Lipschitz geometry of surface germs in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document}: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document} is a topological knot (or link) in S3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^3$$\end{document}. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document} and knot theory. Namely, for any knot K, we construct a surface XK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_K$$\end{document} in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document} such that: the link at the origin of XK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{K}$$\end{document} is a trivial knot; the germs XK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_K$$\end{document} are outer bi-Lipschitz equivalent for all K; two germs XK\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{K}$$\end{document} and XK′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{K'}$$\end{document} are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in R4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^4$$\end{document}, even when they are topologically trivial and outer bi-Lipschitz equivalent.

In Section 3 we define (β 1 , β 2 )-bridges and the saddle move, closely related to the broken bridge construction in [3]. A one-bridge surface germ is a surface germ containing a single (β 1 , β 2 )-bridge and metrically conical outside it. The saddle move relates the metric problem of ambient semialgebraic bi-Lipschitz equivalence of two one-bridge surface germs in R 4 with the topological problem of isotopy of two knots in S 3 corresponding to the links at the origin of the surfaces obtained from these onebridge surface germs by the saddle moves (see Definition 3.3). That is why topological knot invariants, such as the Jones polynomial, yield metric knot invariants, which can be used to recognize ambient semialgebraic bi-Lipschitz non-equivalence of surface germs.
Although one-bridge surface germs are the simplest examples of not Lipschitz normally embedded surfaces, they have rather non-trivial ambient Lipschitz geometry. Another version of Universality Theorem (Theorem 3.13 below) states that, for any two knots K and L, one can construct a one-bridge surface germ X K L such that: 1. The link at the origin of X K L is isotopic to L; 2. For any knots K and L, all surface germs X K L are outer bi-Lipschitz equivalent; 3. Surface germs X K 1 L and X K 2 L are ambient semialgebraic bi-Lipschitz equivalent only if the knots K 1 and K 2 are isotopic.
In Section 4 we consider the Jones polynomial of the link at the origin L = L S(X ) of a surface germ S(X ) obtained from a one-bridge surface germ X by the saddle move (see Definition 3.3). Since the isotopy class of L is an ambient semialgebraic Lipschitz invariant, its Jones polynomial becomes an ambient Lipschitz invariant of X . If X = X K ,i is a "twisted" surface constructed in [3] (see also Theorem 3.5) and K is a trivial knot, then L is a torus link. Its Jones polynomial is computed completely (see Corollary 4.2 and Remark 4.3) and determines the number i of twists. This shows that Jones polynomial can be used to prove ambient bi-Lipschitz non-equivalence of metric knots.
If we do not suppose the surface germ to be a one-bridge surface germ, we obtain a stronger version of Universality Theorem (Theorem 3.14 below). It states that, for any two knots K and L, and any two rational numbers α > 1 and β > 1, one can construct a surface germ X αβ K L such that: 1. The link at the origin of X αβ K L is isotopic to L; 2. For a fixed knot K , the tangent link of X αβ K L (i.e., the intersection of the tangent cone with the unit sphere) is isotopic to K ; 3. All surface germs X αβ K L are outer bi-Lipschitz equivalent for fixed α and β.
All sets, functions and maps in this paper are assumed to be real semialgebraic. We use semialgebraic bi-Lipschitz equivalence, because we refer to the theorem of Valette [13]. Our results are also true for subanalytic bi-Lipschitz equivalence of subanalytic surface germs, and we expect them to remain true in any polynomially bounded ominimal structure over R. The Universality Theorem 3.1 was announced without a proof in the expository paper [4]. Definition 2.5 A semialgebraic germ (X , 0) ⊂ R n is called outer metrically conical if there exists a germ of a bi-Lipschitz homeomorphism H : (X , 0) → C(L X ), where C(L X ) is a straight cone over L X . The map H is called a conification map. A germ (X , 0) is called ambient metrically conical if there exists a germ of a bi-Lipschitz homeomorphism H : R n → R n , such that H (X , 0) = C(L X ). The map H is also called a conification map. A germ (X , 0) is called outer (ambient) semialgebraic metrically conical if a corresponding conification map can be chosen to be semialgebraic.

