Title: Theory / Unimportant / Math / Topology / Knot book
Tags: récupéré MissingMedia examplesSwitchExample1 fromMath hasSeenConjDef closed mathToSee fromAlgorithm definitionToAnswer exampleToQuestion hasNotNewEffectReturnsMeaning questionToInterval toBinaryRelation

Page: 2
Section: 1
Name: Unknot
Linking number: 0
Name2: Trivial knot
Stick number: 3
A-b notation: \(0_1\)

Page: 2
Section: 1
Name: A trefoil knot

Question: How to compute, by hand, a simpler version of \(k\)
Answer: Draw \(k\) on blackboard, redraw it part by part.
Page: 2
Chapter: 1
Variables: \(K\) a knot
Context: Knot
Kind: Ex

Name: A projection of \(k\)
Definition: A 2D representation of \(k\). Each intersection represents the part below by erasing it
Page: 2
Chapter: 1
Variables: \(k\) a knot

Page: 3
Section: 1
Name: Figure-eight knot
Rational: yes
Tangle: 22

Name: The crossings of \(P\)
Definition: The place were the knot crosses itself in the picture
Page: 3
Chapter: 1
Variables: \(k\) a knot,
\(P\) a projection of \(k\)

Smaller0: \(P\) have at most two crossing
Greater0: \(P\) is the trivial knot
Page: 3
Chapter: 1
Variables: \(k\) a knot,
\(P\) a projection of \(k\)
Index: 2
Kind: Ex

Page: 4
Chapter: 1
Name: Central question of knot theory

Name: Topology
Definition: Study of the properties of geometric objects that are preserved under deformations.
Page: 6
Chapter: 1

Name: An alternating knot
Definition: A knot with an alternating projection
Page: 7, 54
Chapter: 1, 2
Section: 1, 4
Index: , 30
Kind: , Ex
Example: Trifoil
Example2: Figure of eight

Name: An alternating projection
Definition: crossing alternates between over and under
Page: 7, 54
Chapter: 1, 2
Section: 1, 4
Variables: \(K\) a knot
Index: , 30
Kind: , Ex
Definition2: its signed planar graph has a single sign
Example: Trifoil
Conjdef: A projection such that, equivalently
Example2: Figure of eight

Question: Taking a projection and changing some crossing leads to
Answer: the unknot.
Page: 7
Chapter: 1
Kind: Ex

Name: The composition of \(J\) and \(K\)
Definition: Taking a projection of \(J\) and one of \(K\). Removing a segment on the outside of those, and linking the the end of those segments without having them crossing, respecting the orientation.
Page: 7
Chapter: 1
Section: 2
Variables: \(J,K\) orientated knots
Notation: \(J\#K\)

Page: 8
Section: 2
Chapter: 1
Name: A composite knot
Definition: The result of composing of two non trivial knot.

Name: A factor knot of \(K\)
Definition: A prime knot such that, once composed with another knot, they form \(K\)
Page: 8
Chapter: 1
Section: 2
Variables: \(K\) a knot

Question: \(K\#\text{unknot}\)
Answer: equals \(K\)
Page: 8
Chapter: 1
Section: 2
Variables: \(K\) a knot,

Name: A prime knot
Definition: A knot which is not composed
Page: 9
Chapter: 1
Section: 2
Example: Trifoil knot
Example2: Figure-eight knot
Example3: Unknot

Name: An orientation on \(K\)
Definition: A direction to travel around the knot
Page: 10
Chapter: 1
Section: 2
Variables: \(K\) a knot

Name: An oriented knot
Definition: A knot with an orientation
Page: 10
Chapter: 1
Section: 2

Name: An invertible knot
Definition: A knot which can be deformed into the same knot with the reverse orientation
Page: 11
Chapter: 1
Section: 2

Name: An ambient isotopy
Definition: A continuous deformation of the knot through 3D space, without any part of the knot crossing.
Page: 12
Chapter: 1
Section: 3
Context: Knot theory

Name: A planar isotopy
Definition: A deformation of the 2D plane on which the representation of the knot lie.
Page: 12
Chapter: 1
Section: 3

Name: A Reidemeister move
Definition: Type I:
Page: 13
Chapter: 1
Section: 3
Definition2: Type II:
Conjdef: Either
Definition3: Type III:
Intuition: One of the way to change a projection of the knot that will change the relation between the crossings.

