VVERTEX DISTORTION OF LATTICE KNOTS
MARION CAMPISI AND NICHOLAS CAZET
Abstract.
The vertex distortion of a lattice knot is the supre-mum of the ratio of the distance between a pair of vertices along theknot and their distance in the (cid:96) -norm. We show analogous resultsto those of Gromov, Pardon and Blair-Campisi-Taylor-Tomova aboutthe distortion of smooth knots hold for vertex distortion: the ver-tex distortion of a lattice knot is 1 only if it is the unknot, andthat there are minimal lattice-stick number knot conformationswith arbitrarily high distortion. Introduction A polygonal knot is a knot that consists of line segments called sticks .A lattice knot is a polygonal knot in the cubic lattice L = ( R × Z × Z ) ∪ ( Z × R × Z ) ∪ ( Z × Z × R ) . All knots and curves in this paper aretaken to be tame.The vertex set of a lattice knot K , denoted V ( K ), is the set of points p ∈ K ∩ ( Z × Z × Z ). The vertex set of a tame lattice knot is finite.For smoothly embedded knots one can assign a value called the dis-tortion : δ ( K ) = sup p,q ∈ K d K ( p, q ) d ( p, q ) . Here, we define the vertex distortion of a lattice knot K in R as δ V ( K ) = sup p,q ∈ V ( K ) d K ( p, q ) d ( p, q ) = max p,q ∈ V ( K ) d K ( p, q ) d ( p, q )where d K ( p, q ) denotes the shorter of the two injective paths from p to q along K and d ( p, q ) = (cid:80) i =1 | p i − q i | , the (cid:96) − metric. Unlikethe distortion of general rectifiable curves, as introduced by Gromov[4], the supremum may be replaced with the maximum in our vertexdistortion of lattice knots: the set V ( K ) is finite. We can turn this intoa knot invariant by defining for each lattice knot type [ K ] δ V ([ K ]) = inf K ∈ [ K ] δ V ( K )where the infimum is taken over all lattice conformations K represent-ing the knot type [ K ]. a r X i v : . [ m a t h . G T ] S e p ERTEX DISTORTION OF LATTICE KNOTS 2
The distortion of smooth knots has proved to be a challenging quan-tity to analyze. Gromov showed that δ ( K ) ≥ π with equality if andonly if K is the standard round circle [4]. Moreover, Denne and Sullivanshowed that δ ( K ) ≥ π whenever K is not the unknot [3].In 1983, Gromov [4] asked if there is a universal upper bound on δ ( K ) for all knots K . Pardon [5] answered this question negativelywhen he showed that the distortion of a knot type is bounded below bya quantity proportional to a certain topological invariant, called repre-sentativity. Blair, Campisi, Taylor, and Tomova showed that distortionis bounded below by bridge number and bridge distance and exhibitan infinite family of knots for which their bound is arbitrarily strongerthan Pardon’s.We show that analogous to [4], Theorem 1.1.
For any conformation K of a knot in the cubic lattice, δ V ( K ) = 1 only if K is the unknot. and like [5] and [2] Theorem 1.2.
There exists a sequence of reduced, minimal lattice-sticknumber knots, T p,p +1 , with δ V ( T p,p +1 ) → ∞ as p → ∞ . This paper is structured as follows: Section 2 contains relevant defi-nitions, and background. Theorem 1.1 is proved in Section 3. Section4 tabulates a family of knots used to prove Theorem 1.2. Section 5 and6 establish properties of these knots. In Section 7, the knots are usedto prove Theorem 1.2. 2.
Definitions
For the duration of the paper, all knots K are taken to be orientatedlattice knots, and [ K ] is the class of lattice knots isotopic to K . Definition 2.1.
The stick number of a conformation K , denoted s CL ( K ),is the number of sticks, i.e. connected line segments, in the conforma-tion of K . Definition 2.2.
The edge length of a conformation K , denoted e L ( K ),is the number of unit length edges in the conformation K .Endow a lattice knot with an orientation. Then, there is a bijectionbetween the set of edges and the set of vertices by sending each edgeto its terminal point. Thus, | V ( K ) | = e L ( K ). Definition 2.3.
The minimal lattice-stick number or lattice stick index of [ K ] is s CL ([ K ]) = min K ∈ [ K ] s CL ( K ). ERTEX DISTORTION OF LATTICE KNOTS 3
Definition 2.4.