Remark 2.6
Notice that the definition makes sense for semialgebraic germs of any dimension, not only for surface germs.

Definition 2.7
An arc in a semialgebraic germ (X , 0) is a germ of a semialgebraic embedding γ : [0, ) → X such that γ (0) = 0. Unless otherwise specified, arcs are parameterized by the distance to the origin, i.e., γ (t) = t. We identify an arc with its image in X .

Definition 2.8
Let f ≡ 0 be (a germ at the origin of) a semialgebraic function defined on an arc γ . The order α of f on γ (notation α = or d γ f ) is the value α ∈ Q such that f (γ (t)) = ct α + o(t α ) as t → 0, where c = 0. If f ≡ 0 on γ , we set or d γ f = ∞. For any two arcs γ and γ in X one can define two orders of contact: inner and outer.

Definition 2.9
The outer order of contact tor d(γ , γ ) is defined as Here d p is a definable pancake metric (see [5]) equivalent to the inner metric. These two orders of contact are rational numbers such that 1 ≤ itord(γ , γ ) ≤ tord(γ , γ ).

Definition 2.10
Let β > 1 be a rational number. Consider the space R 3 with coordinates (x, y, z). For a fixed t ≥ 0, let Z t = {|x| ≤ t, |y| ≤ t} be a square in the x y-plane {z = t} and let Z = t≥0 Z t . Let W + t be the subset of Z t bounded by the line segment I + t = {|x| ≤ t, y = t} and the union J + t of the two line segments connecting the endpoints of Fig. 1a) and let Note that the tangent cone of W is the set {|x| ≤ |y| ≤ z} and the tangent cone of B β is the surface germ {|x| = |y| ≤ z}. Definition 2.11 Let 1 < β 1 ≤ β 2 be two rational numbers. For a fixed t ≥ 0, let Z t = {|x| ≤ t, |y| ≤ t, z = t} and Z = t≥0 Z t be as in Definition 2.10. In the x y-plane {z = t} consider the points (see Fig. 2) Let us connect the points p 1 (t), p 2 (t), p 3 (t), p 4 (t) by three line segments, and definē J + t as the union of these three segments. Let U + t ⊂ Z t be the convex hull ofJ + t . Let P +

Fig. 2
The set U t Fig. 2), and let Note that the set U has the same tangent cone at the origin as W , while the tangent cone at the origin of P is the positive z-axis. Note also that, for β 1 = β 2 = β, the (β, β)-bridge is outer bi-Lipschitz equivalent to the β-bridge.

Definition 2.12
Let X be a semialgebraic surface germ in R 4 with the link at the origin homeomorphic to a circle in S 3 . We say that X is a one-bridge surface germ if 1. There exists a semialgebraic bi-Lipschitz C 1 embedding : The union X ∪ (Z ) is Lipschitz normally embedded in R 4 and ambient semialgebraic metrically conical: there exist a semialgebraic bi-Lipschitz homeomorphism H :

Definition 2.13
Let α > 1 and β > 1 be rational numbers. Consider the space R 3 with coordinates (x, y, z). For a fixed t ≥ 0, let Z α t = {|x| ≤ t α , |y| ≤ t} be a rectangle in the x y-plane {z = t}. Let W α+ Fig. 3 Hornification of the cone over a knot R 3 (see [8] for details). Two diagrams are equivalent if they can be related by a finite sequence of Reidemeister moves.
The following result is a special case of the finiteness theorem of Hardt (see [6]).

Theorem 2.15
Let X be a semialgebraic surface germ. Then, for small t > 0 and for any plane R 2 , such that the projections of the links X ∩ S t are generic, the diagrams of the links X ∩ S t are equivalent.

