satisfied, f ( u ) = f ( u ) ∗ f ( o ) , (5.6) f ( a ) = R ( f ( a ) , f ( a )) , (5.7) f ( a ) = R ( f ( a ) , f ( a )) . (5.8)The conditions above are illustrated in Figure 6. Definition 5.10.
Let (
S, X, ∗ , R , R , · ) be a shadow singquandle. An ( S, X ) -coloring of D isa map f × φ : A ( D ) × R ( D ) → S × X satisfying the following conditions, • f is an S -coloring of D . • φ ( R ( D )) ⊂ X . • For a ∈ A ( D ) and x , x ∈ R ( D ) the following φ ( x ) · f ( a ) = φ ( x ) . (5.9)The condition above is illustrated in Figure 6.When there is no confusion we will refer to an ( S, X )-coloring by a shadow coloring of D . s = f ( u ) s = f ( o ) s ∗ s = f ( u ) φ ( x ) φ ( x ) = φ ( x ) · f ( a ) f ( a ) f ( a ) f ( a ) f ( a ) = R ( f ( a ) , f ( a )) f ( a ) = R ( f ( a ) , f ( a )) Figure 6.
Arcs and regions of diagram D .We will denote a region coloring by a box around the shadow element and we will denote anarc coloring by an element of a singquandle without a box. Note that the conditions required forthe set X to be an S -set for some oriented sinquandle, are the conditions needed to guaranteethat shadow colorings are well defined at crossings, see Figure 7. Proposition 5.11.
Let L be a singular link diagram and ( S, X, ∗ , R , R , · ) be a singquandleshadow. Then for each singquandle coloring of L by S and each element of X there is exactlyone shadow coloring of L . Proof.
Consider a singquandle coloring of L and choose a region of L . Any element of X can beassigned to the chosen region, and any choice determines a unique shadow color of each regionby following the rule in Figure 6. (cid:3) INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 10 s s s s ∗ s x x · s ( x · s ) · s x · s ( x · s ) · ( s ∗ s )= s s s s ∗ s x x · s ( x · s ) · ( s ∗ s ) x · s ( x · s ) · s = s s R ( s , s ) R ( s , s ) x x · s ( x · s ) · s x · R ( s , s ) ( x · R ( s , s )) · R ( s , s )= Figure 7.
Shadow coloring at positive, negative, and singular crossings.
Definition 5.12.
Let L be singular link diagram and ( S, X ) is a singquandle shadow. The shadow counting invariant , ( S,X ) ( L ), is the number of shadow colorings of L by ( S, X ). Example 5.13.
Let ( S, ∗ , R , R ) be the singquandle with S = Z and operations definedby x ∗ y = 3 x − y , x ¯ ∗ y = 7 x − y , R ( x, y ) = 4 x + 6 y , and R ( x, y ) = 8 x + 2 y . We candefine a shadow structure by X = Z with the map · s : Z → Z for each s ∈ S defined by x · s = 2 + x + 2 x . We also represent this shadow operation by the shadow matrix . We will compute a shadow coloring for the following K listed as 3 k in[20]. This singular knot has one coloring by S given by s s s s x i x i · s x i · s x i · s ( x i · s ) · s ( x i · s ) · s ( x i · s ) · s Figure 8.
Shadow coloring of 3 k . INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 11
Hom( SQ (3 k ) , S ) = { ( s → , s → , s → , s → } . For this coloring we also obtain one shadow coloring for each element of X . Therefore, we havethe following four shadow colorings by the above shadow singquandle.
00 00 11 111 00 00 20 002 00 00 33 333 00 00 02 220
Figure 9.
Shadow colorings of 3 k by singquandle shadow X .Therefore, ( S,X ) (3 k ) = 4 . The following corollary implies that the shadow counting invariant does not contain anyinformation not contained by the singquandle counting invariant. We obtain the followingresult by noticing that for each coloring of an oriented singular link by an oriented singquandlewe have a different shadow coloring for each element in the S -set. Corollary 5.14.
The shadow counting invariant of a singular link L by the S -shadow ( S, X )is given by ( S,X ) ( L ) = | X | S ( L ) , where S ( L ) is the singquandle counting invariant.We can define the following polynomial for a sinquandle shadow to obtain a singquandleshadow invariant. Definition 5.15.
The shadow singquandle polynomial , denoted by sp(
S, X ), of the shadowsingquandle (
S, X, ∗ , R , R , · ) is the sumsp( S, X ) = X x ∈ X t r ( x ) , where r ( x ) = |{ s ∈ S ; x · s = x }| . Furthermore, If ( S ′ , Y ) is a subshadow of ( S, X ), then the subshadow singquandle polynomial of ( S ′ , Y ) isSubsp(( S ′ , Y ) ⊂ ( S, X )) = X x ∈ Y t r ( x ) , where r ( x ) = |{ s ′ ∈ S ′ ; x · s ′ = x }| . Example 5.16.