For a lattice knot, the endpoints of each stick arevertices of K ; call such a vertex a critical vertex of the lattice knot.Since our lattice knots are taken to be orientated, each stick has awell-defined initial and final vertex, v and v f respectively. The point v f − v is a coordinate triple with zeros in all but one coordinate; thenonzero coordinate value defines the stick type : x + , y + , z + , x − , y − , or z − . Definition 2.5.
Let K be an orientated lattice knot. Let v and v f be the initial and terminal vertices of a stick S ⊂ K . We have that v f − v = ( a, b, c ) ∈ Z . Then S is an x + -stick if a > x − -stick if a <
0. A y + -, y − -, z + -, and z − -stick is defined analogously where b or c is positive or negative.Heuristically, a stick is parallel to the x -, y -, or z -axis with an orien-tation dictating whether it has an increasing or decreasing, x -, y -, or z -coordinate, respectively, while all other coordinates are held constant.The coordinate that is changing, along with whether it is increasing ordecreasing, gives the stick type.Every lattice knot can be described as a sequence of stick lengthsand a paired sequence of stick types. Beginning at a critical vertex,the orientation induces an ordering of the sticks types. This orientationalso induces a sequence of stick lengths. Every lattice knot can beisometrically translated such that any critical vertex translates to theorigin, therefore our convention will be for this construction to beginat the origin. x y z z + , x + , y + , z − , x − , y − , z + , x + , y + , z − , x − , y − Figure 1.
Stick length sequence with pairing stick typesequence of the trefoilThe tabulation of stick lengths paired with the stick type sequencein Figure 1 gives a construction for the trefoil. The j th row of the x -column reads the length of the j th x -stick, similarly for the y - and z -columns. Beginning at the origin, we see the stick type sequence isinitiated by z + -, x + -, and y + -sticks. Since these are the first x -, y -, and z -sticks, we read the lengths of these sticks from the first row, 3, 2, 1, ERTEX DISTORTION OF LATTICE KNOTS 4 respectively. The following stick type is z − . Since this is the second z -stick traversed, we read its length from the second row of the table.Figure 2 illustrates the prescribed construction of Figure 1. A squareis placed at the critical vertex lying at the origin, the initial point ofthis construction, and an orientation arrow is shown above the originexhibiting that the first stick in the sequence is a z + -stick. The verticesof the knot are shown with dots. Figure 2.
Trefoil constructed from the tabulation inFigure 1In general, there are as many rows in the tabulation of stick lengths,as seen in Figure 1, as the largest total of either x -, y -, z -sticks. If aknot has more z -sticks than x - or y -sticks, then the x and y columnswill be padded with zeros, beginning when the row exceeds the numberof x - or y -sticks. No such padding will be necessary for the knots wetabulate. Definition 2.6. A staircase walk from a ∈ L to b ∈ L is a piecewiselinear, continuous curve r : [0 , → L given by r ( t ) = ( x ( t ) , y ( t ) , z ( t )) such that r (0) = a , r (1) = b and each of x ( t ), y ( t ), and z ( t ) is, independently, nondecreasing or nonincreasing.All possible staircase walks between (1 , ,
0) and (3 , ,
0) are shownin Figure 3; since the entry-wise difference of these two points has nochange in the z -coordinate, all walks are subsets of the xy -plane. ERTEX DISTORTION OF LATTICE KNOTS 5
Figure 3.
Staircase walks between (1 , ,
0) and (3 , , a = ( a , a , a ) , b = ( b , b , b ) ∈ Z , the number of staircase walksfrom a to b is d ( a , b )!( | a − b | )!( | a − b | )!( | a − b | )! . It is important to note that a staircase walk represents a most ef-ficient path in the lattice from one vertex to another, i.e. for a , b ∈ V ( K ), all staircase walks from a to b are of length d ( a, b ) and no latticepath from a to b has a shorter length. Lemma 2.7.
A path from a ∈ Z to b ∈ Z , in L , has length d ( a , b ) if and only if it is a staircase walk.Proof. Suppose r is a path in L from a = ( a , a , a ) ∈ Z to b =( b , b , b ) ∈ Z with length d ( a , b ). Through an isometry, we mayassume that a i < b i for all i ∈ { , , } . The sum of the x + -sticklengths must be at least b − a . Likewise, the sum of the y + - and z + -stick lengths must be at least b − a and b − a , respectively. Thelength of the path is the sum of the stick lengths. Let s be the sum ofthe negative stick lengths. Then the length of the path is greater thanor equal to ( b − a ) + ( b − a ) + ( b − a ) + s = d ( a , b ) + s . Sincethe length of path is d ( a , b ), s = 0 implying that no negative sticksexist. Therefore, all coordinate functions of r are nondecreasing, i.e. r is a staircase walk.Suppose r is a staircase walk in L from a = ( a , a , a ) ∈ Z to b = ( b , b , b ) ∈ Z . Through an isometry, we may assume that a i < b i for all i ∈ { , , } , i.e. all coordinate functions of r are nondecreasing.Therefore, this path will be comprised of only x + -, y + -, and z + -sticks. ERTEX DISTORTION OF LATTICE KNOTS 6
Thus, the sum of the x -stick lengths is b − a . Likewise, the sum of the y -, and z -stick lengths is b − a and b − a , respectively. The lengthof the path is d ( a , b ) = ( b − a ) + ( b − a ) + ( b − a ). (cid:3) Definition 2.8.
