Ribbon distance bounds from Bar-Natan Homology and $α$-Homology

Gujral, Onkar Singh

doi:10.48550/arxiv.2011.01190

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Further Properties and Generalizations Of λ1

Torsion Orders1

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2021

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“…Then, further papers followed: Caprau-González-Lee-Lowrance-Sazdanović-Zhang generalized Alishahi and Dowlin's work for Q to the fields F p for all odd primes p [CGL + 20]. Gujral [Guj20], using α-homology, defined an invariant ν which can be seen to equal our invariant u G , and showed that it provides a lower bound for the ribbon distance between knots; this was a generalization of earlier work by Sarkar [Sar20]. Here, the ribbon distance between two smoothly concordant knots K and J is the minimal k such that there is a sequence K = K 1 , .…”

Section: Further Properties and Generalizations Of λmentioning

confidence: 99%

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Khovanov homology and rational unknotting

Iltgen¹,

Lewark²,

Marino³

2021

Preprint

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Building on work by Alishahi-Dowlin, we extract a new knot invariant λ ≥ 0 from universal Khovanov homology. While λ is a lower bound for the unknotting number, in fact more is true: λ is a lower bound for the proper rational unknotting number (the minimal number of rational tangle replacements preserving connectivity necessary to relate a knot to the unknot). Moreover, we show that for all n ≥ 0, there exists a knot K with λ(K) = n. Along the way, following Thompson, we compute the Bar-Natan complexes of rational tangles.

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Section: Further Properties and Generalizations Of λmentioning

confidence: 99%

“…In a recent paper, Gujral [Guj20] introduced a lower bound ν for the ribbon distance in terms of the maximal order of (2X −(α 1 +α 2 ))-torsion in the α-homology of a knot. We have that α-homology, first described in [KR20], is equivalent to our Z[G]-homology, and the invariant ν is equal to u G .…”

Section: Torsion Ordersmentioning

confidence: 99%