Definition 2.16
Let F K ⊂ S 3 be a smooth semialgebraic embedded surface diffeomorphic to S 1 × [−1, 1], such that the two components K and K of the boundary ∂ F K of F K are isotopic to the same knot K and the linking number (see [9]) of the components K and K is zero. The surface F K is called a characteristic band of the knot K . Let Y K and X K be the cones over F K and ∂ F K , respectively. These cones are called characteristic cones of the knot K .
The standard β-hornification β : For an arc γ ⊂ R 4 , a conical neighborhood of γ is the image V 1 (γ ) of a semialgebraic bi-Lipschitz map : V 1 → R 4 such that γ is the image of the positive t-axis. Fig. 3). We may assume, by Valette's theorem, that β preserves the distance to the origin. For a subset S of V 1 (γ ), the set β (S) is called a β-hornification of S to γ .

Metric knots
Theorem 3.1 (Universality Theorem) Let K ⊂ S 3 be a knot. Then one can associate to K a semialgebraic one-bridge surface germ (X K , 0) in R 4 so that the following holds: 1. The link at the origin of each germ X K is a trivial knot; 2. All germs X K are outer bi-Lipschitz equivalent; 3. Two germs X K 1 and X K 2 are ambient semialgebraic bi-Lipschitz equivalent only if the knots K 1 and K 2 are isotopic.
Proof Let F K ⊂ S 3 be a characteristic band of the knot K , and let Y K and X K be the corresponding characteristic cones (see Definition 2.16). Let S K ⊂ F K be a slice (see Definition 2.17). Let ϕ K : as the corresponding mapping of the cones: Note that K is a bi-Lipschitz homeomorphism. Let W ⊂ R 3 be the set in Definition 2.10, and let Then X K is a one-bridge surface germ, part of the surface germ X K inside M K being replaced by a β-bridge B β (see Fig. 4). Let us show that X K satisfies the conditions of Theorem 3.1.
1) The link at the origin of X K is a trivial knot, because it bounds the closure of F K \ S K homeomorphic to a disk.
2) Let K 1 and K 2 be any two knots. Let : By the definition of the maps K 1 and K 2 (see (1) 3) Note that, for any knot K , the link of the tangent cone C 0 X K of the set X K is the union of two knots isotopic to K , with a single common point. Thus if K 1 and K 2 are not isotopic, then the tangent cones C 0 X K 1 and C 0 X K 2 are not ambient topologically equivalent. This contradicts Sampaio's theorem [12] (see also Theorem 1.1) which implies that tangent cones of ambient Lipschitz equivalent semialgebraic sets are ambient Lipschitz equivalent. In our case, the links of the tangent cones are not even ambient topologically equivalent.
This concludes the proof of Theorem 3.1.
Definition 3.2 A surface germ X K obtained by the above construction is called a bandbridge surface germ corresponding to the knot K and a β-bridge (or a (β 1 , β 2 )-bridge as in the proof of Theorem 3.5 below).