The minimal bounding box of K is the box, [ x , x ] × [ y , y ] × [ z , z ], of smallest volume that contains K . The points { ( x i , y j , z k ) : i, j, k ∈ { , }} are corners of the minimal bounding box.The minimal bounding box of a lattice knot will have integer end-points for all intervals in its product. Lemma 2.9.
A vertex v = ( v , v , v ) ∈ V ( K ) is a corner of theminimal bounding box of a lattice knot K if and only if v , v , and v areupper or lower bounds on the set of x -, y -, and z -values, respectively,of K ’s vertices.Proof. Let [ x , x ] × [ y , y ] × [ z , z ] be the minimal bounding box ofa lattice knot K . Every vertex in K has an x -coordinate between x and x , a y -coordinate between y and y , and a z -coordinate between z and z .Let v be a corner of the minimal bounding box. Then v is of theform v = ( v , v , v ) = ( x i , y j , z k ), for i, j, k ∈ { , } . If i = 1 , j = 1, or k = 1, then v , v , or v bounds, from below, their respective coordinatevalues of the vertices. If i = 2 , j = 2, or k = 2, then v , v , or v bounds,from above, their respective coordinate values of the vertices.Let v = ( v , v , v ) ∈ V ( K ). If v is less than or equal to all the x -values of vertices, then v = x from the definition of the minimalbounding box. If v is greater than or equal to all the x -values ofvertices, then v = x . Apply the same argument to v and v . Thus, v is of the form v = ( x i , y j , z k ), for i, j, k ∈ { , } and is necessarily acorner of the minimal bounding box. (cid:3) Lemma 2.10.
Let K be a lattice knot. Then e L ( K ) is even.Proof. Since a knot is a closed curve, the sum of the x + -stick lengthsmust equal the sum of the x − -stick lengths, likewise for the y - and z -sticks. This implies that the sum of x + - and x − -stick lengths is even,similarly for the y - and z -sticks. The edge length of a knot is the sumof all stick lengths. Therefore, the edge length is the sum of three evenpositive integers. (cid:3) ERTEX DISTORTION OF LATTICE KNOTS 7 Conformations with δ V = 1 Theorem 3.1.
Let K be a lattice conformation with e L ( K ) = (cid:96) . Then δ V ( K ) ≤ (cid:96) .Proof. δ V ( K ) ≤ max p,q ∈ V ( K ) d K ( p, q ) ≤ (cid:96) (cid:3) Theorem 3.2.
For any [ K ] , δ V ([ K ]) ≥ and δ V ([ U ]) = 1 .Proof. For any lattice knot K and any pair of points p and q in V ( K ), d K ( p, q ) ≥ d ( p, q ). Thus, δ V ( K ) ≥ δ V ([ K ]) ≥
1. If[ K ] = [ U ], consider the conformation U = ∂ [0 , × { } . For any pairof points p and q in V ( U ), d K ( p, q ) = d ( p, q ). Therefore, δ V ( U ) = 1and δ V ([ U ]) = 1. (cid:3) Two points v and v (cid:48) ∈ V ( K ) are antipodal if the two distinct injectivepaths r : [0 , → K and r : [0 , → K from v to v (cid:48) are of equal length.We now prove Theorem 1.1. Proof.
