I am well acquainted with the notion of absence of evidence, thank you; I touched on this point above, stating that, although absence of evidence does count as points against the case I make, positive evidence makes stronger points. Were this not the case, then physicists wouldn't be searching for the Higgs Boson; they'd be restricted to theories which are readily explained by only the particles we have evidence of.

A disproof of the Book of Mormon, then, must rest upon just that: disproof. With that in mind, let us examine further those points raised in the link you provided.

Archaeological Fallacies
First, four technologies are mentioned which were "unknown to Mesoamerica": chariots, steel swords, bellows, and silk.

An explanation of the word 'chariot' can be found here.

Many explanations have been made re: steel swords; the reference made in this case comes from the book of Ether, speaking of the Jaredites. I offer the below quote as a counter:

In light of contemporary conditions in Mesoamerica, one can understand this passage a number of ways. Although the blades of most macuahuitls in Mesoamerica were made from obsidian, the Aztecs are known to have had war clubs studded with iron instead of the usual obsidian. There are even examples in Mesoamerica of ceremonial macuahuitls with feathers replacing the obsidian blades.

Various types of material, including iron, replaced the usual obsidian of the macuahuitl, and such a weapon could thus be described as a sword with a metal "blade." Another possibility is to equate this Jaredite steel with the "steel" of the King James translation of the Old Testament, which actually refers to the Hebrew word for "bronze."

Finally, we need to understand that Mosiah translated Ether's plates into social and linguistic concepts with which he was familiar. Mosiah, as king, possessed Laban's sword, a steel weapon that was passed down as one of the insignia of royalty. In translating Ether's record, Mosiah might thus have given the Jaredite kings steel swords, like the one he himself possessed, because in Mosiah's society a king was expected to have a steel sword as his royal weapon.

Bellows are only mentioned in the locale of the old world, not in America, making this a non-point.

Regarding silk: An LDS publication, and a non-LDS publication, "Silkworm of the Aztecs" by Richard S. Peigler, Ph.D., Curator of Entomology, in Museum Quarterly, Vol. 2, No. 1 (Spring, 1993): pp. 10-11 (published by the Denver Museum of Natural History, both show evidence of silk in the Americas.

A note on cities in America comes again from Jeff Lindsay:

As for the account dealing with peoples in the New World, Book of Mormon geography best fits the Isthmus of Tehuantepec (southern Mexico, Guatemala), where a number of sites, cities, etc., have been tentatively correlated with Book of Mormon locations. The best treatment of this is in John Sorenson's An Ancient American Setting for the Book of Mormon. He offers fascinating correlations, very strong (in my opinion), though only a small fraction of the archaeological work has been done that is needed to confirm most of the specific proposals. The Lamanites in the Book of Mormon may correlate well with a part of the early Mayan civilization or one of the other cultural groups in ancient Mesoamerica. The peoples described in the Book of Ether could very well be part of the Olmec peoples from the same area. A number of Mayan legends and the few surviving writings provide interesting parallels with Book of Mormon concepts. Could write much more on this if you're interested. Bottom line: yes, there are real places and there were real people described in what is truly an authentic ancient record. But we are in our infancy when it comes to understanding Mesoamerica. Until scholars are able to do more work there, the argument from silence should be applied with caution.

Further, as noted above, the details of Lehi's journey through the Arabian peninsula have been well correlated with actual places, some with names matching those found in archaeological studies.

Anthropological fallacies

Most stunning of all, the BoM never once indicates that the American continent was anything but uninhabited when the refugees from Jerusalem arrived.

I'm sorry, but this is plainly wrong. We have known for quite some time that the Nephites were not the only inhabitants of ancient America; the Jaredites are an example attested in the Book of Mormon.

Biological fallacies
My goodness, what an intriguing question this is. I'll defer to Jeff Lindsay, who has done much work on this subject, and who has cited many good primary sources, lest there be a complaint against my using his work too many times.

