Jones polynomial and knot transitions in topological semimetals

TL;DR

This paper introduces the use of the Jones polynomial as a topological invariant to analyze and classify knot structures in topological semimetals, revealing how nodal line evolutions relate to changes in knot topology.

Contribution

It develops a novel framework applying the Jones polynomial to topological semimetals, linking local nodal line interactions to global knot topology changes.

Findings

Jones polynomial captures global knot topology of nodal lines

Nodal chain semimetals can evolve into Hopf-link configurations

Theory extends to non-Hermitian exceptional line semimetals

Abstract

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this paper, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf-link. We extend our theory to 3D non-Hermitian multi-band exceptional line semimetals.

Figures

Tables

Table 1. Table 1: The Jones polynomials represent all the generated phases L n 1 ​ n 2 ​ n 3 ​ n 4 subscript 𝐿 subscript 𝑛 1 subscript 𝑛 2 subscript 𝑛 3 subscript 𝑛 4 L_{n_{1}n_{2}n_{3}n_{4}} evolving from the nodal chain in Fig. 2 (a), where n i = 0 , ± 1 subscript 𝑛 𝑖 0 plus-or-minus 1 n_{i}=0,\pm 1 . As ∇ → 𝒌 ​ det [ h 0 ​ ( 𝒌 ) ] ≠ 0 subscript → ∇ 𝒌 delimited-[] subscript ℎ 0 𝒌 0 \vec{\nabla}_{\bm{k}}\det[h_{0}(\bm{k})]\neq 0 at each TP, the evolution is limited to the three possible generated phases marked by the red color.

Links or knots	Jones polynomial	$n={\sum }_{i=1}^4{n}_i$
Unknot	1	$\left|n\right|=1$
Unlink	$-t^{-1/2}-t^{1/2}$	$\left|n\right|=0$
Hopf-link	$-t^{5\mathrm{sign}\left(n\right)/2}-t^{\mathrm{sign}\left(n\right)/2}$	$\left|n\right|=2$
Trefoil knot	$-t^{4\mathrm{sign}\left(n\right)}+t^{3\mathrm{sign}\left(n\right)}+t^{\mathrm{sign}\left(n\right)}$	$\left|n\right|=3$
Solomon’s knot	$-t^{9\mathrm{sign}\left(n\right)/2}-t^{5\mathrm{sign}\left(n\right)/2}+t^{3\mathrm{sign}\left(n\right)/2}-t^{\mathrm{sign}\left(n\right)/2}$	$\left|n\right|=4$

Equations

$${\mathcal{H}}_0(\mathbf{k})=h_0(\mathbf{k}){\tau }_{+}+h_0^{†}(\mathbf{k}){\tau }_{-},$$

$$\det [h_0(\mathbf{k})]=\det [h_0^{†}(\mathbf{k}){]}^{\ast }={\gamma }_0^r(\mathbf{k})+i{\gamma }_0^i(\mathbf{k})=0,$$

$$\nu =\frac{i}{2\pi }{\oint }_{\Gamma ({\mathbf{K}}_0)}d\mathbf{k}\cdot {\nabla }_{\mathbf{k}}\left(\ln \det [h_0(\mathbf{k})]\right),$$

$${\mathcal{H}}_1(\mathbf{k},\lambda )=\lambda h_1(\mathbf{k},\lambda ){\tau }_{+}+\lambda h_1^{†}(\mathbf{k},\lambda ){\tau }_{-},$$

$$t^{-1}J\left(L_{+}\right)-tJ\left(L_{-}\right)+\left(t^{-1/2}-t^{1/2}\right)J\left(L_0\right)=0,$$

$$\mathbf{T}({\mathbf{k}}_0)={\vec{\nabla }}_{\mathbf{k}}{\gamma }_0^r({\mathbf{k}}_0)\times {\vec{\nabla }}_{\mathbf{k}}{\gamma }_0^i({\mathbf{k}}_0).$$

$$f(E,\mathbf{k})=\det [E-{\mathcal{H}}_{nH}(\mathbf{k})]={\Pi }_{i=1}^N[E-E_i(\mathbf{k})],$$

$${\Delta }_f(\mathbf{k})=\prod _{i<j}[E_i(\mathbf{k})-E_j(\mathbf{k}){]}^2=0,$$

Full text

*Introduction—*Topological phases of matter have been attracting extensive attention in the field of condensed matter physics Hasan and Kane (2010); Qi and Zhang (2011); Bansil et al. (2016); Chiu et al. (2016); Armitage et al. (2018); Fang et al. (2016). Although the topological invariants of gapped phases are defined globally, they can be locally analyzed by studying the low energy theories of some gapless points in the Brillouin zone (BZ) from a critical phase Bernevig and Hughes (2013); Hughes (2009). For example, the Chern number can be calculated by analyzing the mass terms around all the Dirac points Bernevig and Hughes (2013); Bernevig et al. (2006). In this sense, all the gapped phases can be generated from those critical gapless phases by adding different types of perturbations Chiu (2014); Chiu et al. (2016).

