New invariants for virtual knots via spanning surfaces
We define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
World Scientific Publishing
2024
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| _version_ | 1811140015225307136 |
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| author | Juhasz, A Ogasa, E Kauffman, LH |
| author_facet | Juhasz, A Ogasa, E Kauffman, LH |
| author_sort | Juhasz, A |
| collection | OXFORD |
| description | We define three different types of spanning surfaces for knots in thickened surfaces.
We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct
a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface
from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology
for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative.
Finally, we use the Behrens–Golla δ-invariant to obstruct a knot from being a stabilization of
another. |
| first_indexed | 2024-04-23T08:25:22Z |
| format | Journal article |
| id | oxford-uuid:47481e71-dff0-4cf8-bc43-8d86fc878884 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:15:15Z |
| publishDate | 2024 |
| publisher | World Scientific Publishing |
| record_format | dspace |
| spelling | oxford-uuid:47481e71-dff0-4cf8-bc43-8d86fc8788842024-07-18T10:48:46ZNew invariants for virtual knots via spanning surfacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:47481e71-dff0-4cf8-bc43-8d86fc878884EnglishSymplectic ElementsWorld Scientific Publishing2024Juhasz, AOgasa, EKauffman, LHWe define three different types of spanning surfaces for knots in thickened surfaces. We use these to introduce new Seifert matrices, Alexander-type polynomials, genera, and a signature invariant. One of these Alexander polynomials extends to virtual knots and can obstruct a virtual knot from being classical. Furthermore, it can distinguish a knot in a thickened surface from its mirror up to isotopy. We also propose several constructions of Heegaard Floer homology for knots in thickened surfaces, and give examples why they are not stabilization invariant. However, we can define Floer homology for virtual knots by taking a minimal genus representative. Finally, we use the Behrens–Golla δ-invariant to obstruct a knot from being a stabilization of another. |
| spellingShingle | Juhasz, A Ogasa, E Kauffman, LH New invariants for virtual knots via spanning surfaces |
| title | New invariants for virtual knots via spanning surfaces |
| title_full | New invariants for virtual knots via spanning surfaces |
| title_fullStr | New invariants for virtual knots via spanning surfaces |
| title_full_unstemmed | New invariants for virtual knots via spanning surfaces |
| title_short | New invariants for virtual knots via spanning surfaces |
| title_sort | new invariants for virtual knots via spanning surfaces |
| work_keys_str_mv | AT juhasza newinvariantsforvirtualknotsviaspanningsurfaces AT ogasae newinvariantsforvirtualknotsviaspanningsurfaces AT kauffmanlh newinvariantsforvirtualknotsviaspanningsurfaces |