A long-standing question by S. Lando asks whether the $\mathfrak{sl}(2)$-weight system extends to a unique 4-invariant of graphs. We show that, in full generality, the answer to this question is negative. However, for certain specializations of the weight system, extensions do exist. Explicit formulae for computing two such specializations of the weight system are already known. We construct recurrence relations for one additional such extension and discuss the last remaining specialization, which conjecturally admits an extension. We also study the polynomial coefficients of the $\mathfrak{sl}(2)$-weight system and resolve the questions concerning their extension.
We prove that the algebra of multiple Eisenstein series is an $\mathfrak{sl}_2$-algebra. In particular, we show that it is graded by weight, which also holds after taking complex linear spans. Further, we identify it as a graded $\mathbb{Q}$-algebra with the associated graded algebra of $q$-analogues of multiple zeta v...
We construct a family of finite local rings whose annihilating-ideal graphs are naturally described by orthogonality of subspaces of $\mathbb{F}_2^n$. For $n=4$ we determine the clique and chromatic numbers exactly and obtain \[ \omega(\AG(R_4))=5<6=\chi(\AG(R_4)). \] Thus $\AG(R_4)$ is not weakly perfect, and the Behb...
For $n\geq3$, the group $\mathrm{SL}_n(\mathbb Z)$ has property $(T)$, so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when th...
We show that the subfactor planar algebra with principal graph $D_\infty$ is the Brauer planar algebra with bubble constant $\delta=2$. The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At $\delta=2$, it relates to the category of representations of the orthogonal group $O(2)$. We...
Let $\mathcal F=\{2K_2,C_4,C_5\}$, so that $\operatorname{Forb}(\mathcal F)$ is the class of split graphs. Resolving a conjecture of Wang and Tang, we prove that the class of split graphs is generalized spectrally closed and that it admits no walk-realizable $\mathcal F$-supporter of any finite order.
We prove that there is no uniform description of $\mathrm{LG}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita--Terasaka knots by $\mat...
Matthew Harper, Ben-Michael Kohli· 0 citations
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