Homologija Hovanova

delo diplomskega seminarja

Žiga Gladek (Author), Sašo Strle (Mentor)

Abstract

Delo se v osnovi tiče najbolj temeljnega problema teorije vozlov tj. klasifikacije vozlov. Vozel si lahko predstavljamo kot krožnico $S^1$, vloženo v evklidski prostor ${\mathbb R}^3$. Bolj splošno lahko govorimo o spletih, ki si jih lahko predstavljamo kot končno mnogo med seboj prepletenih vozlov. Za dva poljubna spleta se nato lahko vprašamo, ali je možno prvega deformirati v drugega, ne da bi ga medtem kjerkoli pretrgali? Odgovor na to vprašanje pa lahko pogosto dobimo s pomočjo spletnih invariant. To so preslikave na množici spletov, za katere velja, da poljubnima spletoma, ki se ju da deformirati drug v drugega, priredijo isto sliko. Če torej invarianta spletoma pripiše različni sliki, lahko nemudoma zaključimo, da se to ne da. Dandanes poznamo veliko primerov invariant, ena od teh je tudi homologija Hovanova, ki je dejansko invarianta na množici orientiranih spletov. Za njeno konstrukcijo so bistvena orodja iz algebraične topologije, poleg tega pa se izkaže, da je v tesni zvezi s še eno invarianto, imenovano Jonesov polinom.

Keywords

matematika;vozli;spleti;homologija;kobordizmi;verižni kompleksi;

Data

Language:	Slovenian
Year of publishing:	2021
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[Ž. Gladek]
UDC:	515.1
COBISS:	91911427
Views:	924
Downloads:	103
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Khovanov homology
Secondary abstract:	This thesis is in essence about the most fundamental problem of knot theory, that is, knot classification. We can imagine a knot as $S^1$ embedded in the three dimensional Euclidean space ${\mathbb R}^3$. More generally, we can talk about links. We can imagine these as finitely many knots, which are in some way interlaced. Given two arbitrary links, we can ask ourselves if it is possible to deform one into the other without cutting or tearing it anywhere in the process. We can often give an answer to this question with the use of link invariants. Link invariants are maps on the set of links, which map any two links that can be deformed to one another to the same image. Therefore, if a link invariant assigns to two links different images, we can conclude that it is in fact not possible to deform one to the other. Nowadays we know of many invariants. One of them is Khovanov homology, which is actually an invariant on the set of oriented links. Its construction relies heavily on tools from algebraic topology, and it turns out that there is a beautiful connection between this homology and another link invariant called Jones polynomial.
Secondary keywords:	mathematics;knots;links;homology;cobordism;chain complex;
Type (COBISS):	Final seminar paper
Study programme:	0
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja
Pages:	59 str.
ID:	14158036