I think the readme on the download site might have the answer:
The HLA typing data of 1,267 individuals related to the 1000 Genomes Project (Table 1) covers 14 populations
encompassing 4 major ancestral groups. After specific PCR amplification, exons were sequenced by Sanger technique.
The sequences were compared to available sequence information in the HLA allele database on exons 2 and 3 for
class I and on exon 2 class II genes, therefore any polymorphism occurring in exon 4 of class I allele or exon 3
of class II gene was not investigated. Typing ambiguities between alleles were allowed since HLA-A, HLA-B, HLA-C
gene products have identical sequences in exon 2 and exon 3 antigen recognition sites. Similarly, for class II genes,
typing ambiguities occur if HLA-DRB1, HLA-DQB1 gene products have identical sequences in exon 2 antigen recognition sites.
(See Appendix S1 in supplemental material for further details). The Allele Database version used in the report is
IMGT 2.26.0 (Jul 2009), effective Feb 2010.
ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/technical/working/20140725_hla_genotypes/README_20140702_hla_diversity
The original paper has the same paragraph
http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0097282
I guess that since the sequences for exons 2 and 3 of HLA-A (in this case as it's Class I) are the same for the 7 HLA types mentioned, you would need to sequence exon 4 to obtain the exact HLA allele.
HGVS nomen from IPD-IMGT/HLA
A01:01:01:01* NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 1077C>T]
A01:01:01:02N* NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 1077C>T]
A01:04N (A01:04:01:01N) NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 627_628dupC; ; 1077C>T]
A*01:22N NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 750G>C; 751delG]
A*01:32 NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 622C>A]
A01:34N (A01:01:38L) NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 705G>A; 1077C>T]
A*01:37 NM_002116.7:c.[203G>A; 271G>A; 282G>C; 299T>C; 301G>A; 341C>A; 385T>C; 489G>A; 521C>T; 527A>C; 538T>C; 539T>G; 545C>T; 555T>G; 559A>C; 560C>G; 570G>C; 571T>G; 755C>T; 1077C>T]