The topological nodal line semimetals protected by chiral symmetry or space-time inversion symmetry can host stable one-dimensional (1D) degeneracy line in the 3D BZ Fang et al. (2016). These nodal lines can form loops Hořava (2005); Burkov et al. (2011); Chen et al. (2015); Kim et al. (2015); Mullen et al. (2015); Yu et al. (2015); Fang et al. (2015); Bian et al. (2016); Chan et al. (2016); Schoop et al. (2016), chains Bzdušek et al. (2016); Yu et al. (2017); Yan et al. (2018); Feng et al. (2018); Gong et al. (2018); Liu et al. (2018); Shao et al. (2018); Lou et al. (2018); Yi et al. (2018); Fu et al. (2018); Singh et al. (2018); Wu et al. (2018); Kim et al. (2019); Chen et al. (2019); Lian et al. (2019), links Chen et al. (2017); Yan et al. (2017); Chang and Yee (2017); Zhou et al. (2018); Wang et al. (2018); Chang et al. (2017); Sun et al. (2017); Li et al. (2018); Ezawa (2017) or knots Ezawa (2017); Bi et al. (2017). Their topological properties are not only captured by the local charge Fang et al. (2016) but also described by the global knot invariant Sun et al. (2017); Lian et al. (2017); Bzdušek and Sigrist (2017); Tiwari and Bzdušek (2019); Yang and Hu (2019). Two nodal knot semimetals (hereafter, knot refers both link and knot) belong to the same (topological equivalence) classes, if their nodal lines can be deformed to each other by non-broken bending and stretching without crossing each other Kauffman (1991). Being analogue to a Dirac point as a topological phase transition, a touching point (TP), where two nodal lines touch together, might be a knot transition between two distinct knot classes. If we start from this critical phase, by adding different types of symmetry allowed perturbations, as the TPs are removed, different trivial and nontrivial nodal knot semimetals can be generated, which are dubbed as generated phases. A question naturally arises whether the knot topology of the nodal lines can be characterized by analyzing the local evolutions around TPs. The answer to this question provides a guide to analyze the possible generated phases emerging from nodal chain semimetals Bzdušek et al. (2016); Yu et al. (2017); Yan et al. (2018); Feng et al. (2018); Gong et al. (2018); Liu et al. (2018); Shao et al. (2018); Lou et al. (2018); Yi et al. (2018); Fu et al. (2018); Singh et al. (2018); Wu et al. (2018); Kim et al. (2019); Chen et al. (2019); Lian et al. (2019), which are symmetry protected critical phases with multiple TPs.

In this paper, we first show that the Jones polynomial Kauffman (1991); Hepworth ; SM (1) can faithfully characterize the orientated nodal knot semimetals protected by chiral symmetry. Similar to the transition of Chern insulator, we also show that the transition of Jones polynomial can be analyzed by studying the local evolutions around all the TPs in a critical phase as shown in Fig. 1. Furthermore, the low energy theory of the TP provides an additional constraint to the local evolutions and excludes the emergence of some generated phases. To demonstrate this theory, we use a nodal chain semimetal with four TPs as an example to show all of the possible generated phases, such as the emergence of a Hopf-link with the non-zero linking number Yan et al. (2017). In the end, we extend the recipe of the knot topology analysis to non-Hermitian exceptional line semimetals Xu et al. (2017); Cerjan et al. (2018); Carlström and Bergholtz (2018); Yang and Hu (2019); Carlström et al. (2019); Moors et al. (2019); Carlström et al. (2019); Wang et al. (2019); Kawabata et al. (2019), since Hermitian chiral symmetric systems and non-Hermitian systems share the identical mathematical structures.

