Expressing signatures using Heavyside theta’s at their jump points; also with $\omega$.

ω2[v_][p_]:=Module[{q=Expand[p],n,c},
If[q===0,0,c=Coefficient[q,ω,n=Exponent[q,ω]];c (v^n)+ω2[v][q-c(((ω+(ω^(-1))))^n)]]];

ω2[v][-5+(1/(ω^3))-(3/(ω^2))+(5/ω)+5 ω-3 (ω^2)+(ω^3)]

θ[x_]/;NumberQ[x]:=HeavisideTheta[x];
sign[ℰ_]:=Module[{num,den,v,p,rs,d,k},
{num,den}=NumeratorDenominator[ℰ];{num,den}/=(ω^(Exponent[num,ω]/2));
p=Factor[Times@@(ω2[v]/@{num,den})/.v->2(2(u^2)-1)];
rs=Solve[p==0,u,Reals];
If[rs==={}, Return[Sign[p/.u->0]]];
rs=Union@(u/.rs);
Sign[Coefficient[p,u,Exponent[p,u]]](((-1))^(Exponent[p,u]))+Sum[
k=1; While[(d=RootReduce[D[p,{u,k}]/.u->r])==0, ++k];
If[EvenQ[k],0,2Sign[d]]θ[u-r],
{r,rs}]
]

sign[1+(u^2)]

f=Expand[(1/2) (4+sign[-2 (-1+2 (u^2))]+sign[(2/3) (-3+4 (u^2))]+sign[((-7+8 (u^2))/(2 (-3+4 (u^2))))]+sign[-((2 (8-23 (u^2)+16 (u^4)))/(-7+8 (u^2)))]+sign[-(((((-3+4 (u^2)))^2) (11-28 (u^2)+16 (u^4)))/(-44+155 (u^2)-176 (u^4)+64 (u^6)))]+sign[((-44+155 (u^2)-176 (u^4)+64 (u^6))/(8-23 (u^2)+16 (u^4)))]+sign[-((-11+76 (u^2)-128 (u^4)+64 (u^6))/(11-28 (u^2)+16 (u^4)))]+sign[-(((-29+160 (u^2)-256 (u^4)+128 (u^6)) (11-170 (u^2)+544 (u^4)-640 (u^6)+256 (u^8)))/(4 (-1+2 (u^2)) (((-3+4 (u^2)))^2) (-11+76 (u^2)-128 (u^4)+64 (u^6))))]+sign[-(((-3+4 (u^2)) (-23+152 (u^2)-256 (u^4)+128 (u^6)))/(11-228 (u^2)+864 (u^4)-1152 (u^6)+512 (u^8)))]+sign[-(((-3+4 (u^2)) (-23+152 (u^2)-256 (u^4)+128 (u^6)) (11-228 (u^2)+864 (u^4)-1152 (u^6)+512 (u^8)))/(2 (-29+160 (u^2)-256 (u^4)+128 (u^6)) (11-170 (u^2)+544 (u^4)-640 (u^6)+256 (u^8))))])
]
Plot[f,{u,-1,1},Exclusions->None]

sign[((1-3 ω+5 (ω^2)-5 (ω^3)+5 (ω^4)-3 (ω^5)+(ω^6))/(ω (1-2 ω+(ω^2)-2 (ω^3)+(ω^4))))]

ℰ=-(((-29+160 (u^2)-256 (u^4)+128 (u^6)) (11-170 (u^2)+544 (u^4)-640 (u^6)+256 (u^8)))/(4 (-1+2 (u^2)) (((-3+4 (u^2)))^2) (-11+76 (u^2)-128 (u^4)+64 (u^6))))

p=Expand[Numerator[ℰ]Denominator[ℰ]]

roots=Union@(u/.Solve[p==0,u,Reals])

Table[RootReduce[p/.u->r]>0,{r,roots}]

RootReduce@Table[p/.u->r,{r,roots}]

Sum[
k=1; While[(d=RootReduce[D[p,{u,k}]/.u->r])==0, ++k];
If[EvenQ[k],0,Sign[d]]θ[r],
{r,roots}]

KasSig[Knot[10,1]]

Table[K->Expand@KasSig[K],{K,AllKnots[{3,8}]}]//Column

Cos[5ArcCos[x]]//TrigExpand