A genetic map is a linear representation of the genes of a chromosome deciphered from the distances between marker loci. The distance is usually a representation of recombination fraction also called as map units. The measurement unit is a centimorgan.
Recombination fraction is the proportion of recombinant chromosomes between the two loci. 1 centimorgan is the distance between two loci in which 1% recombination is found.

Direct adoption of recombination fractions as distance between genes can be applied to loci which are situated closer on a chromosome. However when the distance increases the chances of crossovers are higher and hence the simple adaptation is not sufficient to calculate the distance between the loci. Also the double or even numbered cross overs result in the same progeny as the parental line and hence go unnoticed and doesn't get counted among the recombinants. This largely underestimates the recombination fraction and hence distorts the genetic map. There are issues for mapping three or more points in a genome since the recombination fractions are not additive in nature.

Interference is the effect of one recombination event on the adjacent crossover sites of a gene.

A map function was thus introduced as an error correction methodology in construction of genetic maps. It is a mathematical relation between the probability of recombination and map units. However, the existing map functions do have some limitations and need to be further modified or analyzed based on the observable data.

Map functions relate the distance between loci and the recombination fraction by the equation
R= M (d) where M is the mapping function, r- recombination fraction and d- distance between pairs of loci on a chromosome.

1. Haldane's Mapping function

This is the simplest of the lot assuming the number of crossovers to be in Poisson distribution. This also assumes interference to be nil.

dM = -1/2 ln (1-2r)
Where dM is the distance between the marker loci, r is the recombination frequency
dM is expressed in Morgans. From this, r can be calculated as

r = ½ (1-exp (-2dM))

For smaller distances where recombination frequencies are more predictable, dM= r. when the distance is larger r can take values up to ½.
The disadvantage is the non conformity of the recombination data to the expected Poisson distribution which is underlying the mechanism of Haldane's mapping function. Map distances and recombination frequencies are also found to follow no predictable relations.

2. Kosambi's map function

Kosambi's function considers the number of double crossovers and interference. The interference level is similar to that found in humans. The function is depicted as

d = ¼ ln[(1+2r)/(1-2r)]

where d- distance between markers and r is the recombination fraction. d is calculated as 'Kosambi estimate which can be converted into centiMorgans by multiplying with 100 for construction of linkage maps.

It can also be represented as
r = (1/2) ( e4d -1) / ( e4d +1)
But the Kosambi's can not be extended to more than three loci while calculating joint recombination probabilities. When the recombination fraction is ½, then the Haldane's and Kosambi's mapping functions are equivalent.

Kosambi's map function can also be represented as
rAC = rAB + rBC - 2CrABrBC
rAC is the recombination frequency between A and C
rAB is recombination frequency between A and B
rBC is the recombination frequency between B and C
C is the Coefficient of coincidence.

The interference, I is nil when C is 1 since I = 1-C.

3. Carter-Falconer function:

Although the Kosambi's function considered the interference between the recombination events, the mapping function could not be extended to all situations and organisms. The number of recombinants is not always the same. It varied between the different sex and different organisms. For example the female chromosomes are more prone to recombination at certain stretches of the genome. The figure is usually lower in heterogametic sex.

Carter and Falconer function allows for higher levels of interference according to the linkage studies conducted in mice.
It is represented as
m = ¼ {1/2 [ln(1+2r)-ln(1-2r)] + tan-1 (2r)}

4. Morgan map function:
Morgan's map function assumes a single crossover between adjacent loci.
For example, consider mapping of three points A, B, and C which are in sequence in a gene, the recombination frequency between A and C is not equal to sum of recombination frequencies between AB and BC. The mapping distances are additive but the recombination fractions are not. The distance AC as calculated from recombination frequency depends on the interference.

r (AC) = r(AB)+r(BC)

Thus the distance on map is considered to be equivalent to the recombination fraction which is not always true.
The genetic mapping software such as LINKAGE, QTL Cartographer, Mapmaker etc have provisions for calculating different map functions and making a comparative analysis while generating linkage maps. Genetic algorithms which combine these functions with corrections are also widely used.