Let S = Z = { , , , , , } with singquandle operations defined by x ∗ y =5 x − y = x ¯ ∗ y , R ( x, y ) = 2 x + y and R ( x, y ) = 5 x + 4 y . Consider the shadow structuredefined by X = Z = { , } with shadow matrix (cid:20) (cid:21) . Note that we can compute r ( x ) for each x ∈ X by going through the row of the shadow matrixand counting the occurrences of the row number. Therefore, r (1) = 6 and r (0) = 6, and theshadow singquandle polynomial of ( S, X ) issp(
S, X ) = 2 t . INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 12
We will consider two types of subshadows. We will first consider a subset of X closed underthe action of S . When we consider the subshadow ( S, Y ) ⊂ ( S, X ), where Y = { } , note thatwe can check from the shadow matrix that Y is closed under the action of S . The subshadow( S, Y ) has the following subshadow singquandle polynomialSubsp(
S, Y ) = t . Next, we consider the subshadow ( S ′ , Y ) ⊂ ( S, X ), where S ′ is the subsingquandle consist-ing of { , , } and Y = { } . A straightforward computation shows that S ′ is closed underthe singquandle operations. Furthermore, we can check from the shadow matrix that Y isclosed under the action of S ′ . The subshadow ( S ′ , Y ) has the following subshadow singquandlepolynomial, Subsp( S ′ , Y ) = t . We now prove that the shadow singquandle polynomial is an invariant of shadow singquan-dles.
Proposition 5.17.
Let (
S, X ) and ( S ′ , Y ) be two singquandle shadows. If ( S, X ) and ( S ′ , Y )are isomorphic, then they have equal shadow polynomials, sp( S, X ) = sp( S ′ , Y ). Proof.
Suppose the pair φ : X → Y and f : S → S ′ is a shadow singquandle isomorphism. Then r ( φ ( x )) = r ( x ) and the contribution to sp( S, X ) from x ∈ X is the same as the contribution of φ ( x ) ∈ Y to sp( S ′ , Y ). Since φ and f are bijective maps satisfying equations (3.6), (3.7), (3.8)and (5.3), the result follows. (cid:3) The shadow polynomial can be used to distinguish and classify shadow singqundles. In thefollowing example, we distinguish two shadow singquandles.
Example 5.18.
Let ( S, ∗ , R , R ) be a singquandle with S = Z , x ∗ y = 5 x − y = x ¯ ∗ y , R ( x, y ) = 3 x + 4 y , and R ( x, y ) = 4 x + 3 y . Let X = Z with shadow matrix . The (
S, X, ∗ , R , R , · ) is a singquandle shadow with shadow polynomialsp( S, X ) = 4 t . Let W = Z with shadow matrix . The (
S, W, ∗ , R , R , • ) is a singquandle shadow with shadow polynomialsp( S, W ) = 2 + 2 t . We see that the shadow singquandle polynomial is an effective invariant of singquandle shad-ows.In this section, we see that by simply computing the shadow counting invariant of an orientedsingular link we do not obtain any more information than that obtained from the singquandlecounting invariant. Therefore, in the following section we will enhance the shadow countinginvariant in order to obtain a stronger invariant.
INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 13 Enhanced Shadow Counting Invariant
In this section, we will jazz up the shadow counting invariant from the previous section. Wewill combine the S -shadow counting invariant and the shadow polynomial in order to define anenhanced shadow singquandle invariant for singular link. Definition 6.1.
Let f × φ be a shadow coloring of an oriented singular link diagram D . Theclosure of the set of shadow colors under the action of the image subsingquandle Im( f ) ⊂ S of f × φ is a subshadow called the shadow image of f × φ , which we denote by om( f × φ ) . Definition 6.2.
Let (
S, X ) be an S -shadow and let L be an oriented singular link with diagram D . The singquandle shadow polynomial invariant of L with respect ( S, X ) is SP ( L ) = X f × φ ∈ shadow coloring u Subsp(om( f × φ ) ⊂ ( S,X )) . Examples
In this section, we present two examples in which we show that the shadow sinquandlepolynomial is an enhancement of the singquandle counting invariant. In the first example, weinclude a pair of singular knots with the same singquandle counting invariant and the samesingquandle polynomial but are distinguished by the singquandle shadow polynomial invariant.The computations were performed by
Mathematica and python independently and checked byhand.