Linguistic fallacies
I once again defer to Jeff Lindsay:

One of the most interesting evidences of transoceanic contact between the Old and New Worlds is the Bat Creek Hebrew inscription found by a Smithsonian expedition in Tennessee in 1889. (The Bat Creek Stone and other interesting oddities of archaeology, including pre-Columbian maize in India, can be seen at the Archaeological Outliers site.) Anti-Mormon writers such as the Tanners have spent much effort trying to argue that the writing on the Bat Creek Stone is not Hebrew. However, non-LDS scholar J. Huston McCulloch has now shown that the Bat Creek inscription, once thought to be Cherokee, "fits significantly better as Paleo-Hebrew" (J. Huston McCulloch, "The Bat Creek Inscription: Cherokee or Hebrew?" Tennessee Anthropologist, Vol. 13, Fall 1988, p. 116, as cited by Matthew Roper, Review of Books on the Book of Mormon, Vol. 4, 1992, p. 212). McCulloch's recent work confirms Cyrus Gordon's original hypothesis about the inscription, namely, that it was from between 70 A.D. and 135 A.D. and represented Old World writing (Science Vol. 2, May 1971, pp. 14-16, as cited by Paul R. Cheesman, BYU Studies, Vol. 13, No. 1, p. 85). Carbon-14 dated wood and brass bracelets associated with the inscription date to between A.D. 32 and A.D. 769 (Ibid., pp.107-12, 116) - definitely before Columbus. Cyrus Gordon, a respected non-LDS scholar, wrote:

The Bat Creek Inscription is important because it is the first scientifically authenticated pre-Columbian text in an Old World script or language found in America; and, at that, in a flawless archaeological context. It proves that some Old World people not only could, but actually did, cross the Atlantic to America before the Vikings and Columbus....The discredited pre-Columbian inscriptions in Old World scripts or languages will have to be reexamined and reevaluated, each on the merits of the evidence, case by case. (Cyrus Gordon, "A Hebrew Inscription Authenticated," in Lundquist and Ricks, eds., By Study and Also by Faith, 1:71,80, as cited by Roper, op. cit.; for more on this controversial issue, see also J. Huston McCulloch, "The Bat Creek Inscription: Did Judean Refugees Escape to Tennessee?" Biblical Archaeology Review, July/August 1993, pp. 46-53, 82, and the differing view of P. Kyle McCarter, Jr., "Let's Be Serious about the Bat Creek Stone," Biblical Archaeology Review, July/August 1993, pp. 54-55, 83.)

While critics will repeat old arguments that the Bat Creek Stone is a forgery, it is important to recognize that "there is absolutely no indication that the inscription is a forgery, in the first place, other than the circular, and therefore unscientific, argument that being Hebrew, it must surely be a fake" (J. Huston McCulloch, "The Bat Creek Stone: A Reply to Mainfort and Kwas," Tennessee Anthropologist, Vol. 18, No. 1, Spring 1993, p. 16, emphasis added, as cited by Matthew Roper, FARMS Review of Books, Vol. 9, No. 1, 1997, p. 142). David H. Kelly has also found serious evidence of several pre-Columbian inscriptions of European origin: "We need to ask . . . where we have gone wrong as archaeologists in not recognizing such an extensive European presence in the New World" (David H. Kelly, "Proto-Tifnagh and Proto-Ogham in the Americas," Review of Archaeology, Vol. 2, Spring 1990, p. 10, as cited by Roper, op. cit.). More evidence for scholarly acceptance of Old World scripts in the ancient Americas can be found in W.R. McGlone et al., Ancient American Inscriptions: Plow Marks or History? (Long Hill, Mass.: Early Sites Research Society, 1993, as cited by Sorenson, 1993, p. 21) and Jacques de Mahieu, "Corpus des inscriptions ruiniques d'Amerique du Sud," Kadath 68, Brussels, 1988, pp. 11-42 (cited by Sorenson, 1993, p. 21). More relevant research has tentatively identified hundreds of possible links between Uto-Aztecan languages (in Book of Mormon territory) with the ancient Hebrew language (work by Brian D. Stubbs, including "A Curious Element in Uto-Aztecan," The Epigraphic Society Occasional Papers, Vol. 23, 1998 [according to second-hand sources - I have not yet read this article]; "Elements of Hebrew in Uto-Aztecan: A Summary of the Data," F.A.R.M.S. paper, 1988; "Looking Over vs. Overlooking Native American Languages: Let's Void the Void," Journal of Book of Mormon Studies, Vol. 5, No. 1, Spring 1996, pp. 1-49).

It Takes a Thief...
I will not deign to justify this. Any examination of the story, from either side, will show that this is neither a rigorous disproof or in fact unreasonable.

Lost in Translation
B. F. Sperry writes a response here to the question of Harris vs. Anthon. As for the Book of Abraham, I would be remiss not to refer you to Jeff Lindsay's excellent three-part piece.