Definition 3.3
Consider the (β 1 , β 2 )-bridge B β 1 β 2 = t≥0J t (see Definition 2.11). The setJ t has two componentsJ + t andJ − t , each of them consisting of three line segments connecting the points p 1 (t), p 2 (t), p 3 (t), p 4 (t) and p 1 (t), p 2 (t), p 3 (t), p 4 (t), respectively (see Fig. 2). LetĴ t be the set obtained by replacing the line segments Fig. 5a). Let S β 1 β 2 = t≥0Ĵ t . Let X be a one-bridge surface germ (see Definition 2.12). Replacing B = (B β 1 β 2 ) ⊂ X with S = (S β 1 β 2 ), we obtain a new surface germ S(X ). This defines the saddle move operation applied to X . Lemma 3.4 Let X 1 and X 2 be semialgebraic ambient bi-Lipschitz equivalent onebridge surface germs. Then the surface germs S(X 1 ) and S(X 2 ), obtained by the saddle move applied to X 1 and X 2 , are ambient topologically equivalent, the links at the origin L S(X 1 ) and L S(X 2 ) are isotopic as topological links in S 3 , and the diagrams of the links L S(X 1 ) and L S(X 2 ) are equivalent.
Proof Let Z ⊂ R 3 be as in Definitions 2.10 and 2.11. Let 1 : Z → R 4 and 2 : Z → R 4 be bi-Lipschitz embeddings such that B 1 = 1 (B β 1 β 2 ) ⊂ X 1 and B 2 = 2 (B β 1 β 2 ) ⊂ X 2 . Since X 1 and X 2 are one-bridge surfaces, we can suppose that X 1 ∪ 1 (Z ) and X 2 ∪ 2 (Z ) are straight cones over their links. Let H : R 4 → R 4 be a bi-Lipschitz homeomorphism isotopic to identity such that H (X 1 ) = X 2 . By Valette's Theorem [13] (see also Theorem 1.2) we may suppose that H preserves the distance to the origin, and that the maps 1 and 2 send each section Z t of Z to the sphere S t of radius t centered at the origin.
Let P 1 = 1 (P) and P 2 = 2 (P), where P = t≥0 P t ⊂ B β 1 β 2 (see Definition 2.11) and let P 1 (t) = 1 (P t ) = P 1 ∩ S t and P 2 (t) = 2 (P t ) = P 2 ∩ S t . Since the tangent cone C 0 P of P is the positive z-axis, the tangent cones C 0 P 1 and C 0 P 2 of P 1 and P 2 are rays in R 4 . For a small positive , let N t ⊂ S t be a ball of radius t centered at the point C 0 P 2 ∩ S t , and let N = t≥0 N t be a conical -neighbourhood of C 0 P 2 . Note that P 2 ⊂ N ∩ X 2 ⊂ B 2 for small > 0.
The saddle move operation applied to X 1 replaces The saddle move operation applied to X 2 replaces . Note that the boundary points q 2 (t),q 2 (t),q 3 (t),q 3 (t) of Q t are the same as the boundary points of H ( P 1 (t)), and the boundary points v 2 (t), v 2 (t), v 3 (t), v 3 (t) of V t are the same as the boundary points of P 2 (t). In particular, all these points belong to the bridge B 2 of X 2 , and to the t-ball N t (see Fig. 6). Note that tord( Q 2 , ) as functions of t. Note that the order of all of these functions at the origin is β 2 . Let N 2 be the family of Fig. 6 The images by the map H in the proof of Lemma 3.4 balls N 2,t on S t centered at v 2 (t) with the radius tβ forβ ∈ (β 1 , β 2 ), and let N 3 be the family of balls N 3,t on S t centered at v 3 (t) with the radius tβ . Clearly N 2,t ∩ N 3,t = ∅ and also are homeomorphic to segments, there exists a homeomorphismH 2 : N 1 → N 1 isotopic to identity, such that : 1.H 2 maps the sections z = t to the sections z = t.

2.H 2 is identity on the boundary of
Similarly, there exists a homeomorphismH 3 : N 2 → N 2 isotopic to identity, such that : 1.H 3 maps the sections z = t to the sections z = t.