*Nodal line semimetals protected by chiral symmetry—*We start with a general $2N-$bands Bloch Hamiltonian preserving chiral symmetry

[TABLE]

where $\tau_{\pm}=(\tau_{x}\pm i\tau_{y})/2$, $h_{0}(\bm{k})$ is an $N\times N$ matrix, and chiral symmetry operator $\mathcal{S}=\tau_{z}$. Due to chiral symmetry, the Hamiltonian obeys $S\mathcal{H}_{0}(\bm{k})S^{-1}=-\mathcal{H}_{0}(\bm{k})$. Since $\det[\mathcal{H}_{0}(\bm{k})]$ is the product of all the energies, the locations of the nodal lines at $E=0$ are determined by

[TABLE]

and chiral symmetry leads to at least 2-fold degeneracy in the nodal lines. In other words, the two constraints of $\gamma_{0}^{r}(\bm{k})=0$ and $\gamma_{0}^{i}(\bm{k})=0$ determine two surfaces in the 3D BZ respectively so that their crossings form the nodal lines.

A topological invariant characterizing each individual nodal line is given by the winding number SM (1)

[TABLE]

where $\bm{K}_{0}$ is a point located at the nodal lines and $\Gamma(\bm{K}_{0})$ is a closed-loop enclosing the nodal line and centered at $\bm{K}_{0}$. If the winding number is non-zero, the integral path is not contractible so that these nodal lines are topologically protected and can not be gapped in the presence of any weak chiral-symmetric perturbations. Since the winding number can change its sign under the reverse of the integral path, the direction of the nodal line is given by the normal vector of the oriented integral path corresponding to the positive winding number. By marking an arrow pointing this normal direction along the nodal line aligned with the counterclockwise integral path, one can assign an orientation to these nodal lines.

Critical phase and generated phases— Now we show that all the generated phases from a critical phase with perturbations can be determined by local evolutions around every TP. Consider that the line node system is in a critical phase with $m$-TPs, which can be labeled by $L_{T_{1}...T_{m}}$, as shown in Fig. 2 (a) with $m=4$. Then we can ask the following question: by adding a general form of perturbation respecting chiral symmetry

[TABLE]

where $\lambda$ is an external parameter, what type of a nodal knot as a generated phase can be generated by the perturbation? Since the perturbation is weak, only the local evolutions around the TPs finally determine the linking or knotting properties of the nodal line. To systematically study the generated phases, we project the critical phase in the 3D BZ (Fig. 2(a)) into a 2D plane and deform the projection to the diagram in Fig. 2(b) based on the periodic boundary condition of the 3D BZ. According to the directions of the nodal lines near the TPs, there exist two different types of TPs and the corresponding local evolutions $L_{0/+/-}$, namely type I/II TP and type I/II local evolutions as shown in Fig. 1(b1)/(b2) SM (1). In this regard, the generated phases evolving from the critical phase (multiple TPs) can be labeled by $L_{n_{1}...n_{m}}$, where $n_{i}=0,\pm$ represent the local evolutions near the i-th TP $T_{i}$; Fig. 2(c)-(e) show several possible generated phases. We note that $L_{n_{1}...n_{m}}$ are the 2D projection representation of the 3D knots, which is known as knot diagram SM (1). Although different projection planes lead to distinct knot diagrams of the same knot, the invariant, which will be given later, is independent of the choice of the projection plane Kauffman (1991).

Jones polynomial— Having obtained all the perturbation generated phases, we define the corresponding knot invariant to classify them. In knot theory, the topology of inequivalent knots can be distinguished by distinct knot polynomials Kauffman (1991). We specifically use the Jone polynomial $J(L_{\#})$ to characterize knots $L_{\#}$ in the nodal line semimetals. The reason to choose this polynomial is that the Jones polynomial can distinguish the orientations of the knots Jones (1985) from the directions of the winding numbers as well as reveals that the knot topology connects potential physical observables by using Chern-Simons theory Witten (1989); Lian et al. (2017); Fröhlich and King (1989). Mathematically, the Jones polynomials Kauffman (1991) is a Laurent polynomial $\mathbb{Z}[t^{1/2},t^{-1/2}]$, which satisfies (i) the so-called skein relation

[TABLE]

where $L_{+}$, $L_{-}$ and $L_{0}$ are three oriented knots that are identical except in the small red region as shown in Fig. 1(b1); (ii) initial condition $J(O)\equiv 1$, where $O$ represents an unknot (ring). Any two equivalent orientated knots have the same Jones polynomial. Based on the definition, the skein relation only relates the Jones polynomials of type I local evolution around type I TP. After extending the skein relation from type I to type II (Fig. 2 (b2)) SM (1), we can connect all the generated phases by using Eq. 5. Hence, $J(L_{n_{1}...n_{m}})$ can be calculated systematically via the skein relation and the initial condition $J(O)=1$.

