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| Main Authors: | , , , , , , , , , , , , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.08589 |
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| _version_ | 1866912326616088576 |
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| author | Clingenpeel, Ben Dai, Zongzheng Diraviam, Gabriel Jaber, Kareem Kar, Krishnendu Liu, Ziyun Miklethun, Teo Nagampoozhy, Haritha Perry, Michael Samuelson-Lynn, Moses Seamans, Eli Wright, Ana Xie, Nicole Zou, Ruiqi Zupan, Alexander |
| author_facet | Clingenpeel, Ben Dai, Zongzheng Diraviam, Gabriel Jaber, Kareem Kar, Krishnendu Liu, Ziyun Miklethun, Teo Nagampoozhy, Haritha Perry, Michael Samuelson-Lynn, Moses Seamans, Eli Wright, Ana Xie, Nicole Zou, Ruiqi Zupan, Alexander |
| contents | Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If both $J$ and $J'$ are partial knots for different symmetric union presentations of the same ribbon knot $K$, the knots $J$ and $J'$ are said to be symmetrically related. Lamm proved that if $J$ and $J'$ are symmetrically related, then $\det J = \det J'$, asking whether the converse is true. In this article, we give a negative answer to Lamm's question, constructing for any natural number $m$ a family of $2^m$ knots with the same determinant but such that no two knots in the family are symmetrically related. This result is a corollary to our main theorem, that if $J$ is the partial knot in a symmetric union presentation for $K$, then $\text{col}_p(J) \leq \text{col}_p(K) \leq \frac{(\text{col}_p(J))^2}{2}$, where $\text{col}_p(\cdot )$ denotes the number of $p$-colorings of a knot. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_08589 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Colorings of symmetric unions and partial knots Clingenpeel, Ben Dai, Zongzheng Diraviam, Gabriel Jaber, Kareem Kar, Krishnendu Liu, Ziyun Miklethun, Teo Nagampoozhy, Haritha Perry, Michael Samuelson-Lynn, Moses Seamans, Eli Wright, Ana Xie, Nicole Zou, Ruiqi Zupan, Alexander Geometric Topology 57K10 Motivated by work of Kinoshita and Teraska, Lamm introduced the notion of a symmetric union, which can be constructed from a partial knot $J$ by introducing additional crossings to a diagram of $J \# -\!J$ along its axis of symmetry. If both $J$ and $J'$ are partial knots for different symmetric union presentations of the same ribbon knot $K$, the knots $J$ and $J'$ are said to be symmetrically related. Lamm proved that if $J$ and $J'$ are symmetrically related, then $\det J = \det J'$, asking whether the converse is true. In this article, we give a negative answer to Lamm's question, constructing for any natural number $m$ a family of $2^m$ knots with the same determinant but such that no two knots in the family are symmetrically related. This result is a corollary to our main theorem, that if $J$ is the partial knot in a symmetric union presentation for $K$, then $\text{col}_p(J) \leq \text{col}_p(K) \leq \frac{(\text{col}_p(J))^2}{2}$, where $\text{col}_p(\cdot )$ denotes the number of $p$-colorings of a knot. |
| title | Colorings of symmetric unions and partial knots |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2504.08589 |