Smaller0: \(P\) and \(Q\) presents the same knot
Greater0: \(P\) can be obtained from \(Q\) by Reidemeister moves and planar isotopy
Page: 14
Chapter: 1
Section: 3
Variables: \(P,Q\) projection of knots
Smaller to greater: iff

Question: The number of Reidemeister moves to change the projection \(P\) to the projection \(Q\) is
Answer: unknown
Page: 16
Chapter: 1
Section: 3
Variables: \(K\) a knot
Let \(P,Q\) projection of \(K\) with \(n\) knots

Name: A link
Definition: A set of knotted loops all tangled up together.
Page: 17
Chapter: 1
Section: 4

Page: 17
Section: 4
Chapter: 1
Name: The Whitehead link
Linking number: 0

Page: 17
Section: 4
Chapter: 1
Name: The Booromean ring
French: les anneaux Boroméen

Name: A link of \(n\) components
Definition: A link made of \(n\) knots.
Page: 17
Chapter: 1
Section: 4

Name: A splittable link
Definition: A link which can be deformed such that they lie on different side of a plane in the 3D space.
Page: 17
Chapter: 1
Section: 4

Smaller0: \(P\) and \(Q\) are projection of the same link
Greater0: \(P\) and \(Q\) have the same number of components
Page: 18, 20, 25
Chapter: 1
Section: 4
Variables: \(P,Q\) projecton of links
Greater1: \(P\) and \(Q\) have the same linking number
Greater2: \(P\) and \(Q\) are both tricolorable or both not-tricolorable

Page: 18
Section: 4
Chapter: 1
Name: The unlink of two components
Linking number: 0
Variables: \(P,Q\) projecton of links
Name2: The trivial link

Page: 18
Section: 4
Chapter: 1
Name: The hopf link
Linking number: \(\pm 1\)

Name: The linking number of \(M\) and \(N\)
Definition: The sum, divided by two, of the +1 and -1 generated according to the following rules.
Page: 19
Chapter: 1
Section: 4
Variables: \(L\) an orientate link,
\(M\) and \(N\) two components.

Question: Reversing the orientation of one knot
Answer: multiply the linking number by -1
Page: 19
Chapter: 1
Section: 4
Variables: \(L\) an orientate link,
\(M\) and \(N\) two components.

Name: An invariant
Definition: A property of link which is unchanged by invariant isotopies
Page: 21
Chapter: 1
Section: 4

Smaller0: \(M\) and \(N\) are splittable
Greater0: the linking number of \(N\) and \(M\) is 0
Page: 21
Chapter: 1
Section: 4
Variables: \(L\) an orientate link,
\(M\) and \(N\) two components.

Name: A Brunnian link
Definition: A nontrivial link such that the removal of any component leaves a set of unlinked circles.
Page: 22
Chapter: 1
Section: 4
Example: Borromean ring

Name: A strand in \(P\)
Definition: A piece of the link that goes from one undercrossing to another with only overcrossings in between.
Page: 23
Chapter: 1
Section: 5
Variables: \(K\) a link,
\(P\) a projection of \(K\)

Name: A tricolorable link
Definition: each strand is colored in one of three color
Page: 23
Chapter: 1
Section: 5
Definition2: at least two colors are used
Conjdef: A link such that
Definition3: at each crossing, 1 or 3 colors are used
Property: Invariant under ambiant isotopy

Name: A not tricorolable link
Page: 25
Chapter: 1
Section: 5
Example: The unknot

Name: Example of not tricolorable link
Page: 26
Chapter: 1
Section: 5
Index: 29
Kind: Ex
Example: Whitehead link

Name: The stick number of \(K\)
Definition: The least number of straight sticks necessary to make \(K\)
Page: 29
Chapter: 1
Section: 5
Variables: \(K\) a knot
Abbreviation: \(s(K)\)

Question: \(s(K)\) is at least
Answer: \(\frac{5+\sqrt{(25+8(c(K)-2))}}2\)
Page: 30
Chapter: 1
Section: 5
Variables: \(K\) a knot
Index: 38
Kind: Ex

Question: The stick number \(s(K)\) is at most
Answer: \(2c(K)\) with \(c(K)\) the minimum crossing number
Page: 30
Chapter: 1
Section: 5
Variables: \(K\) a knot
Index: 38
Kind: Ex

