TY - JOUR
T1 - The Alexander and Jones polynomials through representations of rook algebras
AU - Bigelow, Stephen
AU - Ramos, Eric
AU - Yi, Ren
PY - 2012/10
Y1 - 2012/10
N2 - In the 1920's Artin defined the braid group, B n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.
AB - In the 1920's Artin defined the braid group, B n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.
KW - Alexander polynomial
KW - Hecke algebra
KW - Jones polynomial
KW - Knot invariants
KW - planar rook algebra
KW - representations of braid groups
UR - http://www.scopus.com/inward/record.url?scp=84866626355&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84866626355&partnerID=8YFLogxK
U2 - 10.1142/S0218216512501143
DO - 10.1142/S0218216512501143
M3 - Article
AN - SCOPUS:84866626355
SN - 0218-2165
VL - 21
JO - Journal of Knot Theory and its Ramifications
JF - Journal of Knot Theory and its Ramifications
IS - 12
M1 - 1250114
ER -