Example 7.1.
Let (
S, X, ∗ , R , R , · ) be the shadow singquandle with S = Z , X = Z andoperations x ∗ y = 3 x − y = x ¯ ∗ y , R ( x, y ) = 7 x + 6 y , R ( x, y ) = 2 x + 3 y , and shadow matrix . The following two K listed as 4 k and 5 k in [20]. We obtain thefollowing coloring equations from the singular knot 4 k , s = s ¯ ∗ s = − s + 3 s s = s ¯ ∗ s = 3 s − s s = R ( s , s ) = 7 s + 6 s s = R ( s , s ) = 2 s + 3 s s = s ¯ ∗ s = − s + 3 s s = s ¯ ∗ s = 3 s − s . The choice of om( f × φ ) to denote the shadow image of f × φ was derived from the french word ombre forshadow INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 14
From these equations we obtain the colorings listed below. In the following list we identify eachcoloring f ∈ Hom( SQ (4 k ) , S ) with the 6-tuple ( f ( s ) , f ( s ) , f ( s ) , f ( s ) , f ( s ) , f ( s )).Hom( SQ (4 k ) , S ) = { (1 , , , , , , (1 , , , , , , (2 , , , , , , (2 , , , , , , (3 , , , , , , (3 , , , , , , (4 , , , , , , (4 , , , , , , (5 , , , , , , (5 , , , , , , (6 , , , , , , (6 , , , , , , (7 , , , , , , (7 , , , , , , (0 , , , , , , (0 , , , , , } . We obtain the following coloring equations from the singular knot 5 k , s = s ¯ ∗ s = − s + 3 s s = s ∗ s = 3 s − s s = R ( s , s ) = 7 s + 6 s s = R ( s , s ) = 2 s + 3 s s = s ¯ ∗ s = 3 s − s s = s ∗ s = 2 s + 3 s s = s ¯ ∗ s = − s + 3 s . From these equations we obtain the colorings listed below. In the following list we identify eachcoloring f ∈ Hom( SQ (5 k ) , S ) with the 6-tuple ( f ( s ) , f ( s ) , f ( s ) , f ( s ) , f ( s ) , f ( s )).Hom( SQ (5 k ) , S ) = { (2 , , , , , , , (2 , , , , , , , (2 , , , , , , , (2 , , , , , , , (4 , , , , , , , (4 , , , , , , , (4 , , , , , , , (4 , , , , , , , (6 , , , , , , , (6 , , , , , , , (6 , , , , , , , (6 , , , , , , , (0 , , , , , , , (0 , , , , , , , (0 , , , , , , , (0 , , , , , , } . Therefore, both singular knots have the same singquandle counting invariant S (4 k ) =16 = S (5 k ). Therefore, by Theorem 5.14 we obtain that the two singular knots have thesame shadow counting invariant ( S,X ) (4 k ) = 96 = ( S,X ) (5 k ). Furthermore, the twosingular knots have the same singquandle polynomial φ Ssqp (4 k ) = 4 u s t s t s t + 4 u s t s t s t +8 u s t s t s t = φ Ssqp (5 k ). However, the singquandle shadow polynomial invariant distinguishesthe two singular knots: s s s s s s x SP (4 k ) = 24 u t + 24 u t + 48 u s s s s s s s x SP (5 k ) = 48 u t + 24 u t + 24 u t Figure 10.
Singular knots 4 k and 5 k and corresponding SP invariant. INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 15
Example 7.2.
Let (
S, X, ∗ , R , R , · ) with S = Z , X = Z , x ∗ y = 5 x − y = x ¯ ∗ y , R ( x, y ) =5 x + 10 y , R ( x, y ) = 2 x + y , and shadow matrix . We compute the shadow polynomial for the following singular knots derived from the classicaltrefoil knot s s s s xx · s x · s x · s ( x · s ) · s K s s s s s xx · s x · s x · s ( x · s ) · s K s s s s s s xx · s x · s x · s ( x · s ) · s K Figure 11.
Colorings of the singular knots derived from the trefoil. S ( S,X ) SP Singular knot4 32 4 u t + 4 u t + 24 u K , K u t + 8 u t + 8 u + 12 u K In this example we have a collection of singular knots all with the same singquandle countinginvariant with respect to X . If we then compute the shadow singquandle polynomial invariantwe can distinguish K from K and K .8. Acknowledgments
The authors of this paper would like to thank Sam Nelson for fruitful discussion. The authorswould also like to thank Hamza Elhamdadi for providing python code to verify the examplesin Section 7.
INGQUANDLE SHADOWS AND SINGULAR KNOT INVARIANTS 16
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