Name: Dowker notation for \(K\)
Definition: if \(i\) is even and undercrossing, label \(-i\),
Page: 35
Chapter: 2
Variables: \(K\) an orientated knot
Notation: The series of even number associated to \(1,\dots, 2n-1\), where \(n\) is the number of crossing
Index: 4
Kind: Fig
Definition2: otherwise, label it \(i\)
Conjdef: Select an undercrossing, label it 1.
Follow along the orientation
Label the \(i\)-th intersection as follow

Page: 37
Implementation: For \(i=1\) to \(2n-1\):
-if \(i\) is already drawn, draw a line from \(i-1\) to \(i\). By hypothesis, it can be done without crossing any other line
-otherwise, draw a line from \(i-1\) to some place. Call this place \(i\), and add the element linked to \(i\) on it
Section: 2
Effect: A presentation of \(K\)
Variables: \(K\) a knot
Let \(\begin{array}{lll}1&\dots& 2n-1\\i_1&\dots&i_{n}\end{array}\) being Dowker notation for \(K\)
Index: 4
Kind: Fig

Smaller0: there is \(j<n\) such that \(\{i_1,\dots,i_j\}=\{1,\dots,j\}\)
Greater0: each knot represented by \(D\) is composite of a knot with \(j\) intersection and a knot with \(n-j\) intersections
Page: 38
Chapter: 2
Variables: \(D=\begin{array}{lll}1&\dots& 2n-1\\i_1&\dots&i_{n}\end{array}\) Dowker notation
Smaller to greater: iff

Smaller0: two successive numbers are linked to two successive numbers
Greater0: a type II Reidemeister move removes those crossing
Page: 40
Chapter: 2
Variables: \(D=\begin{array}{lll}1&\dots& 2n-1\\i_1&\dots&i_{n}\end{array}\) Dowker notation
\(i\) odd
Index: 9
Kind: Ex
Smaller to greater: iff

Name: A tangle in \(P\)
Definition: A region in the projection plane surounded by a circle, such that the link crosses the circles exactly four times.
Page: 41
Chapter: 2
Section: 3
Variables: \(L\) a link
\(P\) a projection

Page: 41, 46
Definition: The relation on tangle where \(T_2\) can be obtained from \(T_1\) by a sequence of Reidemeister moves keeping the four endpoints of the strings in the tangle remaining fixed with string not leaving the tangle
Chapter: 2
Section: 3
Variables: \(L_1,L_2\) links
\(P_1,P_2\) projections,
\(T_1,T_2\) tangles in \(P_1\) and \(P_2\)
Name: \(T_1\) is equivalent to \(T_2\)
Context: Knot
Definition2: The relation on tangle where both tangle have the same conway notation

Name: The \(\infty\) tangle
Page: 42
Chapter: 2
Section: 3
Variables: \(T_i\) tangles,
\(K_i\) the link, obtained from \(T_i\), by linking the two north (south) edges together
Index: 17
Kind: Fig

Name: The \(n\) tangle
Definition: The tangle obtained by taking the two left edges and doing \(n\) left-handed twists, before letting the right edges leaving
Page: 42
Chapter: 2
Section: 3
Variables: \(n\in\mathbb Z\)
Index: 17
Kind: Fig

Name: The \(i_1,\dots, i_n\) tangle
Definition: \(T_\epsilon\) be the \(\infty\)-tangle
Page: 43
Chapter: 2
Section: 3
Variables: \(i_1,\dots, i_n\) a sequence of relative integer
Definition2: \(T_{i_1,\dots, i_n}\) is obtained as combining:
-a NW-SE symmetry on \(T_{i_1,\dots, i_{n-1}}\)
and on his right
-a \(i_n\) tangle.
Conjdef: By recursion, let

Name: A rational tangle
Definition: A \(i_1,\dots, i_n\) tangle
Page: 44
Chapter: 2
Section: 3
Definition2: A tangle obtained by twisting any number of time the two bottom or the two east strings

Name: The continued fraction generated by those numbers
Definition: \(i_1\) if \(n=1\)
Page: 46
Chapter: 2
Section: 3
Variables: \(i_1,\dots, i_n\)
Notation: \([i_1,\dots i_n]\)
Definition2: \(i_1+\frac{1}{[i_2,\dots, i_n]}\) if \(n>1\)
Conjdef: Either

Name: Positive slope
Page: 44
Chapter: 2
Section: 3

Name: Negative slope
Page: 44
Chapter: 2
Section: 3

Name: An a link
Definition: A link, obtained by closing noth(south) edges of an a tangle
Page: 46, 48
Chapter: 2
Section: 3
Variables: a an adjective related to tangle

