二桥纽与(3,n)环面纽的无迹SU(2)特征标及Z/4瞬子分次
Traceless $\mathrm{SU}(2)$ characters and $\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots
AI总结：
本文研究二桥纽与(3,n)环面纽的无迹SU(2)特征标及Z/4瞬子分次，验证Atiyah-Floer纽猜想枕形侧的表示论数据，证明相关二分性，计算链复形与同调并重现枕形微分。
AI中文摘要：
我们整理并尽可能独立验证了Atiyah-Floer纽猜想的枕形（辛）侧所依赖的表示论数据，涉及二桥纽与(3,n)环面纽。对二桥纽b(p,q)，我们给出简短自包含证明：每个不可约无迹SU(2)表示均为二面体的；这些是在子午线角cos(2πk/p)处的(p-1)/2个二面体特征标，与q无关，且无迹Riley多项式为显式乘积φ_p(u)=∏ₖ(u+4sin²(πk/p))，首一、次数为(p-1)/2，常数项为det K。这清晰解释了Hedden-Herald-Kirk定理——该族纽的枕形同调等于约化奇异瞬子纽同调I^♮，以及为何阻碍一般猜想的八字形泡状和边界上同调在结构上无作用。对(3,n)环面纽，我们计算了完整的无迹特征簇并证明二分性：恰好有(det-1)/2个特征标是二面体的，故当n为奇数时，每个不可约无迹特征标均为非二面体的。通过双分支覆盖Σ(2,3,n)，我们从Fintushel-Stern指标与等变ρ不变量出发，独立计算了生成元的Z/4谱流分次，并与Poudel-Saveliev及Anvari的计算结果校准；当n为奇数时，分次均匀分布在1和3之间，得到链复形IC^♮(T(3,n))=(1+a,a,a,a)，其中a=-σ/4。当n≡1 mod 6时，同调等于该链复形（微分零），但当n≡5时，同调更小2且非零：I^♮的秩始终为∑ᵢ|Δ_{T(3,n)}|，而T(3,5)=P(-2,3,5)=10₁₂₄的秩为7而非9。我们重现了8₁₉=T(3,4)的第一个非零枕形微分，并将其识别为二桥纽上不存在的角部八字形双角。
英文摘要：
For every torus knot, the rank of the differential of the Daemi-Scaduto S-complex of singular instanton homology is determined by the signature and the Alexander polynomial; this follows from theorems of Li-Ye and Daemi-Scaduto. For the torus knots $T(3,n)$ the differential has rank zero or one according to $n$ modulo $6$, and its rank equals Daemi and Scaduto's Frøyshov-type invariant; they proved this in two of the four residue classes and for $n=4,5$, and Hedden, Herald and Kirk computed it for $n\le38$. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their $\mathbb{Z}/4$ gradings for two-bridge knots, whose irreducible traceless characters are all binary dihedral, and for $T(3,n)$, where at most one is. We also give a proof through Khovanov homology, using Schütz's computation for three-braids, of Li and Ye's computation of the rank of the instanton homology of $T(3,n)$, and compare the differential of the knot $8_{19}$ with the single pillowcase bigon of Hedden, Herald and Kirk.
