Detecting Causality with Conjugation Quandles over Dihedral Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912569230360576 |
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| author | Fan, Zining |
| author_facet | Fan, Zining |
| contents | We study whether quandle colorings can detect causality of events for links realized as skies in a $(2+1)$-dimensional globally hyperbolic spacetime $X$. Building off the Allen--Swenberg paper in which their $2$-sky link was conjectured to be causally related, they showed that the Alexander--Conway polynomial does not distinguish that link from the connected sum of two Hopf links, corresponding to two causally unrelated events. We ask whether the Alexander--Conway polynomial together with different types of quandle invariants suffice. We show that the conjugation quandle of the dihedral group $D_5$, together with the Alexander--Conway polynomial, does distinguish the two links and hence likely does detect causality in $X$. The $2$-sky link shares the same Alexander--Conway polynomial but has different $D_5$ conjugation-quandle counting invariants. Moreover, for $D_5$ the counting invariant alone already separates the pair, whereas for other small dihedral groups $D_3$, $D_4$, $D_6$, $D_7$ even the enhanced counting polynomial fails to detect causality. In fact we prove more that the conjugation quandle over D5 distinguishes all the infinitely many Allen-Swenberg links from the connected sum of two Hopf links. These results present an interesting reality where only the conjugation quandle over $D_5$ coupled with the Alexander--Conway polynomial can detect causality in $(2+1)$ dimensions. This results in a simple, computable quandle that can determine causality via the counting invariant alone, rather than reaching for more complicated counting polynomials and cocycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Detecting Causality with Conjugation Quandles over Dihedral Groups Fan, Zining Geometric Topology General Relativity and Quantum Cosmology Primary 57K12, Secondary 57K10, 83C75 We study whether quandle colorings can detect causality of events for links realized as skies in a $(2+1)$-dimensional globally hyperbolic spacetime $X$. Building off the Allen--Swenberg paper in which their $2$-sky link was conjectured to be causally related, they showed that the Alexander--Conway polynomial does not distinguish that link from the connected sum of two Hopf links, corresponding to two causally unrelated events. We ask whether the Alexander--Conway polynomial together with different types of quandle invariants suffice. We show that the conjugation quandle of the dihedral group $D_5$, together with the Alexander--Conway polynomial, does distinguish the two links and hence likely does detect causality in $X$. The $2$-sky link shares the same Alexander--Conway polynomial but has different $D_5$ conjugation-quandle counting invariants. Moreover, for $D_5$ the counting invariant alone already separates the pair, whereas for other small dihedral groups $D_3$, $D_4$, $D_6$, $D_7$ even the enhanced counting polynomial fails to detect causality. In fact we prove more that the conjugation quandle over D5 distinguishes all the infinitely many Allen-Swenberg links from the connected sum of two Hopf links. These results present an interesting reality where only the conjugation quandle over $D_5$ coupled with the Alexander--Conway polynomial can detect causality in $(2+1)$ dimensions. This results in a simple, computable quandle that can determine causality via the counting invariant alone, rather than reaching for more complicated counting polynomials and cocycles. |
| title | Detecting Causality with Conjugation Quandles over Dihedral Groups |
| topic | Geometric Topology General Relativity and Quantum Cosmology Primary 57K12, Secondary 57K10, 83C75 |
| url | https://arxiv.org/abs/2509.03544 |