Let K be a lattice knot with δ V ( K ) = 1, and let v = ( v , v , v ) ∈ V ( K ). By Lemma 2.10, e L ( K ) = 2 (cid:96) for some positive integer (cid:96) . There-fore, there exists a point v (cid:48) ∈ V ( K ) antipodal to v . Note,1 ≤ d K ( v , v (cid:48) ) d ( v , v (cid:48) ) ≤ max p,q ∈ V ( K ) d K ( p, q ) d ( p, q ) = δ V ( K ) = 1 . This implies the shortest path from v to v (cid:48) has a length equal to d ( v , v (cid:48) ). Since the points v and v (cid:48) are antipodal, both paths are ofequal length. Therefore, r and r are both staircase walks, by Lemma2. Thus, if a coordinate function of r is nondecreasing, then the samecoordinate function of r is nondecreasing, likewise if the coordinatefunction is nonincreasing: Assume that the x -coordinate function of r is nondecreasing. Then the x -value of r (1) is greater than or equalto v . Since r (0) = r (0) and r (1) = r (1), the x -value of r (1) isgreater than or equal or equal to the x -value v . Thus, r has a non-decreasing x -coordinate function. An analogous argument applies foreach component function and whether said function is nondecreasingor nonincreasing.Assume that the x -coordinate functions of r and r are nondecreas-ing. Then all points in the image of r and r have x -coordinatesgreater than or equal to the x -value of v . Since K is the union of the ERTEX DISTORTION OF LATTICE KNOTS 8 image of r and r , all vertices of K have an x -coordinate greater thanor equal to v . If the x -coordinate functions were nonincreasing, thenall vertices of K would have an x -coordinate less than or equal to v .A similar argument applies to the y - and z -coordinate functions of thepaths. Thus, the x -, y -, and z -coordinates of all vertices are boundedby v , v , and v , respectively; therefore, we have that v and v (cid:48) arecorners of the minimal bounding box, Lemma 2.9.Since each vertex is a corner of the minimal bounding box of K , K is contained in the boundary of the minimal boundary box. Theboundary of the minimal boundary box is ambiently isotopic to S andthe only knot embeddable in such a surface is the unknot. (cid:3) Corollary 3.3.
The only lattice knot conformations of vertex distor-tion equalling one, up to isometry, are shown in Figure 4.
Figure 4.
Lattice conformations with vertex distortionequalling one 4.
Lattice Torus Knots
In [1] the authors illustrated a triplet of lattice knots but no sticklength or verification of knot type was given. In this section, we willverify a tabulation to be that of a ( p, p + 1) − torus knot with similargeometry to that of [1]For positive integers p >
2, the tabulation of our T p,p +1 knots aregiven in Figure 5. Once verified as ( p, p + 1)-torus knots, these knotswill be used to prove Theorem 1.2. ERTEX DISTORTION OF LATTICE KNOTS 9 x y z p − p −
12 3 p p −
23 3 p − p −
34 4 p p −
45 4 p − p −
56 5 p p −
67 5 p − p − z + , x + , y + , z − , x − , y − ,... ... ... ... z + , x + , y + , z − , x − , y − , . . .2 p − p − p z + , x + , y + , z − , x − , y − p − p − p − p − p p p − p p − p − p + 1 p p − p p − p p p Figure 5.
Stick length sequence with pairing stick typesequence of T p,p +1 The sequence of stick types has 6 p terms and is periodic with period6. There are 2 p x -sticks, 2 p y -sticks, and 2 p z -sticks.First, let us verify that this tabulation of stick types and lengthsconstitutes a polygonal closed curve. The sum of z + -stick lengthsmust equal the sum of the z − -stick lengths, similarly for the x - and y -sticks. The sequence of stick types dictates that the sum of the z + -stick lengths, denoted (cid:80) | z + | , is the sum of the odd row entries in the z -column of Figure 5 while (cid:80) | z − | is the sum of the even row entries.Thus, (cid:88) | z + | = (2 p −
1) + (2 p −
3) + (2 p −
5) + · · · + 3 + 1 = p , and (cid:88) | z − | = [(2 p − p − p − · · · +4+2]+ p = [ p ( p − p = p . Likewise, (cid:88) | y + | = [ p − p − · · · + p −
1] + 2 p − p − ] + 2 p − p , ERTEX DISTORTION OF LATTICE KNOTS 10 (cid:88) | y − | = [ p + p + · · · + p ] + p = [ p ( p − p = p , (cid:88) | x + | = [2+3+4+ · · · + p ]+ p = [( p +2)( p − / p = p / p/ − , and (cid:88) | x − | = [3+4+5+ · · · + p +1]+1 = [( p +4)( p − / p / p/ − . Therefore, these sticks form a closed curve, and the total length ofthe curve is 5 p + 3 p − . Each y - and z -stick lies in a plane whose x -coordinate is some integer a ; these sticks exist in the x - level a of the knot. Definition 4.1.