To demonstrate the approach of obtaining the explicit form of the Jones polynomial, we consider the evolution of the nodal chains with 4 TPs in the semimetals as shown in Fig. 2 (a). This nodal chain semimetal is used as an example through this manuscript. Due to the orientations of the nodal lines, each local evolution near the TP can transit to three configurations $L_{+},L_{-}$ and $L_{0}$ of type I in Fig. 1(b1). First, we start with two unknots $L_{+-0+}$ and $L_{+-0-}$ with $J(L_{+-0+})=J(L_{+-0-})=1$. The skein relation (5) at TP $T_{4}$ connects the two unknots and the unlink (two separated loops) $L_{+-00}$ as shown in Fig. 2(c); therefore, $J(L_{+-00})=-t^{-1/2}-t^{1/2}$. Secondly, knowing the Jones polynomials of the unlink and the unknot, we have $J(L_{+00-})=-t^{-1/2}-t^{1/2}$ and $J(L_{+000})=1$ and then obtain the polynomial of the Hopf-link $J(L_{+00+})=-t^{5/2}-t^{1/2}$ by the skein relation at $T_{4}$ as illustrated in Fig. 2(d). Thirdly, the skein relation at $T_{2}$ also connects an unknot $L_{+-0+}$, a Hopf-link $L_{+00+}$, and a trefoil knot $L_{++0+}$ as shown in Fig. 2(e); hence, trefoil knot invariant is given by $J(L_{++0+})=-t^{4}+t^{3}+t$. By following these rules, we can have the Jone polynomials for the $3^{4}$ configurations of $L_{n_{1}n_{2}n_{3}n_{4}}$ listed in Table 1. In this model, the topology of the generated phases can be simply determined by the summation of the local diagram around every TP, which is similar to the transition of Chern number as shown in Fig. 1 (a). By using this example, it is not difficult to extend our analysis to any generic critical phase with several TPs. Only the local evolution around the TP plays an essential role in determining the topology of generated phases.

Physical constraint— We show that not only the orientations of the lines but also the energy dispersions limit the possibilities of the local evolutions near the TPs. To show the limitation from the dispersions, we first study the conditions for the emergence of TPs in the Hamiltonian (1). Mathematically, these TPs are considered as singularity points of the nodal lines SM (1), which are defined by the vanishing of the tangent vector along the nodal lines at the points. Since the nodal line is located at the intersection of two surfaces $\gamma_{0}^{r}(\bm{k})=0$ and $\gamma_{0}^{i}(\bm{k})=0$, for a point $\bm{k}_{0}$ on the nodal line, the tangent vector $\bm{T}(\bm{k}_{0})$ is perpendicular to the two normal directions $\vec{\nabla}_{\bm{k}}\gamma_{0}^{r}(\bm{k}_{0})$ and $\vec{\nabla}_{\bm{k}}\gamma_{0}^{i}(\bm{k}_{0})$, where $\vec{\nabla}_{\bm{k}}=(\partial_{k_{x}},\partial_{k_{y}},\partial_{k_{z}})$. In this regard, the tangent vector at point $\bm{k}_{0}$ along the nodal line is given by

[TABLE]

Since the TP in the nodal line belongs to singularity point, the momentum $\bm{k}_{\rm{TP}}$ at the TP obeys $\gamma_{0}^{r}(\bm{k}_{\rm{TP}})=\gamma_{0}^{i}(\bm{k}_{\rm{TP}})=\bm{T}(\bm{k}_{\rm{TP}})=0$. To have $\bm{T}(\bm{k}_{\rm{TP}})=0$, the TP evolution is classified as the three cases: (i) the two gradients are parallel ($\vec{\nabla}_{\bm{k}}\gamma_{0}^{r}(\bm{k}_{\rm{TP}})=c\vec{\nabla}_{\bm{k}}\gamma_{0}^{i}(\bm{k}_{\rm{TP}})\neq 0$), (ii) one of the gradients vanishes, and (iii) both vanish as shown in Fig. 3.

