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arXiv 2012.148012020-12-29math.GT

The concordance crosscap number and rational Witt span of a knot

Stanislav Jabuka

备注：: 31 pages, 3 figures
发表信息：: Pacific J. Math. 318 (2022) 375-400
DOI：: 10.2140/pjm.2022.318.375

英文摘要：

The concordance crosscap number $γ_c(K)$ of a knot $K$ is the smallest crosscap number $γ_3(K')$ of any knot $K'$ concordant to $K$ (and with $γ_3(K')$ defined as the least first Betti number of any nonorientable surface $Σ$ embedded in $S^3$ with boundary $K'$). This invariant has been introduced and studied by Zhang using knot determinants and signatures, and has further been studied by Livingston using the Alexander polynomial. We show in this work that the rational Witt class of a knot can be used to obtain a lower bound on the concordance crosscap number, by means of a new integer-valued invariant we call the Witt span of the knot.

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