The x-level a , for a ∈ Z , of a lattice knot K is theintersection of the plane x = a and K . The y-level a and z-level a of K is defined analogously for planes y = a and z = a .Each level of a lattice knot will be empty or contain a set of disjointarcs and/or points.If each level of our closed lattice curve contains no double points,then the curve is simple. The n th partial sum of the the y -stick lengthsequence will give the y -level containing the n th y -stick’s terminal crit-ical vertex.If the terms of the partial sum sequence of y -stick lengths are alldistinct, then T p,p +1 contains just one arc in each y -level, similarly forthe x - and z -stick lengths.Moreover, a sole arc in a level of our closed curves could not intersectitself; this is a result of the stick type sequence cycling z -, x -, y -sticksconsecutively. Each arc will be a figure “ L ”.Thus, double points of our T p,p +1 closed curves could only occur onlevels that represent repeated values in the partial sum sequence of agiven stick type’s length sequence.The partial sum sequence of the y -stick length sequence is p − , − , p − , − , p − , − , . . . , , − p, p, . Ordering the values of this sequence in nondecreasing order, we obtain1 − p, − p, . . . , − , − , , , , . . . , p − , p − , p and observe that thereare no repeated values of the sequence.The partial sum sequence of the z -stick length sequence is ERTEX DISTORTION OF LATTICE KNOTS 11 p − , , p − , , p − , , . . . , , − p, p, . Ordering the values of this sequence in nondecreasing order, we obtain0 , , , , . . . , p − x -stick length sequence is2 , − , , − , , − , , − , . . . , , − p, , − p, , . Excluding the 2 p − x -levels excluding x -level2 contain just one arc. We will show that x -level 2 does not containany double points, illustrating that no level of T p,p +1 contains a doublepoint.The value 2 is repeated p − x -stick length sequence. Therefore, x -level 2 will have p − y − z − -stickarcs, as in Figure 6. Figure 6. x -level 2 of T , The initial critical vertex of a y − z − -stick arc, in this plane, has a y and z value one less than the previous stick’s initial critical vertex.Order the initial critical vertices of these arcs following the orientationof the closed curve; this gives the sequence v , v , . . . , v p − seen inFigure 6. Each column of stick lengths in Figure 5 can be used to definea sequence of vectors. Let z i = (0 , , z i ) where z i is the value in the i th row of the z -column in the table of lengths; we define x i = ( x i , , y i = (0 , y i ,
0) likewise. Then, v = z + x = (2 , , p − . We can
ERTEX DISTORTION OF LATTICE KNOTS 12 then define a recursive sequence v n = v n − + y − − z − − x − − y − + z − + x − , for 2 ≤ n ≤ p − . We verify using the stick length sequences, − x − + x − =(0 , ,
0) for 2 ≤ n ≤ p − y − − y − = (0 , − ,
0) for 2 ≤ n ≤ p − − z − + z − = (0 , , −
1) for 2 ≤ n ≤ p −
1. Therefore, we cansimplify the former recursive definition to v n = v n − + (0 , − , − , for 2 ≤ n ≤ p − , and express the sequence in closed form as v n = (2 , − n, p − − n ) , for 1 ≤ n ≤ p − . This verifies our claim that the initial critical vertexof a y − z − -stick arc, in x -level 2, has a y and z value one less thanthe previous stick’s initial critical vertex. This implies that no two y − -sticks on x -level 2 will intersect.All y − -sticks in x -level 2 have a length of p −
1. Therefore, no terminalcritical vertex of a y − -stick in this plane lies above another y − -stick,i.e. the y -value of any y -stick’s terminal critical vertex is greater thanall y -values of each sequential y -sticks in x -level 2. Thus, no z − -stickof a y − z − -arc will intersect a y − -stick nor another z − -stick on x -level2, and, resultantly, x -level 2 contains no double points.Altogether, we have that Figure 5 tabulates a knot. For p = 7, thetabulation constructs the knot in Figure 7. Figure 7. T , ERTEX DISTORTION OF LATTICE KNOTS 13 Torus Verification
In order to verify that these knots are torus knots, we will generatea scalable toroidal polyhedron that T p,p +1 can be embedding into. Figure 8.