Question: The number of component of a rational link is
Answer: 1 or 2
Page: 47
Chapter: 2
Section: 3
Index: 16a
Kind: Ex

Name: Conway notation
Definition: Describing a rational link by its rational tangle
Page: 47
Chapter: 2
Section: 3
Kind: Ex

Name: An integer tangle
Definition: A \(n\) tangle, with \(n\in\mathbb Z\cup\{\infty\}\)
Page: 47
Chapter: 2
Section: 3
Kind: Ex

Name: Rational tangles
Page: 47
Section: 3
Chapter: 2
Under: multiplication by an integer tangle

Name: Multiplying tangles
Page: 47
Chapter: 2
Section: 3
Typ: Non associative, non commutative, without inverse

Name: Adding tangles
Page: 47
Chapter: 2
Section: 3
Typ: Non associative, non commutative, without inverse

Definition: \(t_1+\dots+t_n\)
Page: 47
Chapter: 2
Section: 3
Variables: \(t_1,\dots, t_n\) representation of tangles
Notation: \(t_1,\dots,t_n\)

Name: A pretzel knot
Definition: A knot of the form \(t_1,\dots,t_n\), with \(t_i\) integers
Page: 48
Chapter: 2
Section: 3

Name: An algebraic tangle
Definition: A tangle, obtained by addition, multiplication, on rational tangles
Page: 48
Chapter: 2
Section: 3

Name: Not algebraic tangle
Page: 49
Chapter: 2
Section: 3
Index: 29
Kind: Feg

Name: A mutation of \(K\)
Definition: Taking a tangle in \(K\), and rotating/mirroring it, leaving another tangle intact
Page: 50
Chapter: 2
Section: 3
Variables: \(K\) a knot

Name: A mutant of \(K\)
Definition: A knot obtained by applying a mutation to \(K\)
Page: 50
Chapter: 2
Section: 3
Variables: \(K\) a knot

Name: Alternating knot
Page: 50
Section: 3
Chapter: 2
Variables: \(K\) a knot
Under: mutation

Name: The Kinoshita-Terasaka mutants
Page: 50
Chapter: 2
Section: 3
Variables: \(K\) a knot

Name: Non-trivial knot
Page: 51
Section: 3
Chapter: 2
Variables: \(K\) a knot
Under: mutation
Not2: not
Under2: product
Under3: Reidemeister move

Name: Ambiant isomorphism
Page: 13
Section: 3
Chapter: 1
Under: Reidemeister move
Under2: composition

Name: amphicheiral knot
Definition: A knot equivalenty to its mirror image
Page: 33
Chapter: 2
Section: 1

Name: A mutation of \(K\)
Definition: A knot, similar to \(K\), where a tangle has been selected, and vertical and/or horizontal symmetry has been applied to it before putting the tangle back in \(K\)
Page: 49
Chapter: 2
Section: 3
Variables: \(K\) a knot

Name: A set of mutant knot
Definition: A set containing a knot, and its 3 mutations, where all mutations are applied to the same tangle
Page: 49
Chapter: 2
Section: 3

Name: Shaded link projection of \(P\)
Definition: The unique black and white coloration of the regions of \(P\), such that the exterior is white, and two regions having two crossings in common strand has distinct color.
Page: 52
Chapter: 2
Section: 4
Variables: \(K\) a knot
\(P\) its representation
Construction: Each intersection has exactly 4 neighbors, hence they those four can be coloriable

Name: The graph associated to the knot projection \(P\)
Definition: The vertices are the black part of the shaded link projection. The edges are between region touching on a crossing.
Page: 52
Chapter: 2
Section: 4
Variables: \(K\) a knot
\(P\) its representation

Name: A positive crossing of the shaded link projection
Definition: A crossing of the form
Page: 52
Chapter: 2
Section: 4
Variables: \(K\) a knot
\(P\) its representation
\(P'\) its shaded link projection

Name: A negative crossing of the shaded link projection
Definition: A crossing of the form
Page: 52
Chapter: 2
Section: 4
Variables: \(K\) a knot
\(P\) its representation
\(P'\) its shaded link projection

Name: The signed graph from the knot projection \(P\)
Definition: Taking the graph from the knot projection, adding a sign on each edge, according to the sign of the crossing it represents
Page: 52
Chapter: 2
Section: 4
Variables: \(K\) a knot
\(P\) its knot projection