In particular, for case (i) and (ii), at least one of the two surfaces must have nonzero gradients at $\bm{k}_{\rm{TP}}$, which satisfies $\vec{\nabla}_{\bm{k}}\det[h_{0}(\bm{k}_{\rm{TP}})]\neq 0$ leading to linear dispersions. Since the types of local evolution depends on the choice of projective plane, we fix a special projective plane spanned by the two tangential vectors at the TP of the two nodal lines, which is dubbed as natural projective plane. For case (i) and (ii), the natural projective plane is perpendicular to the normal vector $\vec{\nabla}_{\bm{k}}\gamma_{0}^{r/i}(\bm{k}_{\rm{TP}})$. It can be shown that for the first two cases the TP must be type II in the natural projective plane SM (1), and the local evolution near the TP is limited to the two possibilities shown in the first two rows of Fig. 3. The reason is that the surface with nonzero gradient at the TP (yellow surface) can always be mapped to the natural projective plane (light blue plane) as shown in Fig. 4(a). This forbids the emergence of $L_{-}$ in Fig. 1 (b2). Here we emphasize that the limitation of the local evolution holds only in the natural projective planes. Choosing another projective plane, we have to transfer the constraint from the natural projective plane to the chosen plane. For example, Fig. 4(a) shows only $L_{+}$ and $L_{0}$ of type II are the only two possible local evolutions in the natural projective plane. By changing a different point of view and applying this evolution constraint, Fig. 4(b) in the new projective plane shows that the two possible local evolutions becomes $L_{-}$ and $L_{0}$ of type I in Fig. 1 (b1).

In contrast to the former two cases, in case (iii) neither $\gamma_{0}^{r}(\bm{k})$ nor $\gamma_{0}^{i}(\bm{k})$ possesses linear terms near $\bm{k}_{\rm{TP}}$ as shown in the third row of Fig. 3. By assuming quadratic terms of $\bm{k}$, in the natural projective plane, the line arrangement at the TP is constrained to type I, and there are three possible evolutions as shown in Fig. 3 SM (1).

Nodal chain semimetals— To demonstrate the tools we established for the study of the knot evolution in semimetals, we are back to the nodal chain semimetal with 4 TPs protected by chiral and two mirror symmetries Fang et al. (2016); Yan et al. (2018). The linear terms of the four TPs are nonzero in the direction perpendicular to the two mirror planes. We consider a specific nodal chain described by the chiral symmetric Hamiltonian (1) with $h_{0}(\bm{k})=2\cos 2k_{x}+\cos k_{x}+3\cos k_{y}-3\cos k_{z}-1/10-2i\sin k_{y}\sin k_{z}$ as shown in Fig. 2(a). Although there are two spatially separated nodal chains in this model SM (1), we focus on one in the two mirror planes ($k_{y}=0$ and $k_{z}=0$) and the chain is marked by $L_{n_{1}n_{2}n_{3}n_{4}}$ representing its topology is determined by the evolution of the four TPs $T_{i}$. The linear dispersion near the TP leads to the non-zero gradient $\vec{\nabla}_{\bm{k}}\det[h_{0}(\bm{k}_{\rm{TP}})]\neq 0$. First, consider the local evolutions at $T_{1},\ T_{3}$, which can have only $L_{+}$ and $L_{0}$ of type II in the natural projective planes (blue planes) as shown in Fig. 4(a). In the other view angle for the knot diagrams in Fig. 2(b-d), $L_{-}$ and $L_{0}$ of type I are only two possible evolutions as shown in Fig. 4(b). Similarly, the TPs $T_{2},T_{4}$ in the projective planes can evolve only to $L_{+}$ and $L_{0}$ of type I in the knot diagram. As a result, the generated phases are constrained to be $L_{n_{1}m_{1}n_{2}m_{2}}$, where $n_{i}=0,1$ and $m_{i}=0,-1$. Due to this constraint from the linear dispersion, globally the chain can evolve to an unknot ($|n|=1$), an unlink ($|n|=0$), or a Hopf-link ($|n|=2$) listed in Table 1 with red color. As shown in Fig. 4 (c), we have a nodal Hopf-link under the perturbation Eq. 4 with $\lambda h_{1}(\bm{k},\lambda)=i\lambda\sin 2k_{x}$. Physically, we have two ways to control perturbations. One is to add the pressure of a material that breaks the mirror symmetry. The other way is that in the photonic lattice, the lattice can be designed artificially. Hence the mirror symmetry breaking term can be added in a controlled way Lu et al. (2015). Finally, using the same recipe, we can show that the nodal chain semimetal ($L_{n_{1}m_{1}}$) with 2 TPs cannot evolve to a Hopf-link ($L_{\pm\pm}$) when $\vec{\nabla}_{\bm{k}}\det[h_{0}(\bm{k}_{\rm{TP}})]\neq 0$ at each TP SM (1).

