Polygonum

Keys available at Flora of Pakistan
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Polygonum arenastrum Boreau (Macaronesia, NW. Africa, Temp. Eurasia: Afghanistan, Albania, Algeria, Altay, Amur, Austria, Azores, Baleares, Baltic States, Belarus, Belgium, Bulgaria, Buryatiya, Central European Rus, Chita, Corse, Czechoslovakia, Denmark, East Aegean Is., East European Russia, Finland, France, Germany, Greece, Hungary, Iran, Iraq, Irkutsk, Italy, Kamchatka, Khabarovsk, Krasnoyarsk, Kriti, Krym, Kuril Is., Madeira, Magadan, Manchuria, Mongolia, Morocco, Netherlands, North Caucasus, North European Russi, Northwest European R, Norway, Pakistan, Palestine, Poland, Portugal, Primorye, Romania, Sakhalin, Sardegna, Sicilia, Sinai, South European Russi, Spain, Sweden, Switzerland, Transcaucasus, Tunisia, Turkey, Turkey-in-Europe, Turkmenistan, Tuva, Ukraine, West Himalaya, West Siberia, Yakutskiya, Yugoslavia; Introduced into: Alabama, Alaska, Alberta, Arizona, Arkansas, British Columbia, California, Chile Central, Chile South, Colorado, Connecticut, Delaware, Florida, Georgia, Great Britain, Idaho, Illinois, Indiana, Iowa, Ireland, Kansas, Kentucky, Labrador, Louisiana, Maine, Manitoba, Maryland, Massachusetts, Mexico Northwest, Michigan, Minnesota, Missouri, Montana, Nebraska, Nevada, New Brunswick, New Hampshire, New Jersey, New Mexico, New York, New Zealand North, Newfoundland, North Carolina, North Dakota, Northwest Territorie, Nova Scotia, Ohio, Oklahoma, Ontario, Oregon, Pennsylvania, Prince Edward I., Québec, Rhode I., Saskatchewan, South Carolina, South Dakota, Tennessee, Texas, Utah, Vermont, Virginia, Washington, West Virginia, Wisconsin, Wyoming, Zimbabwe as per POWO)

Polygonum aviculare L. (Temp. Northern Hemisphere, Macaronesia to Eritrea: Afghanistan, Alabama, Alaska, Albania, Alberta, Algeria, Altay, Amur, Arizona, Arkansas, Austria, Azores, Baleares, Baltic States, Belarus, Belgium, British Columbia, Bulgaria, Buryatiya, California, Canary Is., Central European Rus, China North-Central, China South-Central, China Southeast, Chita, Colorado, Connecticut, Corse, Costa Rica, Cyprus, Czechoslovakia, Delaware, Denmark, District of Columbia, East Aegean Is., East European Russia, East Himalaya, Egypt, Eritrea, Ethiopia, Finland, Florida, France, Føroyar, Georgia, Germany, Great Britain, Greece, Hainan, Hungary, Iceland, Idaho, Illinois, Indiana, Inner Mongolia, Iowa, Iran, Iraq, Ireland, Irkutsk, Italy, Kamchatka, Kansas, Kazakhstan, Kentucky, Khabarovsk, Kirgizstan, Korea, Krasnoyarsk, Kriti, Krym, Kuril Is., Labrador, Lebanon-Syria, Libya, Louisiana, Madeira, Magadan, Maine, Manchuria, Manitoba, Maryland, Michigan, Minnesota, Mississippi, Missouri, Montana, Morocco, Myanmar, Nebraska, Nepal, Netherlands, Nevada, New Brunswick, New Hampshire, New Jersey, New Mexico, New York, North Carolina, North Caucasus, North Dakota, North European Russi, Northwest European R, Northwest Territorie, Norway, Nova Scotia, Ohio, Oklahoma, Ontario, Oregon, Pakistan, Palestine, Pennsylvania, Poland, Portugal, Primorye, Qinghai, Québec, Rhode I., Romania, Sakhalin, Sardegna, Saskatchewan, Saudi Arabia, Sicilia, South Carolina, South Dakota, South European Russi, Spain, Sweden, Switzerland, Tadzhikistan, Taiwan, Tennessee, Texas, Tibet, Transcaucasus, Tunisia, Turkey, Turkey-in-Europe, Turkmenistan, Ukraine, Utah, Uzbekistan, Vermont, Virginia, Washington, West Himalaya, West Siberia, West Virginia, Wisconsin, Wyoming, Xinjiang, Yakutskiya, Yemen, Yugoslavia, Yukon; Introduced into: Angola, Argentina Northeast, Argentina Northwest, Argentina South, Bolivia, Brazil South, Brazil Southeast, Cape Provinces, Chad, Chile Central, Chile North, Chile South, Colombia, Ecuador, Falkland Is., Free State, Guatemala, Hawaii, Juan Fernández Is., Kenya, KwaZulu-Natal, Lesotho, Massachusetts, Mexico Central, Mexico Northeast, Mexico Northwest, Namibia, New Zealand North, Newfoundland, Northern Provinces, Paraguay, Prince Edward I., Svalbard, Tristan da Cunha, Uruguay, Zimbabwe as per POWO)