Name: The unknotting number of the projection \(P\)
Definition: The minimal number of crossings to change so that \(P\) becomes the unknot
Page: 57
Chapter: 3
Section: 1
Variables: \(K\) a knot
\(P\) its projection
Abbreviation: \(u(P)\)

Name: The unknotting number of the knot \(K\)
Definition: The minimal unknot number of a projection of \(K\)
Page: 57
Chapter: 3
Section: 1
Variables: \(K\) a knot
Definition2: Given any projection \(P\), the minimal number of sequence of (switching a crossing, and applying an ambiant isomorphism) until the knot become the unknot
Abbreviation: \(u(K)\)

Page: 59
Implementation: Select a point \(p\) and a direction. Move along the line in this direction, each time a new crossing is met, ensure that the current part of the line is above in this crossing.
Section: 1
Chapter: 3
Effect: Transform \(P\) into the unknot by switching the crossing
Variables: \(K\) a knot,
\(P\) its projection
Extra: Imagine that the line goes from \(z=1\) to \(z=0\) while moving along the loop. Then no knot can be created.

Smaller0: the unknot number of \(K\) \(u(K)\)
Greater0: the minimum crossing number \(c(K)\)
Page: 59
Chapter: 3
Section: 1
Variables: \(K\) a knot,
Kind: Ex
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: the unknot number of \(K\) \(u(K)=1\)
Greater0: \(K\) is prime
Page: 60
Chapter: 3
Section: 1
Variables: \(K\) a knot,

Smaller0: \(u(K_1\#K_2)\)
Greater0: \(u(K_1)+u(K_2)\)
Page: 60
Chapter: 3
Section: 1
Variables: \(K_1,K_2\) a knot,
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Name: A \(k\)-move
Definition: replaces two untwisted strings with two strings that twist around each other with \(k\)-crossings in a right-handed manner.
Page: 63
Chapter: 3
Section: 1
Variables: \(k\in\mathbb N\)

Name: A \(-k\)-move
Definition: replaces two untwisted strings with two strings that twist around each other with \(k\)-crossings in a left-handed manner.
Page: 63
Chapter: 3
Section: 1
Variables: \(k\in\mathbb N\)

Name: An overpass
Definition: A part of the knot that goes over at least one crossing, and never goes under a crossing.
Page: 64
Chapter: 3
Section: 2
Variables: \(K\) a knot
\(P\) a projection

Name: A maximal overpass
Definition: An overpass which can not be extended
Page: 64
Chapter: 3
Section: 2
Variables: \(K\) a knot
\(P\) a projection

Name: The bridge number of \(P\)
Definition: the number of maximal overpass of \(P\)
Page: 64
Chapter: 3
Section: 2
Variables: \(K\) a knot
\(P\) a projection
Abbreviation: \(b(P)\)
Typ: non negative integer

Name: The bridge number of \(K\)
Definition: The minimal bridge number of projections of \(K\)
Page: 64
Chapter: 3
Section: 2
Variables: \(K\) a knot
Abbreviation: \(b(P)\)

Smaller0: the bridge number of \(K\)
Greater0: the minimal number of crossing of a projection \(K\)
Page: 65
Chapter: 3
Section: 2
Variables: \(K\) a knot
Index: 12
Kind: Ex
Smaller to greater: is less than
Greater to smaller: is greater than

Name: two-bridge knot
Definition: A knot whose bridge number is 2
Page: 65, 66
Chapter: 3
Section: 2
Name2: Rational knot

Name: An \(n\)-crossing projection of \(K\)
Definition: A projection of \(K\) with \(n\) crossings
Page: 3
Chapter: 1
Variables: \(K\) a knot
\(n\) natural

Name: An \(n\)-crossing knot
Definition: A knot with an \(n\)-crossing projection and no \(n'\)-crossing projection for \(n'<n\)
Page: 3
Chapter: 1
Variables: \(n\) natural

Question: The number of factor decomposition of \(K\) in prime knot is
Answer: exactly 1
Page: 10
Chapter: 1
Variables: \(K\) a knot

Name: The mirror image of \(K\)
Definition: The unique knot whose projection is the symmetric by a line of a projection of K
Page: 14
Chapter: 1
Section: 3
Variables: \(K\) a knot

Name: Amphiceiral knot
Definition: A knot equivalent to its mirror knot
Page: 15
Chapter: 1
Section: 3
Name2: Achiral knot (name used by chemists)

