Epsilon and Upsilon concordance invariants — Examples and Counterexamples

The only Maslov-zero basis terms in the displayed staircase are $x_0$ and $U^{-1}{x}_2$; $x_1$ has odd Maslov parity, so no integral $U$-translate of $x_1$ has Maslov grading zero. Both even-degree terms are cycles, and $\mathrm{\partial }(U^{-1}{x}_1)=x_0+U^{-1}{x}_2$ shows they represent the same nonzero generator. Their coordinates are $(0,1)$ and $(1,0)$, each of $F_1$-weight $1\mathrm{/}2$. The sum is a boundary, so every Maslov-zero cycle representing the normalized generator has filtration $1\mathrm{/}2$. Thus ${\nu }_1=1\mathrm{/}2$ and $\mathrm{{\rm Y}}(1)=-2(1\mathrm{/}2)=-1$.

The displayed basis term $x_2$ has coordinates $(0,-1)$ and weight $F_1(x_2)=-1\mathrm{/}2$, which is less than $1\mathrm{/}2$. It has Maslov grading $-2$, so it is excluded from the Maslov-zero representative set; although it is a cycle, its homology class is not the normalized degree-zero generator. The false rule would return $-2(-1\mathrm{/}2)=+1$, contradicting step 1.1.

Under Countable Choice, [F2] gives that the diagonal grid complex has the same Upsilon value as the original invariant, so step 1.1 gives the stated grid value $-1$. The grid comparison uses the supplier prerequisites recorded in [F2].

In the vertical complex, the differential is $x_1\mapsto x_2$, so its degree-zero homology generator is $x_0$ and first appears at Alexander level $1$. The normalized comparison in [F4], after quotient by the chosen $U$ action, identifies this filtered hat complex with the specified-data complex in The Alexander filtration and tau level; hence $\tau (T_{2,3})=1$. For the exact epsilon maps at $s=1$, the target of $F_1$ contains $U^{-1}{x}_1$ with boundary $x_0$: its other differential term $U^{-1}{x}_2$ has $j<1$ and is killed there. Thus $F_1=0$. The source of $G_1$ contains the degree-zero cycle $x_0$, while $U^{-1}{x}_1$ has $i>0$ and is excluded, so that class survives and maps to the nonzero vertical class. Thus $G_1\mathrm{\ne }0$ and the exact three-case definition gives epsilon $1$ for this complex.

A Maslov-zero cycle representing the normalized total generator is either $x_0$ or $U^{-1}{x}_2$: the Maslov-zero chain group in the displayed basis is spanned by these two cycles, and $\mathrm{\partial }(U^{-1}{x}_1)=x_0+U^{-1}{x}_2$ makes their homology classes equal. Their coordinates are $(0,1)$ and $(1,0)$, so their filtration levels are $t\mathrm{/}2$ and $1-t\mathrm{/}2$. Therefore ${\nu }_t=\min (t\mathrm{/}2,1-t\mathrm{/}2)$ and ${\mathrm{{\rm Y}}}_{T_{2,3}}(t)=-2{\nu }_t$, which gives the stated two linear pieces. They agree at $t=1$, where the slope changes.

The algebraic Laurent dual of the displayed complex has generators $x_0^{\ast },x_1^{\ast },x_2^{\ast }$ at Maslov/Alexander values $(0,-1),(1,0),(2,1)$ and differential $\mathrm{\partial }{x}_0^{\ast }=U{x}_1^{\ast }$, $\mathrm{\partial }{x}_2^{\ast }=x_1^{\ast }$, $\mathrm{\partial }{x}_1^{\ast }=0$. In its vertical complex $x_2^{\ast }$ maps to $x_1^{\ast }$, leaving $x_0^{\ast }$ as generator at Alexander level $-1$. At $s=-1$, the $F_{-1}$ target kills $U{x}_1^{\ast }$ because $i<0$, so $x_0^{\ast }$ becomes a cycle; its only degree-one basis vector is $x_1^{\ast }$ with zero differential, so the class is not a boundary and $F_{-1}\mathrm{\ne }0$. The $G_{-1}$ source has only $x_0^{\ast }$ in degree zero, and its differential $U{x}_1^{\ast }$ is retained; $U{x}_2^{\ast }$ lies above $j=-1$ and is excluded. Hence $G_{-1}=0$ and the exact epsilon convention gives $-1$. The degree-zero chain space has basis $x_0^{\ast },U{x}_2^{\ast }$; their individual differentials are both $U{x}_1^{\ast }$, so the only nonzero cycle is $x_0^{\ast }+U{x}_2^{\ast }$, and it represents the normalized generator. Its two terms have coordinates $(0,-1)$ and $(-1,0)$, giving ${\nu }_t^{\ast }=\max (-t\mathrm{/}2,-1+t\mathrm{/}2)=-\min (t\mathrm{/}2,1-t\mathrm{/}2)$. Thus its Upsilon is the negative of the value in step 1.2 by an explicit cycle computation. Fact [F3] identifies the dual of the admitted trefoil complex from [F4] with its actual reverse-mirror complex, establishing both knot-level values.

Under [F5], the same calculation is obtained from the diagonal grid complex after accounting for its $W^{\otimes (n-1)}$ factors, whose nonzero t-grade shifts are nonpositive on $[0,1]$. The factor is retained and does not change the maximal non-torsion grading.