MTH 392A Topics in Knot theory Prabhash kumar mishra supervisor: Dr. aparna dar

Motive of the project To develop the basic knowledge and understanding of Knots. To develop the skills to find out the difference between two given knots i.e. knot invariants. To get some elementary knowledge about Braids. Get to know about the 3D surfaces which contains the basic knots at their boundaries i.e. Seifert surfaces. To read about Torus knots and it’s elementary properties.

What is a Knot? A knot is a simple closed polygonal curve in R3 i.e. it is a union of the segments [p1, p2], [p2, p3], …….., [pn-1, pn] of an ordered set of distinct points (p1, p2,…., pn) in which each segment intersects exactly two others. Knots are “stick knots” as per the above definition and as shown in the figure but are usually drawn and thought of as smooth.

Simplest knots The simplest knot is an unknot. Then there comes two trefoils which are well known of the basic knots . The two trefoils are not same but the mirror images of each other.

Elementary knot moves On a given knot we may perform the following four operations Basically 1st one says that we can divide and edge into two and vice-versa. 2nd says that if there is a point in space C and if the triangle formed by A, B and C doesn’t intersect the knot then we may remove AB and add two new edges AC and CB and vice-versa.

Equivalent knots A knot K is said to be equivalent to K’ if we can obtain K’ from K by applying the elementary knot moves finite no of times. As in the adjacent figure K1 and K2 are actually equivalent and are called “Perko’s pair”. In appearance the look completely different but we can find K2 from K1 by applying elementary knot mover finite no of times This was shown in 1970 by an American lawyer KA Perko.

Alternating knots and Links Goes “under”  “over” “under”  “over”….. Means we can not find two overpasses or two underpasses in succession. Links: A link is a collection of knots which do not intersect but which may be linked together. A knot can be described as a link with one component. Hopf link

Reidemeister’s moves Reidemeister’s move change the projection of the knot. This actually changes the relation between crossings but not changes the knot. 1) First: Allows us to put in/take out a twist. 2) Second: Allows us to either add two crossings or remove two crossings.

Reidemeister’s moves 3) Third: Allows us to slide a strand of a knot from one side of crossing to other. Theorem: Given two knots K1 and K2, then K1=K2 if you can get K2 from K1 by a series of Reidemeister’s moves.

Linking number The linking number is a way of measuring numerically how linked up two components are. The linking number represents the number of times that each curve winds around the other. Positive crossing +1 Negative crossing -1 So if there are more than two components, add up the link numbers and divide by 2.

Linking number Look at the following example, Rotate under the strands As the left crossing can be removed by R1. Rest all has Positive crossings i.e. +1 so Linking no= 1+1+1+1 2 so the linking number is 2.

The unknotting number The unknotting number of a knot is the minimum number of the times the knot must be passed through itself (Crossing switch) to untie it. As in the above example, we can see that the unknotting number of the trefoil knot is 1.

Bridge number In a knot diagram one or more overpasses in succession form a bridge. So the minimum number of overpasses out of all possible diagrams of the knot is called it’s Bridge number. As for regular trefoil diagram we can see 3 overpasses but in the above figure adjacent to regular diagram there are only 2 overpasses. So bridge number of trefoil knot is 2. (As we can not find any diagram of trefoil knot which has only one overpass so minimum is 2)

How can we tell two knots are different?

Polynomial Any two projections of the same knot yield the same polynomial (true for all types of polynomials people have discovered for knots…Jones, Alexander, Kauffman etc.) If a given type of polynomial is different for two knots, then the knots are distinct .

Alexander polynomial Step 1: Orient the knot K and choose a crossing: Step 2: Compute A(L+), A(L-), and A(L0), where L+ , L- and L0 are following operations to be performed on that crossing. Step 3: Calculate A of the entire knot K using the following 2 rules: a) A( unknot )=0 and A(L+)-A(L-)-(t1/2-t-1/2)A(L0)=0.

Alexander polynomial example Let find Alexander polynomial for trefoil so D is our actual knot and we choose crossing 1 so D is L_ E is L+ F is Lo we can see easily that E is an unknot. Now lets try for F. we choose crossing 2 now. So F is L_ , G is L+ and H is Lo. So we have spilt them into unknots E and H and G consists of two unknots which has Alexander polynomial zero. Finally A(D)= t-1+(1/t)

Jones polynomial Step 1 to step 3 are all same as Alexander Jones skein relation is, And of course, the Jones Polynomial of the Unknot is 1!

For two disjoint unknots and Hopf link

Hopf Link

Seifert surfaces In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is a surface whose boundary is a given knot or link. A Seifert surface must be oriented. Given here is the Seifert surface of Trefoil knot as you can see the yellow boundary. Mobius strip has the unknot at it’s boundary but is not a Seifert surface as not oriented.

Genus and Euler characteristic A closed orientable surface F is topologically equivalent to sphere with several handles on surface. The no of handles is called Genus. g(f). We may divide a closed orientable surface into V points, E edges and F faces then Euler characteristic X(F)= V-E+F

Euler characteristic As for example let us take a tetrahedron, V= 4 E= 6 F= 4 so V-E+F= 4-6+4= 2 Actually for any convex polyhedra it comes out to be 2 always as for example for Cube, octahedron etc. If a surface is closed without a boundary then X(F)= 2-2g(F)

Torus knots Torus knot is a special kind of knot that lies on a surface of an unknotted torus in R3. (p,q)-torus knot: winds q times around a circle in the interior of the torus and p times around it’s axis of rotational symmetry. (2,3)-torus knot i.e.  trefoil knot

More about torus knots We can parameterize any (p,q)-torus knot like this. x = r cos(pβ) y = r sin(pβ) z = -sin(qβ) where r = cos(qβ)+a and 0<β<2π on the surface of the torus (x-a)^2 + z^2 = 1 Properties of torus knots:  A torus knot is trivial if |p| or |q| is equal to 1  A torus link arises when p and q are not co-prime in that case no of components are gcd(p,q).

More about torus knots (p, q) torus knot is equivalent to (q, p) torus knot. This can be proved by moving the strands on the surface of torus. (2, 3) torus knot (3, 2) torus knot but they are actually same.

The Braids theory Braid is a set of n strings, all of which are attached to an horizontal bar at the top and at the bottom( or respectively left and right). Such that each string intersects any horizontal plane between the two bars exactly once. As in the given figure except (c) all are braids. In (c) at a point of time string is going back so if we take a horizontal plane there it’ll cut it more than 1 times.

More about braids The braid shown in right is called the Trivial Braid. Braid permutation is a braid invarient. Let 1,2,3…n are n points on top bar and each goes to i1, i2….in on bottom bar respectively then braid permutation is,

Product of Braids Attach bottom bar of first to top bar of second In general if a, b and c are three braids . Then a*b ≠ b*a, NOT COMMUTATIVE but a*(b*c) = (a*b)*c ASSOCIATIVE

Inverse and unit If we take product of any braid with the trivial braid it remains the same braid so Trivial braid is a unit for braids. a*1 = 1*a = a If we take mirror image of a braid a i.e. a’ and take product it’ll give trivia braid so mirror image is Inverse.  

Elementary transpositions These are the elementary transpositions given for 4-braid. Similarly for n-braids we can find 2(n-1) braids. Any braid can be divided into product of elementary braids as in the following example,

Fundamental relations and Alexander theorem

Alexander theorem It states that “Given an arbitrary knot then it is equivalent to a knot that has been formed from a braid. (we have to define a center point.)’’    