Glossary of Knot Theory

MSC Classification: 57 (Manifolds) 
This glossary contains terms relevant to the study of knot theory.
Rough Guides to Topology 
General Topology  
Algebraic Topology  
Manifold Theory  
Knot Theory 
Main Glossary
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 $\Delta$unknotting operation
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 $\#$unknotting operation
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 $\mu$invariant
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 $2$bridge knot (link)
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 $3$colorability
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 $4$plat
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Aa
 Acheiral knot
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 Adequate knot
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 Addition (of knots/links)
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 Addition (of tangles)
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 \inv AkutsuWadati polynomial
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 Alexander ideal
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 \inv Alexander invariant
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 Alexander module
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 Alexander polynomial
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 Alexander's horned sphere
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 Algebraic link
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 Almostalternating link
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 Alternating link
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 Ambient Isotopy
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 Ampicheiral link
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 AndrewsCurtis link
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 Antoine's horned sphere
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 Antoine's necklace
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 Arborescent link
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 \inv Arf invariant
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Bb
 Bicollar
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 Borromean link/rings)
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 \inv Boundary link
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 \inv Bracket polynomial
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 Braid
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 Braid closure
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 Braid group
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 \inv Braid index
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 Braid notation
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 Braid representation
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 Braided link
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 Branched cover
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 \inv Bridge index/number
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 \inv Bridge length
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 Brunnian link
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Cc
 Cable knot
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 \inv Catalan number
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 Category
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 Chebyshev polynomial
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 Cinquefoil knot
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 Clasp
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 Classification (of knots)
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 Classification (of links)
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 Classification (of rational knots)
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 Classification (of rational tangles)
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 Classification (of tangles)
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 Closed braid
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 Colored link
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 \inv Coloring number
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 Companion knot
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 \inv Complement (of a knot)
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 Component
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 \inv Component number
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 Composite knot
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 Concordance (of knots)
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 Connected sum (of knots)
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 Continued fraction
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 Conway notation
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 \inv Conway polynomial
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 Conway sphere
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 Covering space
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 \inv Crookedness (of a knot)
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 Crossing/crossover
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 Crossing change
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 \inv Crossing number
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 Cyclic branched cover
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Dd
 Dehn surgery
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 Dehn's homology sphere
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 Dehn's lemma
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 Deltaunknotting operation
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 Denominator (of a tangle)
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 \inv Determinant (of a knot)
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 Diagram (of a knot/link)
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 Diagram (of a tangle)
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 DNA topology
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 Double branched cover
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 Doubled knot
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 Dowker notation
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Ee
 Equivalent (links)
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 Essential (loop)
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 Exceptional
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Ff
 \inv Fibered knot
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 Figureeight knot
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 Flat (embedding)
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 Flype
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 Foliation
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 Fourplat
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 Fourier knot
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 Framing (of a knot)
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 Framing (of a torus)
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 Fundamental group (of a knot)
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 Fundamental theorem of surgery
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Gg
 Generalized unknotting number
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 Generator (of a group)
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 \inv Genus (of a knot)
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 Genus (of $3$manifolds
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 \inv Gordian distance/number
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 Granny knot
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 Group (of a knot)
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Hh
 Handlebody
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 Handle surgery
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 Harmonic knot
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 Heegaard splitting/surface
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 Higherdimensional knot
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 Homeomorphism
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 \inv HOMFLY polynomial
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 Homology/homotopy sphere
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 Hopf link
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 Horned sphere
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 Hyperbolic link
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IiJjKk
 Incompressible torus
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 Invariant (of a knot/link)
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 Irreducible knot
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 Isotopy
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 Jones polynomial
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 Jordan curve theorem
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 \inv Kauffman bracket
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 \inv Kauffman polynomial
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 \inv Kauffman bracket skein module
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 Khovanov homology
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 KinoshitaTerasaka knot
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 Kirby moves
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 Knot
 A "loop" in space: a $1$dimensional connected PLsubmanifold of $S^3$.
 Knot table
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Ll
 \inv Laurent polynomial
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 Lens Space
 The space obtained by gluing two solid tori together by a homeomorphism along their boundaries.
 Link
 Several "loops" in space: an arbitrary $1$dimensional PLsubmanifold of $S^3$.
 \inv Linking number
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 Locally flat
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 Locally [un]knotted tangle
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 Longitude (of a knot)
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 Longitude (of a torus)
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 Loop Theorem
 A homotopy element nontrivial in $\del M$ but trivial in $M$ can be represented by a simple closed curve in $\del M$. This statement can be made slightly more general.
 Lover's knot
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Mm
 Mapping class group
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 Markov moves
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 Meridian (of a torus)
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 Mirror image (of a knot/link)
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 Monodromy
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 Montesinos link
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 Multi$\#$ unknotting operation
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 Mutation (of a knot)
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NnOo
 \inv $n$colorability (of a knot)
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 Noncancellation theorem
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 Nugatory crossing
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 Numerator (of a tangle)
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 Obverse (of a knot)
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 Orientable double cover
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 Orientation
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 Oriented link
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PpQq
 Partial sum (of tangles)
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 Pattern knot
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 Peripheral torus
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 Perko pair
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 Piecewise linear
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 PL
 Short for piecewise linear.
