On Certain Branched Cyclic Covers of S3

2020 ◽  
pp. 43-46
Author(s):  
Mark D. Baker
Keyword(s):  
Cyclic Covers ◽  
Branched Cyclic Covers

scholarly journals A remark on branched cyclic covers

1993 ◽  
Vol 87 (3) ◽  
pp. 237-240
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Cyclic Covers ◽  
Branched Cyclic Covers

RECURSION FORMULAS FOR SOME ABELIAN KNOT INVARIANTS

2000 ◽  
Vol 09 (03) ◽  
pp. 413-422 ◽  
Author(s):  
WAYNE H. STEVENS
Keyword(s):  
Recursion Formula ◽  
Alexander Polynomial ◽  
Square Roots ◽  
Knot Invariants ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Recursion Formulas ◽  
Branched Cyclic Covers ◽  
Linear Recursion ◽  
Fold Cover

Let K be a tame knot in S3. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders for the odd-fold covers that contain the square roots of the orders for the even-fold covers quotiented by the order for the two-fold cover. (That these square roots are all integers follows from a theorem of Plans.)

Knots which are branched cyclic covers of only finitely many knots

1985 ◽  
Vol 98 (2) ◽  
pp. 301-304
Author(s):  
Paul Strickland
Keyword(s):  
Cyclic Cover ◽  
Cyclic Covers ◽  
Infinite Cyclic Cover ◽  
Branched Cyclic Covers

In [5] we proved two results: theorem 1, which said that if k was a simple (2q – 1)-knot, q 1, then it was equivalent to the m-fold branched cyclic cover of another knot if and only if there existed an isometry u of its Blanchfield pairing 〈,〉, whose mth power was the map induced by a generator t of the group of covering translations associated with the infinite cyclic cover of k; and theorem 2, which showed that if k were the m-fold b.c.c. of two such knots, then these would be equivalent if and only if the corresponding isometries were conjugate by an isometry of 〈,〉. Using this second result, we present two cases where k may only be the m-fold b.c.c. of finitely many knots.

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