Polygonum cognatum Meisn. (E. Medit. to Mongolia and W. Himalaya: Altay, Inner Mongolia, Iran, Iraq, Lebanon-Syria, Mongolia, North Caucasus, Pakistan, Palestine, Transcaucasus, Turkey, Turkmenistan, West Himalaya, Xinjiang as per POWO)

Polygonum effusum Meisn. (Stamens are generally 5 in P. plebeium… here I can clearly see 8 stamens, distinctly pedicellate flowers exserted from ochrea and equal size tepals. Hence I think your plant is Polygonum effusum Meisn.)
(Afghanistan to Indian Subcontinent: Afghanistan, Assam, Bangladesh, India, Pakistan, West Himalaya as per POWO)

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As per efi thread :
Treatments of Polygonum and related genera (subfamily Polygonoideae) those by Haraldson and Ronse Decraene & Akeroyd, have segregated species of Polygonum in the broad sense adopted into two separate tribes, Polygoneae and Persicarieae. The remainder of Polygonum in the broad sense is generally assigned to Persicaria (or a varying number of segregate genera, such as Aconogonon and Bistorta) which, with Antenoron and Koenigia, comprises the tribe Persicarieae.
Characters for segregation of Two species:
Polygonum: . Flowers axillary, solitary or in small clusters, sometimes forming lax leafy spikes. Perianth (4- or)5-parted, not enlarged in fruit. Stamens 5-8, reduced in number in outer whorl; filaments, especially the inner, strongly dilated at base.
Persicaria: Inflorescence spicate, spikes usually several. Tepals fused for about 1/3 their length. Nectaries free, globular. Styles 1, bifid or trifid.

From Floral Database of Tamilnadu : 

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FRLHT Indian Medicinal Plants Nomenclature Database with images/ herbarium, distribution, local names etc. :

Polygonum thymifolium

I have updated eFI (efloraofindia) page on Aconogonon, Bistorta, Fagopyrum, Fallopia, Koenigia, Persicaria, Polygonum & Rubrivena

Attempts have been made to incorporate most of the species available in India & nearby areas with details & keys directly or through links as far as possible. It’s quite possible that there may be some discrepancy in the accepted names & synonyms taken from other links.

Species discussed so far in efloraofindia are given at the bottom of the page in the form of links against Subpages. On clicking them one can see all the details.

If someone can provide complete list of Indian species with source references it will be wonderful.

Any comments/ corrections are welcome.

In my opinion most of the above genera should be treated under Polygonum sensu lato.  I am working on Indian Polygonaceae but it needs thorough field work. The taxonomic complexes are to be solved with the help of SEM/Molecular studies. I have listed all Indian species but it is not yet published.