2.H 3 is identity on the boundary of
Then we define a homeomorphism H : . This proves that S(X 1 ) and S(X 2 ) are ambient topologically equivalent, and the links at the origin L S(X 1 ) and L S(X 2 ) are isotopic as topological links.  Proof Consider a characteristic band F K ⊂ S 3 , a slice S K ⊂ F K , and characteristic cones Y K and X K (see Definitions 2.16 and 2.17). We construct a band-bridge surface germ with a (β 1 , β 2 )-bridge corresponding to K as follows. Let M K = {tσ : t ≥ 0, σ ∈ S K } ⊂ R 4 be the cone over S K (as in Theorem 3.1). Let K : M K → Z ⊂ R 3 be the map defined in (1): Note that K is a bi-Lipschitz homeomorphism. For 1 < β 1 ≤ β 2 , let U ⊂ R 3 be the set in Definition 2.11. We define The set Y K ,0 is obtained by replacing the set W (see Definition 2.10) with the set U in construction of the set X K in the proof of Theorem 3.1. Let X K ,0 = ∂Y K ,0 be its boundary (see Fig. 7 ). This construction replaces a β-bridge in Theorem 3.1 by a (β 1 , β 2 )-bridge. In particular, the one-bridge surface germ (X K ,0 , 0) satisfies conditions of Theorem 3.1. Let now F K ,i be the set obtained by removing the slice S K from F K , making i complete twists and adding S K back (see Fig. 8a-d). Let Y K ,i be the set obtained from the cone over F K ,i by replacing the set M K (the cone over S K ) with the set U (see show that the link of X K ,i is a trivial knot and the tangent cone of X K ,i is a cone over the union of two knots isotopic to K , pinched at one point. We are going to prove that X K ,i and X K , j are not semialgebraic ambient bi-Lipschitz equivalent if i = j. The result of the saddle move applied to each of these surface germs is a surface germ such that its tangent link is the union of two copies of the knot K , with the linking number of the two copies being twice the number of complete twists. Thus the links S(X K , j ) and S(X K ,i ) are not isotopic when i = j. It follows from Lemma 3.4 that surface germs X K ,i and X K , j are not ambient semialgebraic bi-Lipschitz equivalent when i = j.
Note that the topology of the tangent link of X K ,i does not depend on i. The tangent link is formed by two copies of K pinched at one point.

Remark 3.6
Let X K ,i be the surface germ constructed in the proof of Theorem 3.5. Then the link at the origin of the surface germ S(X K ,i ), obtained from X K ,i by the saddle move, is a subset of F K ,i isotopic to ∂ F K ,i . Proposition 3.7 Let X K ,i be a surface germ constructed in Theorem 3.5, and let S(X K ,i ) be the surface germ obtained by a saddle move applied to X K ,i . If K is a trivial knot, then the link at the origin of S(X K ,i ) is a torus link.
Proof For a small > 0, the boundary of the -neighbourhood of K is an unknotted two-dimensional torus T K ⊂ S 3 . One can define coordinates (φ, ψ) on T K , where φ ∈ K , ψ ∈ S 1 , so that the curvesK = {φ ∈ K , ψ = 0} andK = {φ ∈ K , ψ = π } have the linking number zero. Then F K = {φ ∈ K , 0 ≤ ψ ≤ π } ⊂ T K is a characteristic band of the knot K (see Definition 2.16) bounded by the curvesK and K . If X K ,0 is the surface germ constructed in Theorem 3.5, then the link at the origin of S(X K ,0 ), isotopic to the union ofK andK , is a trivial torus link.
The surgery for constructing a surface germ X K ,i in Theorem 3.5 (see Fig. 8) corresponds to the choice of a coordinate system (φ, ψ i ) on T K such that the band with the linking number 2i. Since the link at the origin of S(X K ,i ) is isotopic to the union ofK i andK i (see Remark 3.6) it is a torus link. Proposition 3.8 Let X 1 and X 2 be two one-bridge surface germs. If the germs are ambient bi-Lipschitz equivalent, then the links of the origin L S(X 1 ) and L S(X 2 ) are isotopic.

Remark 3.9
Saddle move on the level of knot diagrams is described as follows: is replaced by .
Here we are going to define the crossing move, that will be useful for further calculations.

Definition 3.10
We proceed in a similar way to the definition of the saddle move. Consider the subset B of a one-bridge surface X outer bi-Lipschitz equivalent to a (β 1 , β 2 )-bridge B β 1 β 2 = t≥0J t (see Definition 2.11). The setJ t has two componentsJ + t andJ − t , consisting of three line segments connecting the points p 1 (t), p 2 (t), p 3 (t), p 4 (t) and p 1 (t), p 2 (t), p 3 (t), p 4 (t), respectively, in the plane {z = t, w = 0} (see Fig. 2). Let us embed this set to R 4 with coordinates (x, y, z, w). Replacing the line segments [ p 2 (t), p 3 (t)] and [ p 2 (t), p 3 (t)] with the line segment [ p 2 (t), p 3 (t)] and a circle arc in the half-space {w ≥ 0} with the ends at p 2 (t) and p 3 (t), orthogonal to the plane {w = 0} (see Fig. 5b), we replace the setJ t with the setĴ t . Let B β 1 β 2 = t≥0Ĵ t . Note that the surface germs B β 1 β 2 and B β 1 β 2 have the same boundary arcs. Replacing the subset B of X with the subset B outer bi-Lipschitz equivalent to B β 1 β 2 , so that B and B have the same boundary arcs, we get a new surface germ C(X ). This defines a crossing move operation applied to X .

