Bibliographic record
Abstract
In this thesis we delve into the computation of the Gassner invariant for string links, which are a more generalized form than braids, utilizing a (co)homological approach. We restrict this (co)homology invariant, denoted as $\mathcal{G}_h$, to pure braids, leading to the derivation of the Gassner representation. We introduce the concept of "flying cars," which assigns an invariant $\mathcal{C}(L)$ to an $(n+1)$-component string link $L$. This invariant, an $n \times n$ matrix, has entries in the field $\mathbb{Q}(t_0, t_1,\ldots, t_n)$. We establishes a connection between the invariant $\mathcal{C}(L)$ and the homology Gassner invariant $\mathcal{G}_{h}\left(L\right)$ of $L$ through the formula $\mathcal{G}_{h}\left(L\right) = \left(D_n \cdot \mathcal{C}\left(L\right) \cdot D_n^{-1}\right)//\rho_{col}//{m^t}$. Here, $D_n$ is a diagonal matrix, $m^t$ denotes matrix transpose, and $\rho_{col}$ represents column permutation. We prove that $\mathcal{C}(L)$ is indeed an invariant of string links under the Reidemeister moves, thereby directly verifying the invariance of the homology Gassner invariant. Moreover, we provide formulas for the intersection product $\mu:=\langle-, -\rangle: H_1(P;\mathcal{F}) \times H_1(P;\mathcal{F}) \rightarrow \mathcal{F}$, which is defined on the cycles of the homology group $H_1(P;\mathcal{F})$. In this context, $P$ is an $(n+1)$-punctured disk viewed as a subspace of the complement $X$ of an $n+1$ string link, and $\mathcal{F}$ is a local coefficient system on $X$ determined by the abelianization map $\epsilon: \pi_1(X,x_0) \rightarrow \langle t_0, t_1, \cdots, t_n \rangle$. This map takes values in the free abelian group $\langle t_0, t_1, \cdots, t_n \rangle$. We conclude by verifying that the homology Gassner invariant is unitary with respect to this intersection product.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".