 Plumbing
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 Poincar/'e conjecture
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 Poincar/'e manifold
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 Polygonal link
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 \inv Polynomial (of a knot)
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 Pretzel link
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 Prime diagram (of a knot/link)
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 Prime knot
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 Prime tangle
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 Projection (of a knot)
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 Property P
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 Quantum $SU_q(2)$ invariants
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Rr
 $r$parallel link
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 Rational decomposition (of knots)
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 Rational knot (link)
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 Rational tangle
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 Reduced knot
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 \inv Reducible knot
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 Reef knot
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 Reflection (of a knot)
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 Regular isotopy
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 Regular projection
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 Reidemeister moves
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 Relaxation (of a knot/link)
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 Removable crossing
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 Reverse (of a knot)
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 Reversible (knot)
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 \inv Ribbon link
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 Ribbonslice conjecture
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Ss
 Satellite knot
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 SCC
 Short for simple closed curve.
 Sch\:onflies theorem
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 Seifert circuits
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 Seifert form/matrix
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 Seifert surface
 A surface in $S^3$ whose boundary is a knot (this exists for every knot).
 Sign (of a crossing)
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 \inv Signature (of a knot)
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 Singular
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 Singular knot
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 Skein formula/relation
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 \inv Slice knot
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 Slicing disk
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 Slope (of a surgery)
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 Smith conjecture
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 Smooth
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 Sphere Theorem
 If $\pi_2(M) \neq 0$, then there is a noncontractible sphere embedded in $M$.
 Spinning
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 \inv Split/splittable link
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 Split diagram
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 Spun knot
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 \inv Square bracket polynomial
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 Square knot
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 Standard position
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 State (of a knot diagram)
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 Stevedore's knot
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 \inv Stick number
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 \inv Strand passage distance
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 Strongly prime (knot diagram)
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 Sum (of knots)
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 Sum (of tangles)
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 Surgery
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 Suspended knot
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 Suspension
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Tt
 Tame link
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 Tangle
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 \inv Tangle distance
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 Tangle surgery
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 \inv Threecolorability
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 Torsion number
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 \inv Torsion invariant
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 Torus knot/link
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 Torusalternating link
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 Trefoil knot
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 \inv Tricolorability
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 Trivial knot
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 Trivial link
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 Trivial tangle
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 \inv Tunnel number
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 Twist link
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 Twistequivalent
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 Twistspun knot
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 Twisted double
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 Twobridge knot (link)
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UuVvWw
 Universal covering
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 Unknot
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 \inv Unknotting number
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 Unknotting number additivity conjecture
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 Unknotting operation
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 Unlink
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 \inv Unlinking number
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 \inv Vassiliev invariants
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 Viergeflechte
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 \inv Volume (of a knot)
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 \inv $w$signature
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 Whitehead double
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 Whitehead's link
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 Whitehead's manifold
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 \kt Whitehead, ??
 def
 Wild embedding/knot/link
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 Wirtinger presentation
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 \inv Writhe (of a knot)
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XxYyZz
Theorems
In this section, we give some important theorems used in knot theory. Where possible, the statements are given in language rather than notation. In some cases we give two versions of the theorem, one “intuitive” and the other “semantic,” the second usually including extra information. An indication of the proof is also given or, if not, a reference.
Embedding Theorems
The following theorems give ways to translate algebraic information into geometric information. Basically, if the fundamental group is nontrivial (in dimensions one or two), they tell how to find a representative of the group which is an embedding, namely onetoone.
 Dehn's Lemma
 If a disk $D$ is mapped into a \thrmm, such that its boundary is mapped injectively into $M$, then the map can be adjusted on the interior of the disk to make it an embedding.
Proof: corollary of the Loop Theorem.  Loop Theorem
 Intuitive: A homotopy element nontrivial in $\del M$ but trivial in $M$ can be represented by a simple closed curve in $\del M$.
Semantic: If $N$ is a normal subgroup of $\pi_1(S)$ for a surface $S \subset \del M$ and $\ker(\pi_1(S)\to\pi_1(M)) \not \subset N$, then there is a proper embedding of the disk in $M$ whose boundary is not an element of $N$.  Sphere Theorem
 Intuitive: If $\pi_2(M) \neq 0$, then there is a noncontractible sphere embedded in $M$.
Semantic: If $N$ is a $\pi_1(M)$ invariant subgroup of $\pi_2(M)$ with $\pi_2(M) \not \subset N$, then there is an embedding of the sphere in $M$ which is not an element of $N$.
Classification Theorems
 Theorem (lens space classification)
 The lens spaces $L(p,q)$ are classified by (i) $L(1,q)=S^3$ and (ii) $L(p,q)=L(p,q')$ if either $q \equiv \pm q' \mod p$ or $qq' \equiv \pm 1 \mod p.$
Proof: ask when a homeomorphism on the boundary of a torus can be extended to a homeomorphism on the whole torus.
Knot Theorists
 Adams, Colin
 Author of The Knot Book. Has made significant contributions to the connections between knot theory and hyperbolic geometry.
 Alexander, J
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 Artin, Emil
 Credited with inventing braids.
 Conway, John
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 Dehn, ??
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 Heegaard, ??
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 Jones, Vaughan
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 Kauffman, Louis
 Author of Knots and Physics and several other books on knot theory. Namesake of the Kauffman bracket and the Kauffman bracket skein module.
 Khovanov
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 Lickorish, ??
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 Lord Kelvin
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 Perko, Kenneth
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 Poincare/'e, ??
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 Przytycki, Jozef
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 Reidemeister, ??
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 Rolfsen, Dale
 Author of Knots and Links.
 Seifert, ??
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 Tait, P
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 Thurston, William
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 Wirtinger, K
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References
1. D. Rolfsen, Knots and Links.
2. Lickorish.
3. Colin Adams, The Knot Book.
4. Burde and Zieschang.
page revision: 14, last edited: 19 Sep 2009 23:24