Knot invariants coming from pre-image indices (Q2236457)

The authors define an algebraic clasp number \(acp(K,\varphi)\) for any oriented knot by using relative homomorphism indices. The homomorphism index is the key in the definition of the algebraic clasp number \(acp(K,\varphi)\). More results about (relative) homomorphism index and pre-image class can be found in the Section 2 of this paper and [\textit{Y. Gu} and \textit{X. Zhao}, Bull. Belg. Math. Soc. - Simon Stevin 24, No. 4, 725--739 (2017; Zbl 1391.55001)]. This algebraic clasp number is related to the clasp number \(cp(K)\): Corollary 4.3: \(\min\{|acp(K,\varphi)|:\varphi \text{ is a clasp disc of } K\}\leq cp(K)\). Actually, this algebraic clasp number is weaker in distinguishing knots than the clasp number \(cp(K)\). But in [\textit{H. Matsuda}, Osaka J. Math. 40, No. 4, 801--833 (2003; Zbl 1061.57011)], the clasp number \(cp(K)\) was shown to be non-additive. The good thing about the algebraic clasp number \(acp(K,\varphi)\) defined in this paper is that it is more computable, and it is additive. Theorem 5.1: \(acp(K_1\sharp K_2, \varphi_1 \natural \varphi_2)=acp(K_1,\varphi_1)+acp(K_2,\varphi_2)\). The definition of algebraic clasp number is as follows: Definition 4.1: Let \(K\) be an oriented knot in \(S^3\), and \(\varphi:(D^2,S^1)\rightarrow (S^3,K)\) be a clasp disc of \(K\). The algebraic clasp number \(acp(K,\varphi)\) is an integer defined by the equality: \[ \mathcal{L}_3[D^2\times S^1]=(-2\cdot acp(K,\varphi)\cdot j_*^V([(S^3)^2, (S^3)^2-\Delta_{(S^3)^2}]),\cdot), \] where \begin{itemize} \item[(a)] \([D^2\times S^1]\) is the generator of \(H_3((D^2)^2, (S^1)^2)\cong \mathbb{Z}\), \item[(b)] \(\mathcal{L}_3=\mathcal{L}_3(\varphi\times \varphi, (\varphi\times \varphi)|_{(S^1)^2};(\varphi\times \varphi)^{-1}(\Delta_{(S^3)^2})-\Delta_{(D^2)^2},\Delta_{(S^3)^2})\) is the relative homology homomorphism index in dimension \(3\) of the map \(\varphi\times \varphi:((D^2)^2,(S^1)^2)\rightarrow ((S^3)^2, K^2)\) for the pre-image subset \((\varphi\times \varphi)^{-1}(\Delta_{(S^3)^2})-\Delta_{(D^2)^2}\) at the diagonal \(\Delta_{(S^3)^2}\subset (S^3)^2\), \[ H_3((D^2)^2,(S^1)^2)\xrightarrow{\mathcal{L}_3}H_3((S^3)^2, (S^3)^2-\Delta_{(S^3)^2}\cap ((S^3)^2-K^2))\xleftarrow{j_*^V} H_3((S^3)^2,(S^3)^2-\Delta) \] \item[(c)] \(j_*^V(H_3((S^3)^2,(S^3)^2-\Delta))\cong \mathbb{Z}\) is considered as a summand of \[ H_3((S^3)^2, (S^3)^2-\Delta_{(S^3)^2}\cap ((S^3)^2-K^2))\cong \mathbb{Z}^2. \] \end{itemize} And it is worth mentioning that the algebraic clasp number \(acp(K,\varphi)\) is related to the local intersection number of \(\varphi|_{D^2}\) and \(\varphi |_{S^1}\). Theorem 4.2: Let \(\varphi:(D^2,S^1)\rightarrow (S^3,K)\) be a clasp disc of an oriented knot \(K\). Then the algebraic clasp number \(acp(K,\varphi)\) of \(K\) is \[ \frac{1}{2}\sum_{x\in \varphi(\operatorname{int} D^2)\cap \varphi(S^1)}I_x \] where \(I_x\) is the local intersection number of \(\varphi|_{D^2}\) and \(\varphi |_{S^1}=\partial D^2\) at \(x\).