1. Mathematical problems in knot theory
Prabhash Kumar Mishra Supervisor: Dr. Aparna Dar

2. Objective
Main objective of this project is to cover the concepts which can be useful to understand the mathematical problems in knot theory which are open for many years.

3. Tricolorability of knots
A knot diagram is tricolorable if each arc can be colored in such a way with one of the three colors that. 1) At least two colors are used in the diagram. 2) At every crossing either, i) All three colors are used or, ii) Only one color is used.
A knot is Tricolorable if it’s diagrams are tricolorable.

4. Tricolorability of knots
For example Trefoil knot is tricolorable as shown in figure.

5. Tricolorability and Reidemeister’s moves
Tricolorability is an invariant under Reidemeister’s moves. We can explain it from figure given below.

6. HOMFLY Polynomial Step 1: Orient the knot K and choose a crossing.
Step 2: Compute P(L+), P(L-), and P(L0), where L+ , L- and L0 are following operations to be performed on that crossing.
Step 3: Calculate P of the entire knot K using the following 2 rules:
a) P( unknot )=1 and b) l*P(L+)+ l-1 *P(L-)+ m*P(L0) = 0

7. HOMFLY Polynomial
The HOMFLY polynomial of a link L that is a split union of two links L1 and L2 is given by, P(L) = -{l + l-1}/m P(L1) P(L2)
For example if want to find the HOMFLY polynomial for then it will be -{l + l-1}/m .

8. Fenchel’s Theorem
Theorem: The total curvature of a simple smooth closed curve in 3-space is ≥ 2π with equality if it is a plane convex curve.
The proof of this theorem uses Frenet frame, Gauss Bonnet theorem and Gaussian curvature.
To use the Frenet frame we assume that the curvature of the curve never vanishes.
The proof uses the fact that explosion factor of Gauss map M: SS2 of any surface S is actually the Gaussian curvature.

9. Applications of Fenchel theorem
There are some practical questions in knot theory like “What is the shortest/thickest piece of rope that can form a closed knot” which actually uses the Fenchel theorem to work upon.
There is a very important theorem in knot theory about curvature of the knotted curves “Fary-Milnor Theorem” and proving this theorem requires the understanding of Fenchel theorem.

10. Legendrian representative of a knot
Legendrian diagram of a knot is a front projection of the knot in 2-D plane C  R a = y dx – dz
Legendrian diagram is the front projection in xz plane such that a/c = 0  y = dz/dx
i.e. we can get the y co-ordinate of each point by looking at it’s slope.

11. Legendrian representative of a knot
A general Legendrian knot looks like figure 1, where m is a part with m number of twists as in figure 2.

12. Thursten Bennequin number
Thursten-Bennequin number is defined for a rectilinear Legendrian front diagram of a knot. And maximum Tb(f) over all Legendrian representatives of a knot is an invariant for that knot.
Rectilinear front diagram is defined as a knot diagram composed of only horizontal and vertical line segments such that at any crossing, horizontal segment is over vertical Tb(f)= w(f)-c(f)
Where w(f) is the writh and c(f) is the cusp number.

13. Thursten Bennequin number
Writhe is counting of no of crossings with sign +1 if crossing is L+ and -1 if crossing is L- .
Cusp number is number of locally upper right corners of F.
Let us look at this example. This is the right handed trefoil knot.
W(f)= 1+1+1=3
C(f)= 2
Tb(f)= 3-2= 1

14. Connected sum and prime knots
Connected Sum of Knots: To form a connected sum of two knots, cut each knot at any point and join the boundaries of the cut, keeping orientations constant.
Connected sum is independent of location of cut

15. Connected sum and prime knots
Factoring: To factor a knot into two components, select a sphere that intersects the knot (transversely) at two points and separate into two components. Then join the two loose ends of each knot with some path in sphere.
Unlike the connected sum this operation depends on sphere that one choose to factor the knot.

16. Prime knots
A knot is called prime if it can not be represented by a connected sum of two knots such that both of these are knotted (nontrivial).
Any knot which is not prime is called Composite.
Torus knots are prime knots because we don’t get any valid factorization when we cut them with any arbitrary sphere. Either it gives one factor as a trivial knot or it cuts the knot at more than two points.

17. Seifert algorithm
Seifert surface: A Seifert surface Sk of a knot K is an orientable surface that has boundary K.
The adjacent figure is the Seifert surface of trefoil knot.
Seifert algorithm is a process to get the Seifert surface of any given knot.

18. Seifert algorithm
Step 1: Given a projection of a knot, orient it and resolve all crossings by joining incoming segment from one strand to the outgoing component of that strand that crosses it. This will result in collection of circular components called Seifert circles.
Step 2: Now consider each Seifert circle as the boundary of a disk at different level in 3-space.
Step 3: Finally reconnect the disks where crossings were, with a strip giving it a half twist. We will call it bridges.

19. Seifert algorithm

20. Additivity of crossing numbers
It has long been conjectured that crossing number of links are additive under the connected sum of links.
But this is still an open problem for knots and has been open for last 100 years.
The best known result to date is the following: if K1 and K2 are any two alternating links then, Cr(K1#K2) = Cr(K1) + Cr(K2)

21. Additivity of crossing numbers
It has been shown that torus knots are also additive for crossing numbers.
We will take a brief look, how?
Let (K) be the number of components.
c is the number of crossings in given diagram
S is the number of Seifert circles produced by Seifert algorithm.
Then genus g = [c+2-s- (K)]/2
The genus of K denoted by g(K) is then defined to be the minimum genus over the genera of all possible Seifert surfaces of K.

22. Additivity of crossing numbers
Now for any link K1 and K2 , g(K1#K2) = g(K1) + g(K2)
Similarly the bridge number is additive too, bg(K1#K2) = bg(K1) + bg(K2)
Braid index: According to a classical result of Alexander , every link has a closed braid representation. Thus we can define the braid index (br) of a link K as the least no of strings needed in it’s braid representation.
And according to Birman and Menasco, br(K1#K2) = br(K1) + br(K2) - 1

23. Additivity of crossing numbers
Since a closed braid with n strings can not have more than n bridges, so bg(K) ≤ br(K)
If br(K) = n then there will be n closed loop. So minimum no of Seifert circles which will be formed is n.  s(K) = br(k) where s(K) is minimum no of Seifert circles.

24. Additivity of crossing numbers
Deficiency: Deficiency of a link is defined as, d(K) = Cr(K) + 2 – br(K) – 2g(K) - (K)
It can be proved that d(K)≥0 for any K.
And further if d(Ki)=0 for i= 1 to m then, Cr(K1+K2+K3+….+Km) = Cr(K1) + Cr(K2) +…..+ Cr(Km)
For the torus knots deficiency comes out to be zero.
So combining above two points it is clear that Torus knots are additive for crossing numbers.

25. References “Knot theory and its applications”, Kunio Murasugi
“Prime factorization of knots”, Morinigo, Marcos A.
“The additivity of the crossing numbers”, Yuanan Diao
“The Fary-Milnor theorem and Fenchel theorem”, Herman Gluck
“On knots”, Louis H. Kauffman