Torus with embedded T , We will first show that, excluding the final y − - and final z − -stick,each stick type is coplanar. This was verified for the remaining y − - and z − -sticks in pursuit of proving our closed curve had no double points,Figure 6. We will use an analogous method to show coplanarity of theremaining stick types; we will use the collinearity of critical vertices.The convex hull of each stick type will then be used as a face of thepolyhedron.A coplanarity argument for each stick type follows. z + : We will show that all initial critical vertices of z + -sticks arecollinear. Let v n represent the n th z + -stick’s initial critical vertex.Then, v = (0 , ,
0) and v n = v n − + z − + x − + y − − z − − x − − y − , for 2 ≤ n ≤ p . We have z − − z − = (0 , , y − − y − =(0 , − , x − − x − = ( − , , . Therefore, v n = (1 − n, − n, n −
1) = (1 , , −
1) + ( − , − , n, for 1 ≤ n ≤ p . ERTEX DISTORTION OF LATTICE KNOTS 14 y − : Since all initial critical vertices of z + -sticks are terminal criti-cal vertices of y − -sticks, the collinearity of the z + -stick initial criticalvertices implies that the terminal critical vertices of the y − -sticks arecollinear. Thus, the y − -sticks are coplanar. x + : As seen in the earlier verification associated with Figure 6, theinitial critical vertices of the y − -sticks, excluding the final y − -stick, arecollinear with v n = (2 , , p −
2) + (0 , − , − n for 1 ≤ n ≤ p − x + -stick’s terminal critical verticesare collinear and at least all but the final x + -sticks are coplanar.The final x + -stick has an initial point of (1 − p, − p, p ), the penulti-mate x + -stick has an initial point of (2 − p, − p, p ), and the x + -stickprevious to this has an initial point of (3 − p, − p, p ); these initialcritical vertices are collinear. Therefore, the final three x + -sticks arecoplanar implying that all the x + -sticks are coplanar. x − : Since all of the y − -sticks’ terminal critical vertices are collinearand all y − -sticks are of equal length, the y − -sticks have collinear initialcritical vertices. Therefore, all x − -sticks’ terminal critical vertices arecollinear, and all x − -sticks are coplanar.Excluding the final y − - and z − -sticks, the convex hull of a stick typeintersects two other convex hull of stick types; the convex hull of astick type intersects the convex hull of the stick type prior and after inthe stick type sequence. This produces a band with two twists, seen inFigure 9, that T p,p +1 , partially, embeds into. Figure 9. T , superimposed with planar faces ERTEX DISTORTION OF LATTICE KNOTS 15
Importantly, the geometry of this band remains fixed for all T p,p +1 .Meaning, gluing of addition polygonal faces to Figure 9 to create thetorodial polygon of Figure 10, is a general operation. The geometryof Figure 10 welcomes an embedding of the general T p,p +1 . An explicitparameterization of the remaining torodial fragments is cumbersomeand has been omitted to save space.Thus, T p,p +1 is a torus knot for each p , notably a ( p, p + 1)-torusknot. The type of torus knot is verifiable from Figure 10. Figure 10.
Torus with embedded T , Theorem 5.1. s CL ([ T p,p +1 ]) = 6 p Proof.
For any knot type, s CL ([ K ]) ≥ b [ K ], where b [ K ] is the bridgeindex of [ K ] [1]. The bridge index of a ( p, q )-torus knot is min( p, q ), so s CL ([ T p,p +1 ]) ≥ p. Since we verified a lattice knot of 6 p sticks to be aconformation of T p,p +1 , s CL ([ T p,p +1 ]) = 6 p. (cid:3) ERTEX DISTORTION OF LATTICE KNOTS 16 Irreducibility
We would like to eliminate any unnecessary length in the T p,p +1 knots. In general, one may arbitrarily isotope, radially, a portion ofa smooth knot toward infinity. This has the ability to increase thedistortion of the embedding with minimal alterations to its geometryin a neighborhood excluding the adjustment. Figure 11.
Distortion increasing stretchSuch a stretch spoils the geometric interpretation of the configura-tion’s distortion.For lattice knots, the removal of such a stretch has a maximal reduc-tion. The confines of the lattice limits the amount such a perturbationcan be inwardly reduced, before a stick is eliminated. We seek to re-duce the knot in a fashion that decreases the volume of the minimumbounding box.As earlier discussed, each level of our T p,p +1 conformations contains afigure “ L ” arc preceded and followed by parallel sticks, the “legs” of the“ L ”. For our purposes, the above heuristic manifests as a movementof an “ L ” by a reduction of its legs, Figure 12.We not only want to reduce the volume of the minimum boundingbox but also minimize the step length isotopically through the samemotion of minimizing the legs of an “ L ”. Altogether, this strengthensthe result that this class of knot configurations has arbitrarily highdistortion since an aspect of minimality is introduced. ERTEX DISTORTION OF LATTICE KNOTS 17
Figure 12.
