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University of Illinois – Chicago

1. Schneider, Jonathan. Diagrammatic Theories of 1- and 2- Dimensional Knots.

Degree: 2016, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/20811

A meta-theory is described whereby any diagrammatic knot theory may be defined by specifying diagrams and moves. This is done explicitly in dimensions 1 and 2, with more abstract indication of how to extend the meta-theory to higher dimensions. Several examples are given in dimensions 1 and 2, with information about how the theories are related. A topological model for each theory is described. Particular focus is placed on virtual knot theory and welded knot theory, building on work by Kauffman, Satoh, and Rourke, with new results about Rourke's model of welded knots.
*Advisors/Committee Members: Kauffman, Louis H. (advisor), Radford, David (committee member), Takloo-Bighash, Ramin (committee member), Licht, Arthur L. (committee member), Culler, Marc (committee member).*

Subjects/Keywords: knot theory; knot diagrams; surface knot theory; 2-knot theory; virtual knots; virtual knot theory; welded knots

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Schneider, J. (2016). Diagrammatic Theories of 1- and 2- Dimensional Knots. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/20811

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Schneider, Jonathan. “Diagrammatic Theories of 1- and 2- Dimensional Knots.” 2016. Thesis, University of Illinois – Chicago. Accessed July 14, 2020. http://hdl.handle.net/10027/20811.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Schneider, Jonathan. “Diagrammatic Theories of 1- and 2- Dimensional Knots.” 2016. Web. 14 Jul 2020.

Vancouver:

Schneider J. Diagrammatic Theories of 1- and 2- Dimensional Knots. [Internet] [Thesis]. University of Illinois – Chicago; 2016. [cited 2020 Jul 14]. Available from: http://hdl.handle.net/10027/20811.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Schneider J. Diagrammatic Theories of 1- and 2- Dimensional Knots. [Thesis]. University of Illinois – Chicago; 2016. Available from: http://hdl.handle.net/10027/20811

Not specified: Masters Thesis or Doctoral Dissertation

Indian Institute of Science

2. Kulkarni, Dheeraj. Relative Symplectic Caps, Fibered Knots And 4-Genus.

Degree: 2012, Indian Institute of Science

URL: http://hdl.handle.net/2005/2285

The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus.
In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S3 .
We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in S3 . Using this result, we give a criterion for quasipostive fibered knots to be strongly quasipositive.
Symplectic convexity disc bundles is a useful tool in constructing symplectic fillings of contact manifolds. We show the symplectic convexity of the unit disc bundle in a Hermitian holomorphic line bundle over a Riemann surface.
*Advisors/Committee Members: Gadgil, Siddhartha.*

Subjects/Keywords: Symplectic Geometry; Symplectic Capping Theorem; Symlpectic Manifolds; Fibered Knots; 4-Genus Knots; Symplectic Caps; Knot Theory; Contact Geometry; Contact Manifolds; Quasipositive Knots; Symplectic Convexity; Topology; Symplectic Neighborhood Theorem; Seifert Surfaces; Riemann Surface; Geometry

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kulkarni, D. (2012). Relative Symplectic Caps, Fibered Knots And 4-Genus. (Thesis). Indian Institute of Science. Retrieved from http://hdl.handle.net/2005/2285

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Kulkarni, Dheeraj. “Relative Symplectic Caps, Fibered Knots And 4-Genus.” 2012. Thesis, Indian Institute of Science. Accessed July 14, 2020. http://hdl.handle.net/2005/2285.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Kulkarni, Dheeraj. “Relative Symplectic Caps, Fibered Knots And 4-Genus.” 2012. Web. 14 Jul 2020.

Vancouver:

Kulkarni D. Relative Symplectic Caps, Fibered Knots And 4-Genus. [Internet] [Thesis]. Indian Institute of Science; 2012. [cited 2020 Jul 14]. Available from: http://hdl.handle.net/2005/2285.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kulkarni D. Relative Symplectic Caps, Fibered Knots And 4-Genus. [Thesis]. Indian Institute of Science; 2012. Available from: http://hdl.handle.net/2005/2285

Not specified: Masters Thesis or Doctoral Dissertation

Indian Institute of Science

3. Kulkarni, Dheeraj. Relative Symplectic Caps, Fibered Knots And 4-Genus.

Degree: 2012, Indian Institute of Science

URL: http://etd.iisc.ernet.in/handle/2005/2285 ; http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf

The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus.
In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S3 .
We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in S3 . Using this result, we give a criterion for quasipostive fibered knots to be strongly quasipositive.
Symplectic convexity disc bundles is a useful tool in constructing symplectic fillings of contact manifolds. We show the symplectic convexity of the unit disc bundle in a Hermitian holomorphic line bundle over a Riemann surface.
*Advisors/Committee Members: Gadgil, Siddhartha.*

Subjects/Keywords: Symplectic Geometry; Symplectic Capping Theorem; Symlpectic Manifolds; Fibered Knots; 4-Genus Knots; Symplectic Caps; Knot Theory; Contact Geometry; Contact Manifolds; Quasipositive Knots; Symplectic Convexity; Topology; Symplectic Neighborhood Theorem; Seifert Surfaces; Riemann Surface; Geometry

Record Details Similar Records

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kulkarni, D. (2012). Relative Symplectic Caps, Fibered Knots And 4-Genus. (Thesis). Indian Institute of Science. Retrieved from http://etd.iisc.ernet.in/handle/2005/2285 ; http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Kulkarni, Dheeraj. “Relative Symplectic Caps, Fibered Knots And 4-Genus.” 2012. Thesis, Indian Institute of Science. Accessed July 14, 2020. http://etd.iisc.ernet.in/handle/2005/2285 ; http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Kulkarni, Dheeraj. “Relative Symplectic Caps, Fibered Knots And 4-Genus.” 2012. Web. 14 Jul 2020.

Vancouver:

Kulkarni D. Relative Symplectic Caps, Fibered Knots And 4-Genus. [Internet] [Thesis]. Indian Institute of Science; 2012. [cited 2020 Jul 14]. Available from: http://etd.iisc.ernet.in/handle/2005/2285 ; http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Kulkarni D. Relative Symplectic Caps, Fibered Knots And 4-Genus. [Thesis]. Indian Institute of Science; 2012. Available from: http://etd.iisc.ernet.in/handle/2005/2285 ; http://etd.ncsi.iisc.ernet.in/abstracts/2943/G25244-Abs.pdf

Not specified: Masters Thesis or Doctoral Dissertation