Quandle coloring and cocycle invariants of composite knots and abelian extensions (Q2805389)

A quandle is a set \(X\) with an operation \(*\) satisfying (1) \(x*x=x\), (2) \(*y : X \ni x \mapsto x*y \in X\) is a bijection, and (3) \((x*y)*z=(x*z)*(y*z)\). A quandle is called faithful if \(*x \neq *y\) for all pairs \(x,y \in X\) with \(x \neq y\). \textit{L. Vendramin} obtained a list of connected quandles with less than 48 elements [J. Knot Theory Ramifications 21, No. 9, 1250088, 10 p. (2012; Zbl 1333.57026)].NEWLINENEWLINELet \(D\) be a digram of an oriented knot, and \(A\) the set of arcs of \(D\). A map \(c\) from \(A\) to \(X\) is called an \(X\)-coloring if \(x_{\tau}*y_{\tau}=z_{\tau}\) holds at every crossing \(\tau\), where \(y_{\tau}\) is the element of \(X\) assigned by \(c\) to the arc \(a\) going over \(\tau\), and \(x_{\tau}\;(z_{\tau})\) that to the arc in the right (left) hand of \(a\). A coloring is said to be trivial if it is a constant map. Let \(\text{Col}_X(K)\) denote the number of all colorings on any diagram \(D\) of an oriented knot \(K\) by \(X\). As it is well known, this gives a knot invariant. The fundamental quandle of a knot was defined in [\textit{D. Joyce}, J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] and [\textit{S. V. Matveev}, Math. USSR, Sb. 47, 73--83 (1984); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78--88 (1982; Zbl 0523.57006)], and it is known that the fundamental quandles of two knots \(K\) and \(K'\) are isomorphic if and only if \(K=K'\) or \(K=\operatorname{rm}(K')\), where \(\operatorname{rm}(K')\) denotes the reversed mirror image of \(K'\). A coloring can be regarded as a homomorphism from the fundamental quandle to \(X\). Fundamental quandles have infinitely many elements.NEWLINENEWLINEIn the present paper, it is shown in Proposition 3.1 that, for any non-trivial oriented knot \(K\), there is a finite quandle by which \(K\) is colored non-trivially. The concept of an end monochromatic tangle is introduced. A \(1\)-string tangle \(T\) is called end monochromatic with \(X\) if any \(X\)-coloring of \(T\) assigns the same color on the two end points. A knot \(K\) is called end monochromatic with \(X\) if \(K\) becomes an end monochromatic \(1\)-tangle when it is cut at some base point. Quandle colorings on \(K\) and its reversed mirror \(\operatorname{rm}(K)\) cannot distinguish these knots. However, \(K\) and \(\operatorname{rm}(K)\) can be distinguished, if, for some oriented knot \(R\) and some quandle \(X\), \(\text{Col}_X(R \sharp K) \neq \text{Col}_X(R \sharp \operatorname{rm}(K))\), where \(\sharp\) denotes the connected sum.NEWLINENEWLINEThe authors show that this method is very effective by calculating the above coloring numbers for knots with crossing number \(12\) or less and braid index less than \(4\) setting \(R\) as \(3_1, 5_1, 9_1\), and \(X\) as extensions of quandles in Vendramin's list (Example 5.13, Remarks 5.14-16). For this purpose, we need a non-faithful quandle \(X\). (For a finite homogeneous quandle \(X\) and two oriented knots \(K_1, K_2\), if \(K_1\) or \(K_2\) is end monochromatic with \(X\), the equality \(|X|\,\text{Col}_X(K_1 \sharp K_2)=\text{Col}_X(K_1)\,\text{Col}_X(K_2)\) holds (Lemma 4.6). If \(X\) is faithful, then any knot is end monochromatic with \(X\) (Lemma 4.4), and hence \(\text{Col}_X(R \sharp K)=\text{Col}_X(R \sharp \operatorname{rm}(K))\) for any knots \(R\) and \(K\) (Corollary 4.8). The reviewer noticed, and confirmed with the authors, that Lemma 4.7 and Corollary 4.8 need the premise that \(X\) is homogeneous as Lemma 4.6 does.)NEWLINENEWLINEA function \(\phi : X \times X \rightarrow A\) for an abelian group \(A\) is called a quandle \(2\)-cocycle if \(\phi(x,x) = 0\) and \(\phi(x,y)-\phi(x,z)+\phi(x*y,z)-\phi(x*z,y*z)=0\). Then \(E = X \times A\) becomes a quandle by \((x,a)*(y,b)=(x*y,a+\phi(x,y))\), and it is called an abelian extension of \(X\) by \(\phi\). Note that abelian extensions are not faithful. For an oriented knot \(K\), a \(2\)-cocyle invariant is an element of the group ring \({\mathbb Z}[A]\) defined by \(\Phi_{\phi}(K)=\Sigma_{\mathcal C} \Pi_{\tau} \phi(x_{\tau}, y_{\tau})^{\epsilon( \tau )}\) for any diagram of \(K\), where \({\mathcal C}\) is the set of all \(X\)-colorings, and \(\epsilon(\tau)\) is the sign of \(\tau\). If \(\Phi_{\phi} (K)\) is asymmetric, then \(K \neq \operatorname{rm}(K)\) (Corollary 5.10). If \(X\) is a faithful quandle, \(\Phi_{\phi} (K)\) has terms \(k_u u\), \(k_{u^{-1}}u^{-1}\) with \(k_u \neq k_{u^{-1}}\), and \(\Phi_{\phi} (R) = r_e e + r_u u\) with \(r_e = |X|\), then \(\text{Col}_E(R \sharp K) \neq \text{Col}_E(R \sharp \operatorname{rm}(K))\), where \(e\) is the identity element in \(A\), \(u\) is a non-identity element in \(A\), \(|X|\) is the cardinality of \(X\), and \(E\) is the extension of \(X\) by \(\phi\) (Corollary 5.12). Hence we can distinguish \(K\) and \(\operatorname{rm}(K)\) by the numbers of colorings of the composite knots in this case. The authors remarked that often this method has computational advantage over applying Corollary 5.10. Moreover, for a non-identity element \(v \in A\), the coefficient of \(v^{-1}\) is calculated by \((|X|\text{Col}_E(R_v \sharp K) - r_e \text{Col}_E (K))/(r_v |A|)\), if \(K\) is end monochromatic with \(X\), and there is a knot \(R_v\) with \(\Phi_{\phi} (R_v) = r_e e + r_v v\) (Proposition 6.1). Non-faithful quandles and extensions of quandles are investigated in sections 7 and 8.