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A lifting defect for oriented t3 and t̅6 moves

  • Wenbiao Liu
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Abstract

We study oriented t3- and t̅6-moves through the problem of lifting Fox $3$-colorings to colorings by a nine-element Alexander quandle. We define a nonnegative integer lifting defect δ for oriented links and prove that it is invariant under these moves. For knots, the defect equals the rank of an alternating bilinear form obtained from a Seifert matrix, so it is even. We also prove that it is additive under connected sum. Let Ks be the connected sum of s≥ 1 copies of 946. Then δ(Ks)=2s, so these knots represent infinitely many distinct equivalence classes under the oriented moves, and none is equivalent to an unlink. After forgetting orientation, each Ks can be transformed into an unlink with 2s+1 components by 3s ordinary $3$-moves.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026lifting,
  title = {{A lifting defect for oriented $t_3$ and $\bar t_6$ moves}},
  author = {Wenbiao Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34831},
  eprint = {2610.34831},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34831},
}
RIS
TY  - UNPB
TI  - A lifting defect for oriented t_3 and t̅_6 moves
AU  - Wenbiao Liu
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.34831
UR  - https://arxiv.org/abs/2610.34831
AB  - We study oriented t_3- and t̅_6-moves through the problem of lifting Fox $3$-colorings to colorings by a nine-element Alexander quandle. We define a nonnegative integer lifting defect δ for oriented links and prove that it is invariant under these moves. For knots, the defect equals the rank of an alternating bilinear form obtained from a Seifert matrix, so it is even. We also prove that it is additive under connected sum. Let K_s be the connected sum of s≥ 1 copies of 9_46. Then δ(K_s)=2s, so these knots represent infinitely many distinct equivalence classes under the oriented moves, and none is equivalent to an unlink. After forgetting orientation, each K_s can be transformed into an unlink with 2s+1 components by 3s ordinary $3$-moves.
ER  -