Reduction of T , While the retraction of an “ L ” is a reduction of a pair of parallelsticks, of opposite signs, we will first discuss the implications of reduc-ing a sole stick. We will then use this to show that no stick in our T p,p +1 can be reduced in this fashion, implying that no “ L ” can be retracted.Any stick, s i , of the knot can be positioned to satisfy the relativegeometry of Figure 13 . In an attempt to shrink s i , v or v f wouldmove into the interior of the stick, an integer length. For the purposeof this discussion, endow the four sticks in Figure 13 with a clockwiseorientation. Figure 13.
Range of motion planes for shrink action
ERTEX DISTORTION OF LATTICE KNOTS 18
Then, a movement of v in an attempt to shrink s i will move the twosticks prior to s i . If v is slid along s i to v f , the two previous stickswould trace the blue and red plane segments.If v f is slid along s i to v , the succeeding sticks would trace theyellow and purple planes. If either the blue or red plane segmentsintersect the knot at a point that is one away, in the (cid:96) metric, fromthe respective stick that traces the plane, then s i cannot be isotopicallyreduced with the orientation .If the yellow or purple plane intersects the knot at a point that is oneaway from the respective stick, then s i cannot be isotopically reducedagainst the orientation .If s i cannot be reduced with and against the orientation, then s i is isotopically irreducible .For general lattice knots, this condition is sufficient but not necessaryto determine that a given stick cannot be reduced without producinga double point.Our T p,p +1 knots are irreducible in this fashion. To avoid a cumber-some, abstract showcase of this, we will briefly discuss the irreducibilityof the sticks in Figure 6.From earlier discussion, accompanying Figure 6, the z -sticks cannotbe reduced with the orientation and the y -sticks cannot be reducedagainst the orientation. This is easily seen by the position of the criticalvertices. The opposite directions of reduction are more subtle.The z -sticks in this plane cannot be reduced against the orientation.This is true since the sequential y − -stick would intersect the secondsequential x − -stick of a given z -stick in the plane of Figure 6The y -sticks in this plane cannot be reduced with the orientation. Ifthe first y -stick, the stick with initial vertex v in Figure 6, is reducedwith the orientation, then the previous z + -stick would intersect thesecond to last x − -stick. If any other y -stick is reduced with the orien-tation, then the previous x + -stick would intersect the second previous z + -stick.An analogous process continues to show that each stick of our T p,p +1 cannot be reduced.7. Proof of the Main Theorem
Recall that vertex distortion of a lattice knot is defined as δ V ( K ) = max p,q ∈ V ( K ) d K ( p, q ) d ( p, q ) . ERTEX DISTORTION OF LATTICE KNOTS 19
We will find a, b ∈ V ( T p, p +1 ) such that lim p →∞ d K ( a,b ) d ( a,b ) = ∞ . This willprove Theorem 1.2 since d K ( a, b ) d ( a, b ) ≤ max p,q ∈ V ( K ) d K ( p, q ) d ( p, q ) = δ V ( K ) . We will analyze the distortion value of a pair of vertices in the afore-mentioned T p, p +1 knots, see Figure 5. There is a point on the p/ z − -stick that lies one away in space from a point on the final y + -stick,see Figure 14. Figure 14.
Distortion path of T , We will calculate the distortion value of this pair.Since the points lie one away in ambient space, the distortion valuewill be equal to the shorter of the two path lengths. Let us count thelength of the path following the positive orientation, beginning at thepoint on the z − -stick.We begin with a partial z − -stick of length p/ y + -stickof length 3 p/ −
1. Of the remaining sticks, there are p/ y − -sticks;each of these sticks has a length of p . Thus, the total length of the y − -sticks is p / . There are p/ − y + -sticks; each of these sticks has a length of p − y + -sticks is p / − p/ . The x + -sticks have lengths following the sequence: p/ , p/ , . . . , p/ p/ , p ; this sequence has p/ x + -sticks is 3 p / p/ − . ERTEX DISTORTION OF LATTICE KNOTS 20
The x − -sticks have lengths following the sequence: p/ , p/ , . . . , p/ p/ , p/ p/ p/ x − -sticks is 3 p / p/ . The length of the z − and z + sticks together follow the sequence: p − , p − , . . . , , , . The total length of the z -sticks is p / − p/ . Therefore, the length of the curve is9 p / p/ − . Since the total length of the knot is 5 p + 3 p −
2, the second path fromone vertex to the other has a length of11 p / p/ − . This implies the first path is the shorter of the two and that9 p / p/ − p increase. It is easyto computationally check all pairs of points for small p , in fact thedistortion is equal to the lower bound for even p less than 12; anotherformula, 11 p / − p − /
4, is easily verified through computation forsmall odd p .For even p greater than 10, the distortion realizing points do not lieon the largest y + -stick and p/ z − -stick. For even p greater than 12and less than 24, we have verified that one distortion realizing pointlies on the longest y + -stick but the other lies on p/ − z − -stick.There are only four sticks to add to the previous computation whencounting the length of the one-larger-median pair of vertices. Subtract-ing this length from the total length of the knot gives the distortionlower bound of 11 p / − p/ − . This can be verified, computationally, to be the actual distortion foreven p greater than 10 and less than 24.8. Conjecture
The following conjectures naturally arise from the above discussion.