Non-Hermitian exceptional line semimetals— This recipe studying the knot topology can even be extended to the non-Hermitian system Bender (2007); Feng et al. (2017); El-Ganainy et al. (2018); Özdemir et al. (2019); Miri and Alù (2019); Shen et al. (2018); Yao and Wang (2018); Yao et al. (2018); Yokomizo and Murakami (2019); Zhang et al. , namely, the 3D non-Hermitian exceptional line semimetals  Xu et al. (2017); Cerjan et al. (2018); Carlström and Bergholtz (2018); Yang and Hu (2019); Carlström et al. (2019); Moors et al. (2019); Carlström et al. (2019); Wang et al. (2019); Kawabata et al. (2019). While in Hermitian systems the nodal lines require symmetry protection Fang et al. (2016), the non-Hermitian exceptional lines are robust against any small perturbation even in the absence of any symmetries Shen et al. (2018); Yang and Hu (2019). We here focus on a general $N$-band non-Hermitian tight-binding Hamiltonian $\mathcal{H}_{nH}(\bm{k})$ with periodic boundary condition or with no skin modes Yao and Wang (2018); Yao et al. (2018); Yokomizo and Murakami (2019); Zhang et al. . According to the characteristic polynomial of the Hamiltonian

[TABLE]

the condition for the emergence of band degeneracy $E_{i}(\bm{k})=E_{j}(\bm{k})$ requires $f(E,\bm{k})=\partial_{E}f(E,\bm{k})=0$, which is equivalent to SM (1)

[TABLE]

where $\Delta_{f}(\bm{k})$ is the discriminant of the characteristic polynomial $f(E,\bm{k})$ as a function of $E$ SM (1); I.M. Gelfand (1994). For example, if $f(E)=aE^{2}+bE+c$, $\Delta_{f}=b^{2}-4ac$. Hence, the solution of Eq. 8 must be a set of 1D degeneracy lines in the 3D BZ. Using the Sylvester matrix of the characteristic polynomial to build the discriminant SM (1); Woody ; JANSON (2010), we can show the discriminant is a single-valued function of $\bm{k}$. Therefore, the topological charge can be defined by the quantized winding number in Eq. 3 with $h_{0}(\bm{k})=\Delta_{f}(\bm{k})$, and the non-zero winding number protects the degeneracy line and determines the knot orientation. In particular, in presence of arbitratry perturbations, the degeneracies are called stable exceptional lines Kawabata et al. (2019); Yang et al. (2019), where the non-Hermitian Hamiltonian is not diagonalizable Miri and Alù (2019). Since the mathematical structures of the Hermitian chiral symmetric systems and non-Hermitian ones are identical, we follow the same recipe above to characterize the evolution of the degeneracy lines near TPs in the non-Hermitian systems with the identical constraint of the local evolution at each TP.

In summary, topologically-protected lines emerge in several distinct condensed matter systems, such as Hermitian chiral-symmetric semimetals and non-Hermitian systems. We start with nodal lines with several TPs; the Jones polynomial characterizes the knot topology of the lines with orientation in 3D BZ, and the topology essentially is only determined by the local evolution near each TP. The low energy theory limits the line orientation at any TP; furthermore, if $\vec{\nabla}_{\bm{k}}\det[h_{0}(\bm{k}_{\rm{TP}})]$ or $\vec{\nabla}_{\bm{k}}\Delta_{f}(\bm{k}_{\rm{TP}})$ does not vanish at the TP, the corresponding local evolution is limited to two possible ones. Using the nodal chain semimetal with 4 TPs as an example, we can show how can we calculate the Jones polynomial for the generated phases. Our methodology provides general rules to the evolution of the topologically-protected lines with TPs and paves the way toward searching for exotic topological knot-node semimetals.

Acknowledgements— The work is supported by the Ministry of Science and Technology of China 973 program (No. 2015CB921300, No. 2017YFA0303100), National Science Foundation of China (Grant No. NSFC-11888101, 1190020, 11534014, 11334012), and the Strategic Priority Research Program of CAS (Grant No.XDB07000000). C.-K.C. is supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant XDB28000000).