Are there any flights which permits a Sikh to carry a kirpan?

I see many more problems than just the airplanes here. Though it seems to be a small item, I do not know how long yours precisely is. But depending on the type of blade it can already be considered a weapon when it has a rather short blade. For example in this brochure, unfortunately I found only the German one, Swiss authorities explain, that a knife with a blade longer than 5cm is a weapon and is forbidden to be carried in public places:
http://www.ejpd.admin.ch/content/dam/data/sicherheit/waffen/Brosch%C3%BCre/waffenbroschuere-d.pdf

Similar rules may apply in other European countries too. I heard that the UK is quite strict in such things, but I don’t have a source for that, maybe you check it on your own.

Then you have to make sure that your blade is not considered a weapon when you carry it during custom controls, that could bring you into serious trouble as it might be considered smuggling a weapon.

So my advice in this matter:
Inform yourself carefully about the laws at your destination and any country you have to travel through when getting there. It’s no fun to end up in a jail, just because you have overlooked a little detail in some law – especially not outside Western Europe (and already there it’s bad enough). Send mails to the local authorities, better ask two or three different places.

Same also for the air companies. I remember a Jewish friend of mine who had to put a steel Star of David (less than 2cm in diametre!) into his checked-in luggage because it was considered a weapon. If travelling by plane you have to expect that, even though the company allows you to carry it with you, the security personal still takes it away. They’re not always connected to the air company and might have different rules.

I exagerated a bit, I’m aware of that. But I know of the value this has for you, so I strongly advice that you make as sure as you can whether you can carry it the way you need at the places you go. It’s a very unpleasant experience when something so valuable is taken away, even if it’s just temporar

Unfortunately, in this day and age, anything usable as a weapon must be eyed with suspicion. It doesn’t matter if it’s part of your religious garb or artifacts — if it could potentially be used to attack passengers or crew, it’s a weapon and cannot be carried on. If some countries, such as India, want to make exceptions for small ones on domestic flights, that’s their business, but don’t expect that on international flights. About the best accommodation that can be made is to allow you to carry it in checked baggage.

Before 9/11, who would have given a second thought to someone with a boxcutter?

Karlson beautifully took care of the international aspect of this situation wherein unfortunately you cannot carry a kirpan as carry-on on yourself.

But, since Mr.Sardarji is a religious person and it is possible that he would like to make his family happy without making sacrifices with his beliefs I would like to provide more information with regard to the kirpan and how it can be carried on some flights.

As this question of Sikh Answers explains,

The Kirpan (Punjabi: ਕਿਰਪਾਨ, kirpān) is a religious sword worn in a
strap that enables a it to be suspended near one’s waist or tucked
inside one’s belt. A Sikh who wears a kirpan is not wearing it because it is a weapon;
he or she is wearing it because it is part of their officially prescribed religious
uniform.

The kirpan usually looks something like this,

As you can see the object is relatively small and Air India (as well as Jet Airways, it seems it’s a regulation everyone must follow) does allow it in domestic flights,

As long as one can assure that the blade size is less than 6 inches and the total size of the object is less than 9 inches. This provision is specifically for the kirpan and is mentioned specifically in the rules for hand luggage as well.

A ‘Kirpan’ with a total maximum length of 9 inches (22.86 cms.), but a
blade not exceeding 06 (six) inches (15.24 cms.), is permitted for
carriage by a Sikh Passenger on his person, within India or on an
Indian Registered aircraft on Domestic routes of fully Domestic
Flights within India, subject to the requirement of a Public Order.
(BCAS Circular 14/2005).

But unfortunately to conform with international regulations, they also cannot allow it on international flights,

Carriage of ‘Kirpan’ shall not be permitted in the cabin of an
aircraft on an International Flight, either on the domestic or
international sector flown by the passenger. If the ‘Kirpan’ is more
than the length specified above, it shall be carried by the passenger
in his registered baggage.

So my suggestion to Mr.Sardarji,

You can try to convince your family to take a flight to somewhere else in India which can be far from Punjab (India is a big country after all, full of culture!) and still carry your kirpan with you. Maybe if you and your family sit together and talk, you all can reach a suitable decision.

I’m a fellow Sikh, and unfortunately there isn’t much you can do in this situation. You will have to remove your kirpan and place it in check-in luggage as posted by @karlson.

Some sikhs wear these small kirpans in their necklace,

Others don’t travel by plane at all. While traveling do remove kirpan.

You should do ardaas before and after, Guru Sahib sadh bakshand ne.

For whatever reason the subject comes up in context of the Bill introduced in New Zealand with respect to Maori. In addition to that the article also has specific information regading kirpan:

The Sikh Centre brought to the select committee the need to be sensitive to the diverse cultures and beliefs of individuals passing through airport control to ensure they are not unfairly targeted during security checks.

For their part, the Sikh community has made a conscious decision to recommend that their kirpans (ceremonial knives) are removed from under their robes, and included in their check-in luggage. Contrary to perhaps the preconceived view, the kirpan should be appreciated as a symbol of the Sikh religion, not a weapon.

Similar discussion from New Zealand parliament.

In addition kirpans are not allowed on the planes as per new TSA regulations.

One thing in common to all these is that they refer to security rules imposed by ICAO with respect to carrying weapons on airplanes.

And if this is not enough you can take a look at the FAQ from Sikh Coalition Section on Kirpan:

What happens when a Sikh goes on an airplane? Do you have to take the kirpan off?

At the present time, Sikhs put their kirpans into check-in luggage and do not carry it with them on an airplane.