Conjecture. lim p,q →∞ δ V ([ T p,q ]) → ∞ ERTEX DISTORTION OF LATTICE KNOTS 21
Conjecture. δ V ([ K ]) = 1 ⇐⇒ [ K ] = [ U ]Theorem 3.2 gives one direction. Proposition 8.1.
If no lattice knot in [ K ] has vertex distortion 1 while δ V ([ K ]) = 1 , then any sequence of knots whose vertex distortion ap-proaches 1 is of arbitrarily large length.Proof. Since δ V ([ K ]) = inf K ∈ [ K ] δ V ( K ) = 1, there exists a sequence oflattice knots in [ K ] such that lim n →∞ δ V ( K n ) = 1. We have that δ V ( K n ) =max p,q ∈ V ( K n ) d K ( p,q ) d ( p,q ) = d K ( p (cid:48) ,q (cid:48) ) d ( p (cid:48) ,q (cid:48) ) = a n /b n for relatively prime positive integers a n , b n and some pair p (cid:48) , q (cid:48) ∈ V ( K n ). Let α = gcd( d K ( p (cid:48) , q (cid:48) ) , d ( p (cid:48) , q (cid:48) )).Then d K ( p (cid:48) , q (cid:48) ) = αa n and a n ≤ d K ( p (cid:48) , q (cid:48) ).Suppose a n is bounded above by (cid:96) ∈ Z for all n . Then a n ∈{ , , . . . , (cid:96) } for each n . Thus, there are only finitely many possibil-ities for b n , all relatively prime integers of each integer from 2 to (cid:96) .Therefore, { δ V ( K n ) } n ∈ N is a finite set. A sequence that only takeson finitely many values cannot converge to a value outside its range.Since δ V ( K n ) (cid:54) = 1 for all n , we have that a n is unbounded.Thus, lim n →∞ d K ( p (cid:48) , q (cid:48) ) = ∞ .For every (cid:15) > N such that n > N = ⇒ | δ V ( K n ) − | < (cid:15). Also, for every M ∈ N , there is an n (cid:48) > N such that d K ( p (cid:48) , q (cid:48) ) > M forsome distortion realizing points p (cid:48) , q (cid:48) ∈ V ( K n (cid:48) ).Also, Md ( p (cid:48) ,q (cid:48) ) < δ K ( K n (cid:48) ) < (cid:15) + 1 implying M(cid:15) +1 < d ( p (cid:48) , q (cid:48) ). Thus,lim n →∞ d ( p (cid:48) , q (cid:48) ) = ∞ , i.e. take M → ∞ for fixed (cid:15) . (cid:3) The vertex distortion of a lattice knot measures the extremity of howfar geodesic paths along the knot deviate from being staircase walks.We see from Proposition 8.1 that there exists knots of arbitrary largelength whose paths become arbitrarily close to being staircase walkwhenever the lattice knot type has a distortion of one. If δ V ( K ) > K / ∈ [ U ], then Conjecture 8 is true. References [1] COLIN ADAMS, MICHELLE CHU, THOMAS CRAWFORD, STEPHANIEJENSEN, KYLER SIEGEL, and LIYANG ZHANG,
Stick index of knots andlinks in the cubic lattice , Journal of Knot Theory and Its Ramifications (2012Apr), no. 05, 1250041. ERTEX DISTORTION OF LATTICE KNOTS 22 [2] Ryan Blair, Marion Campisi, Scott A. Taylor, and Maggy Tomova,
Distortionand the bridge distance of knots , Journal of Topology (2020Mar), no. 2, 669–682.[3] Elizabeth Denne and John M Sullivan, The distortion of a knotted curve , 2004.[4] Mikhael Gromov,
Filling riemannian manifolds , J. Differential Geom. (1983),no. 1, 1–147.[5] John Pardon, On the distortion of knots on embedded surfaces , Annals of Math-ematics (2011Jul), no. 1, 637–646.
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