Remark 3.11
Crossing move on the level of knot diagrams is described as follows: is replaced by .

Remark 3.12
One can show that, for the fixed orientation on L X , the isotopy class of the resulting knot or link is an ambient bi-Lipschitz invariant. However, in what follows we do not need this result.

Fig. 9 Construction of X K L
The next statement is a modification of the Universality Theorem.

Theorem 3.13
For any two knots K and L, there exists a germ of a semialgebraic one-bridge surface germ X K L such that: 1. The link of X K L at the origin is isotopic to L. 2. For a fixed knot K all surface germs X K L have isotopic tangent links. In particular, surface germs X K 1 L and X K 2 L are ambient semialgebraic bi-Lipschitz equivalent only if the knots K 1 and K 2 are isotopic.
Proof We use the construction from the proof of Theorem 3.1. Consider a characteristic band F K , the characteristic cones Y K and X K (see Definition 2.16). Consider the surface germ X K defined in (2) for the knot K . Let γ ⊂ X K be an arc not tangent to the set V K defined in (2) (i.e., tord(γ , γ ) = 1 for any γ ⊂ V K ). Let V (γ ) be a small conical neighbourhood of γ in R 4 , such that V (γ ) ∩ X K is a Hölder triangle. Let us embed the straight cone Z L over L inside V (γ ) so that its imageZ L does not intersect X K , and its is ambient topologically equivalent to L. Let us choose two arcs γ 1 and γ 2 in X K ∩ V (γ ), and two arcs γ 1 and γ 2 inZ ∩ V (γ ), satisfying the following conditions: a. tord(γ 1 , γ 2 ) = tord(γ 1 , γ 2 ) = 1. b. Replacing the union of the Hölder triangles T (γ 1 , γ 2 ) ⊂ X K and T (γ 1 , γ 2 ) ⊂Z with the union of Hölder triangles T (γ 1 , γ 1 ) ⊂ V (γ ) and T (γ 2 , γ 2 ) ⊂ V (γ ), as shown in Fig. 9, we obtain a semialgebraic set X K L such that X K L ∩ V (γ ) is conical and its link is isotopic to the connected sum of K and L. Note that construction of X K L is similar to the saddle move construction in Definition 3.3.
Let us check that the surface germ X K L satisfies conditions of Theorem 3.13. 1. Since the link of X K is unknotted, the connected sum is isotopic to L. The proof of the fact that, for a fixed knot L, all surface germs X K L are outer bi-Lipschitz equivalent is the same as the proof that all surface germs X K are outer bi-Lipschitz equivalent in the proof of Theorem 3.1.
2. Since X K L is a one-bridge surface germ, its tangent link is the union of two knots with a single common point. One of these two knots is isotopic to K , and the other one is isotopic to the connected sum of K and L. Since the first knot is isotopic to K , condition 2 is satisfied.
The next result is another modification of the Universality Theorem. In contrast to the previous results, we consider surface germs with the metric structure more complicated than one-bridge. Theorem 3.14 For any two knots K and L, and for any two rational numbers α and β such that 1 ≤ α ≤ β, there exists a semialgebraic surface germ X αβ K L such that: 1. For any knots K and L, the link at the origin of X αβ K L is isotopic to L. 2. For any knots K and L, the tangent link of X αβ K L is isotopic to K . 3. For fixed α and β, all surface germs X αβ K L are outer bi-Lipschitz equivalent.
Proof Let F K ⊂ S 3 be the characteristic band of a knot K (see Definition 2.16). It is diffeomorphic to S 1 × [−1, 1], and its boundary has two components K and K isotopic to K . Let (ρ, l), where ρ ∈ S 1 and l ∈ [−1, 1], be coordinates in F K . Let Y K and X K be the corresponding characteristic cones (see Definition 2.16). Then (ρ, l, t) are coordinates in Y K , where t is the distance to the origin. Let Y α K be a subset of Y K defined as follows: Notice that the tangent link of Y α K is a knot isotopic to K . Let S K = {(ρ, l) : |ρ − ρ 0 | ≤ } be a slice of F K (see Definition 2.17) for a small > 0, and let M K be the cone over Let γ ⊂ X αβ K be an arc far from the set W αβ , i.e., tord(γ , γ ) = 1 for any arc γ ⊂ W αβ . Let Z L be the straight cone over L.
be a β-hornification of Z L to γ (see Definition 2.18 and Fig. 3). Let us choose two arcs γ 1 and γ 2 in X K ∩ V β (γ ), and two arcs γ 1 and γ 2 in Z β L,γ ∩ V β (γ ) satisfying the following conditions: a. tord(γ 1 , γ 2 ) = β, tord(γ 1 , γ 2 ) = β. b. If we remove from X K the Hölder triangle bounded by the arcs γ 1 and γ 2 , remove from Z β L,γ the Hölder triangle bounded by the arcs γ 1 and γ 2 , and add to the set X K ∪Z the Hölder triangle obtained as the union of line segments connecting γ 1 (t) and γ 1 (t), and the Hölder triangle obtained as the union of line segments connecting γ 1 (t) and γ 2 (t), we obtain a semialgebraic set X αβ K L with the link isotopic to the connected sum of the links of X K and Z β L,γ (see Fig. 9). Note that construction of X αβ K L is similar to construction of X K L in the proof of Theorem 3.13 and to the saddle move construction in Definition 3.3.
Let us check that the surface germ X αβ K L satisfies conditions of Theorem 3.14. 1. Since X αβ K has a trivial link, the connected sum is isotopic to L. 2. Since Z L is a subset of a β-horn neighbourhood of γ , it corresponds to a single point in the tangent link. Thus the tangent link of X αβ K L is the same as the tangent link of X αβ K , which is isotopic to K . 3. The proof of the fact that the surface germs X αβ K L are outer bi-Lipschitz equivalent for a fixed L is the same as the proof that all surfaces X K L are outer bi-Lipschitz equivalent in the proof of Theorem 3.13.

