class SYMMET |
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**** | check if knot has symmetries in Murasugi's method.
c.f. Murasugi, K.,"On periodic knots." ______Comment.Math.Helv.,46(1971),162-174. $K$ be a knot, $K'$ a knot obtain from $K$ dividing by knot symmetry of period $p$. $p$:prime. $A$ :Alexander polynomial of K. $Ad$ :Alexander polynomials of K'. Then: (a) $ A = Ad^{p} * Ri^{p-1} $ in $ Z_p<t>$ and (b) $Ad | A$ in $Z<t>$. where $R_i = 1 + t + t^2 + t^3 +....+ t^(i-1)$ , $gcd(i,p)=1$ . Search pair p and polynomial Ad which admit (a),(b). So knot K can have period p. (1) If A=1 then p:=every prime, Ad:=1. So display "p=0". Let p:prime. (Set p from prime table and check followings.) (2) Let $A_p$ be $A$ in $Z_p<t>$. $A_p$ be $A (mod) Z_p$. If $A_p$=1 then let $Ad_p=1$ (on $Z_p$), ($Ad_p is $Ad (mod) Z_p$ _ (3) search $R_i$ s.t. $R_i^{p-1} | Ap$ (in $Z_p<t>$). Let $Aq := Ap/Ri^(p-1)$. (in $Z_p<t>$ ) $Aq$ is $Ad^{p}$ in (a). _ (Aftre here compute in the form of Conway polynomial in this program) (4) Search $Ad_p$ s.t. $Ad_p^p = Aq$ (in $Z_p<t>$ ). ( Now, $Ad_p$ satisfy (a).) _ Make $Ad$ satisfy (b), and $Ad_p = Ad (mod) Z_p$ (5) Check if $Ad | A$ in Z<t> . |
shared ApolyP:POLYS_INTI; |
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**** | Alexander mod prime |
shared Cdp1:POLYS_INTI; |
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**** | degree of Deg0, DegDP. copy of CpS0, Cdp. |
shared Cpoly:POLYS_INTI; |
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**** | Conway |
shared CpolyDP,CpolyQ:POLYS_INTI; |
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shared CpolyDP,CpolyQ:POLYS_INTI; |
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shared DegCDP:INT; |
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shared DegM:INT; |
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**** |
shared DegR:INT; |
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shared PolyD:POLYS_INTI; |
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**** | poly. of K' on Z |
shared RPoly:POLYS_INTI; |
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**** | - 1 1 1 1 1 ....1 |
shared TblS:POLYS_INTI; |
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**** | work table TblS,---DegM |
shared bottom:INTI; |
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shared cyclic:BOOL; |
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shared prime:INTI; |
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**** |
shared ApolyP:POLYS_INTI; |
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**** | Alexander mod prime |
shared Cdp1:POLYS_INTI; |
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**** | degree of Deg0, DegDP. copy of CpS0, Cdp. |
shared Cpoly:POLYS_INTI; |
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**** | Conway |
shared CpolyDP,CpolyQ:POLYS_INTI; |
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shared CpolyDP,CpolyQ:POLYS_INTI; |
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shared DegCDP:INT; |
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shared DegM:INT; |
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**** |
shared DegR:INT; |
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shared PolyD:POLYS_INTI; |
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**** | poly. of K' on Z |
shared RPoly:POLYS_INTI; |
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**** | - 1 1 1 1 1 ....1 |
shared TblS:POLYS_INTI; |
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**** | work table TblS,---DegM |
shared bottom:INTI; |
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shared cyclic:BOOL; |
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shared prime:INTI; |
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**** |
Alex2Conway(a:POLYS_INTI,out p:POLYS_INTI) |
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**** | Convert from Alexander poly. to compressed Conway poly. On knot, Deg is even.On Conway poly. s be z^2 , and set p[s], |
Conway2Alex(c:POLYS_INTI, out a:POLYS_INTI) |
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**** | convert from compressed Conway poly. to Alexander poly. On knot, Deg is even.On Conway poly. s be z^2 , and set p[s], |
Filter |
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Filter_SetPoly(d:INT) |
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**** |
_Recursive_call_-- We compute as Conway poly. So, must convert Alexander polynomial to Conway poly. |
Find |
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Restore(Cdp:POLYS_INTI) |
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**** | When Cdp( Conway polynomial for K') is known in Zp[t]. Re-construct Conway polynomial in Z[t]. Note that Cdp|Cpoly in Z[t]. |
Restore_CheckDiv |
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**** | Check if Ad | Aq ( PolyD | Cpoly) and display. |
Restore_SetPolyD:BOOL |
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**** | true if Setset polynomial Cd=PolyD[] from TblS[]Cdp1(x) : Conway poly. of K' mod Z_px=i+bottom.Reconstruct PolyD Conway poly. for K' using TblS[x]=(Cd(x)-Cdp1)DIV prime. |
Restore_SetVal(ite:INT) |
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**** | change x= ite+bottom. And compute Cpoly(x), Cdp1(x). Make Cd(x) and TblS[ite]=(Cd(x)-Cdp1(x)) DIV prime. |
TestSym(apoly:POLYS_INTI) |
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**** | Main function. Test if apoly has Murasugi's condition on prime. |
divide_Zp(Poly0,PolyDv:POLYS_INTI, out PolyQ:POLYS_INTI):BOOL |
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**** | true if PolyDv | Poly0. PolyQ:=Poly0/PolyDv |