Name: A crossing whose linking number is +1
Page: 19
Chapter: 1
Section: 4

Name: A crossing whose linking number is -1
Page: 19
Chapter: 1
Section: 4

Name: \(\infty\) tangle
Page: 42
Chapter: 2

Name: 0 tangle
Page: 42
Chapter: 2

Name: The \(n\) tangle
Definition: Tangle obtained from the 0 tangle by taking the top-left part, putting it over the top right part, \(n\) times.
Page: 43
Chapter: 2
Section: 3
Variables: \(n\) natural

Name: The \(-n\) tangle
Definition: Tangle obtained from the 0 tangle by taking the top-right part, putting it over the top left part, \(n\) times.
Page: 43
Chapter: 2
Section: 3
Variables: \(n\) natural

Name: The \(n_{1,1}\dots n_{1,i_1},\dots,n_{j,1} \dots n_{j,i_j}\)-tangle
Definition: if \(i=0\), the 0 tangle
Otherwise: the sum of
-the \(n_{1,1}\dots n_{1,i_1},\dots,n_{j-1,1} \dots n_{j-1,i_{j-1}}\)-tangle
-the \(n_{j,1}\dots n_{j,i_j}\)-tangle
Page: 46
Chapter: 2
Section: 3
Variables: \(n_{1,1},\dots,n_{1,i_1},\dots,n_{j,1},\dots,n_{j,i_j}\in\mathbb Z\)

Name: The \(n_1 \dots n_m\) tangle
Definition: if \(m=0\), the 0 tangle
Otherwise: the product of
-the \(n_1 \dots n_{m-1}\)-tangle
-the \(n_m\)-tangle
Page: 43
Chapter: 2
Section: 3
Variables: \(n_1,\dots,n_m\in\mathbb Z\)

Name: A conway notation of \(T\)
Definition: \(n_1,\dots,n_m\) where \(T\) is the \(n_1,\dots,n_m\)-tangle
Page: 43
Chapter: 2
Section: 3
Variables: \(T\) a tangle

Name: A rational tangle
Definition: The \(n_1,\dots,n_m\)-tangle for some sequence of integers \(n_1,\dots,n_m\)
Page: 43
Chapter: 2
Section: 3

Name: negative slope
Page: 44
Chapter: 2
Section: 3
Context: knot

Name: The continued fraction of the \(n_1,\dots,n_m\)-tangle
Definition: \(n_m+\frac{1}{n_{m-1}+\frac{1}{\dots\frac{1}{n_1}}}\)
Page: 44
Chapter: 2
Section: 3
Variables: \(n_1,\dots,n_m\in\mathbb Z\)
Context: knot

Name: A rational link
Definition: A link obtain by closing the two top and the two bottom tangles of a rational tangle
Page: 46
Chapter: 2
Section: 3
Context: knot

Name: The product of \(T_1\) and \(T_2\)
Page: 47
Chapter: 2
Section: 3
Variables: \(T_1,T_2\) tangles
Context: knot
Notation: \(T_1\times T_2\)

Name: The sum of \(T_1\) and \(T_2\)
Definition: sending to
Page: 47
Chapter: 2
Section: 3
Variables: \(T_1,T_2\) tangles
Context: knot
Notation: \(T_1+ T_2\)

Name: A pretzel knot
Definition: A knot obtained from a \(n_1,\dots,n_i\)-tangle
Page: 48
Chapter: 2
Section: 3
Context: knot

Name: An algebraic tangle
Definition: A \(n_{1,1}\dots n_{1,i_1},\dots,n_{j,1} \dots n_{j,i_j}\)-tangle
Page: 48
Chapter: 2
Section: 3
Context: knot

Name: An algebraic link
Definition: The link obtained by connecting the two top and the two bottom part of an algebraic tangle
Page: 48
Chapter: 2
Section: 3
Context: knot

Smaller0: each \(n_{i,j}\) are non-negative
Greater0: is alternating
Page: 48
Chapter: 2
Section: 3
Variables: \(n_{1,1},\dots,n_{1,i_1},\dots,n_{j,1},\dots,n_{j,i_j}\in\mathbb Z\)
Index: 20
Kind: Ex
Prefix Greater: the \(n_{1,1}\dots n_{1,i_1},\dots,n_{j,1} \dots n_{j,i_j}\)-tangle
Smaller1: is a knot
Context: knot