Duality of Khovanov-Rozansky link homology : a dissertation presented to the Faculty of Princeton University in candidacy for the degree of Doctor of Philosophy : recommemded for acceptance by the Department of Mathematics
Gornik, Bojan
Predstavimo konstrukcijo ▫$sl(n)$▫ homologije spleta Hovanov-Rozanskega (HR), ki je invarianta spleta v obliki dvojno stopničenega ▫$\mathbb{C}$▫-prostora. Eulerjeva karakteristika invariante glede ...na eno stopničenje je ▫$n$▫-Jonesov polinom. Posplošimo Lee-jev izrek o homologiji spleta Hovanova (▫$n=2$▫) na splošen primer HR homologije: predstavimo sprektralno zaporedje, ki začne s HR homologijo spleta in konvergira k enostavnejši deformirani HR homologiji spleta. Deformirana HR homologija spleta je eksplicitno izračunana: kot ▫$\mathbb{C}$▫-prostor zavisi le od števila komponent spleta in njihovih spletnih števil; v primeru vozla ▫$K$▫ je njeno stopničenje popolnoma določeno z enim samim celim številom ▫$s_n(K)$▫, ki je invarianta vozla. Preučimo obnašanje HR homologije spleta in njene deformirane variante pri zamenjavi spleta ▫$L$▫ z njegovim zrcalnim spletom ▫$-L$▫. Pokažemo da je, glede na njeno dvojno stopničenje, sumand ▫$(i,j)$▫ HR homologije spleta ▫$L$▫ dualen sumandu ▫$(-i,-j)$▫ HR homologije spleta ▫$-L$▫ za vse ▫$i,j \in \mathbb{Z}$▫. Podobno za deformirano homologijo pokažemo, da je ▫$s_n(K)=-s_n(-K)$▫.
Type of material - dissertation
; adult, serious
Publication and manufacture - Princeton : [B. Gornik], 2008
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