Knot invariants
In this section we make slight changes of notations. In the previous sections we used the notation L X for the link at the origin of a surface germ X . Here we are going to use the notation K X if the link at the origin of X is a knot, and L X if it is a topological link with more than one component.
Let  1 Let X be a one bridge surface such that the link of X at the origin is a knot K X . Let K C(X ) be the knot, obtained from K (X ) by the crossing move. Let Y be a one-bridge germ such that the link at the origin of Y is the same knot K Y = K X as the link at the origin of X . Let S(Y ) be a germ obtained from Y by the saddle move. Suppose that Y is such that the link at the origin of the surface S(Y ) is a 2-component link L S (Y ) . If the Jones polynomial J (K C(X ) ) of the knot K C(X ) satisfies J (K C(X ) ) = −t where D K is a diagram of a knot K determined by X , and D K is a diagram (determined by the crossing move) of a knot K C(X ) then X and Y are not semialgebraic ambient bi-Lipschitz equivalent.
Proof Let D K be a diagram of a knot K (X ) determined by X and let D K be a diagram of a knot K C(X ) . Let us orient D K in an arbitrary way. We orient D K so that the intersection, corresponding to the crossing move (see Fig. 10) on the diagram is positive, i.e., it looks like . Let S(X ) be a germ of a surface obtained from X by a saddle move. Let D L be the corresponding diagram of the characteristic link L S(X ) . We orient D L such that the part, corresponding to the saddle move (see Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.