Intro. To Topology W/ Knot Theory

Front 1

Homeomorphic

Back 1

When two sets/spaces are one-to-one and onto with a continuous inverse.

Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

Front 2

Simple Closed Polygonal Curve

Back 2

A polygonal closed curve that does not cross itself.

It is simple if each segment intersects exactly two other segments, intersecting only at endpoints.

Front 3

Knot

Back 3

A simple closed polygonal curve in R^3.

Front 4

Polygonal Curve

Back 4

A curve that is entirely made up of line segments (no arcs).

Front 5

Link

Back 5

A finite collection of pairwise disjoint (mutually disjoint) knots in R^3

Front 6

Vertices

Back 6

When (p_1, ..., p_k) defines a knot and no proper subset of these points defines the same knot, the elements {p_i} are called the vertices of the knot.

Front 7

Unknot / Trivial Knot

Back 7

The unknotted circle.

A knot determined by three nonlinear points.

Front 8

Equivalence

Back 8

Knots K and J are equivalent if there is a sequence of knots K = K_0, K_1, ..., K_n = J where each knot in the sequence is an elementary deformation of the preceding knot.

Front 9

Projection

Back 9

There are may different pictures of the same knot; we call such a picture of a knot a projection of the knot.

A projection of a knot K is a projection of the knot onto some plane.

Front 10

Regular Projection

Back 10

The projection is injective everywhere, except at a finite number of crossing points.

The projection is regular if no three points on the plane project to the same point in the projection and if no vertex in the knot projects to the same point as any other point in the knot would.

Front 11

Knot Diagram

Back 11

A knot diagram for K is a regular projection for K with over and under crossings indicated. The crossings of a knot diagram are the places where the segments in the projection cross.

Front 12

Composite Knot (Connected Sum)

Back 12

The composition of J#K of nontrivial knots.
(This is an analogy to the positive integers, where we call an integer composite if it is the product of positive integers, neither of which is equal to 1)

Front 13

Factor Knots

Back 13

The knots J and K in the composition J#K of nontrivial knots.

Front 14

Prime Knot

Back 14

A knot that is not composite, like the trefoil or figure-eight knot.

Front 15

Orientation / Oriented

Back 15

A knot can be oriented by consistently orienting the segments making it up (or equivalently, ordering the vertices).
One way of forming a composite know is to orient each know, and then connect the knows so that the orientations match up.
Another way is to orient the know and then connect the knots so the orientations do not match up.

Front 16

Invertible

Back 16

A knot is invertible if it can be deformed back onto itself so that a given orientation is taken to the opposite orientation.

Note: If a know K is invertible, then J#K is well defined, no matter what orientations are used.

Front 17

Amphicheiral

Back 17

A knot that is equivalent to its mirror image.
(The knot obtained by changing every crossing to the opposite crossing)

Front 18

(T/F)
A link is called SPLITTABLE if the components of the link can be deformed so that they lie on the same side of a plane in three-space.

Back 18

False.
Lie on different sides of a plane.

Pg. 17

Front 19

Strand

Back 19

In a projection is a piece of the link that goes from one undercrossing to another with only overcrossings in between.

Front 20

Tricolorable

Back 20

If each of the strands in a projection can be colored one of three different colors, so that at each crossing, either three different colors come together or all the same color comes together. We further require that at least two of the colors are used.

Front 21

Tangle

Back 21

A region in the projection plane surrounded by a circle such that the knot or link crosses the circle exactly four times.

Front 22

Equivalent Tangles

Back 22

Two tangles are equivalent if we can get from one to the other by a sequence of Reidemeister moves while the four endpoints of the strings in the tangle remain fixed and while the strings of the the tangle never journey outside the circle defining the tangle.

Front 23

Rational Link

Back 23

Close off the strands of a rational tangle.

Front 24

Rational Tangle

Back 24

A tangle of the form n_1 n_2 ... n_k where the n_i are integers.

Front 25

Adding Tangles

Back 25

Place tangles next to each other and join the strands.

Front 26

Algebraic Tangle/Link

Back 26

A link formed when we connect the NW string to the NE string and the SW String to the SE string on an algebraic tangle.

A tangle formed by the operations of addition and multiplication on regional tangles. (An algebraic link is the associated link)

Front 27

Unknotting Number

Back 27

A number n is the unknotting number if
there is a projection the knot such that changing n crossings in the projection results in the unknot and
there is no projection such that fewer changes would result in the unknot.

Front 28

K-move

Back 28

A local change in the projection that replaces two untwisted strings with two strings that twist around each other with k crossings. k = positive over crossing
-k = negative over crossing

Replacing two untwisted strands in a diagram by two strands that wind around each other k times.

Front 29

K-equivalent

Back 29

Getting from one projection to another projection through a series of k-moves and -k-moves

Front 30

Overpass

Back 30

A subarc of the knot that goes over at least one crossing but never goes under a crossing.

Front 31

Maximal Overpass

Back 31

An overpass that could not be made any longer.

Front 32

Bridge Number

Back 32

Number of maximal overpasses in the projection.

Front 33

Bridge Number of a Knot K

Back 33

The least bridge number of any projection K.

Front 34

Continuous

Back 34

No extreme changes or gap in the function.

Front 35

Continuous Inverse (Bicontinuous)

Back 35

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Front 36

Closed Polygonal Curve

Back 36

A polygon curve that begins and ends in the same location.

For distinct points p and q in R^3, let [p,q] be the line segment joining them. For an ordered set (p_1, p_2, ..., p_k) of points in R^3, the union of the segments [p_i, p_i+1] for 1 <= i <= k-1 together with [p_k, p_1] is called a closed polygonal curve.

Front 37

Topology

Back 37

The study of the properties of geometric objects that are preserved under deformations.

Front 38

Alternating Knot

Back 38

A knot with a projection that has crossings that alternate between over and under as one travels around the knot in a fixed direction. (Like the trefoil or figure-eight knot)

Front 39

(T/F)
By changing some of the crossings from over to under or vice versa, any projection of a know can be made into a projection of the unknot.

Back 39

True

Front 40

(T/F)
By changing the crossings from over to under or vice versa, any projection of a knot can be made into the projection of an alternating knot.

Back 40

True

Front 41

Unlink

Back 41

A union of unknots, all lying in one plane.

Front 42

Elementary Deformation

Back 42

A knot J is an elementary deformation of a knot K if one is formed from the other by adding a single vertex v_0 not on the knot such that the triangle formed by v_0 together with its adjacent vertices v_1 and v_2 intersects the knot only along the segment [v_1, v_2].

A knot J is called an elementary deformation of a knot K if one of the knots is determined by (p_1, ..., p_k) and the other is determined by (p_0, p_1, ..., p_k) where p_0 is a point not collinear with p_1 and p_n and where the triangle spanned by (p_0, p_1, p_n) intersects the knot determined by (p_1, ..., p_k) only in [p_1, p_n].

Front 43

~One Crossing~
Theorem
Claim

Back 43

Thm: If 2 knots have the same diagram, then they are equivalent.
Claim: If a knot is nontrivial (not equivalent to the unknot), then there is more than one crossing in a projection.

Front 44

How do you make the composition of two knots?

Back 44

By removing a small arc from each knot projection (on the outside) and then connecting the four endpoints by two new arcs. The added arcs do not overlap and do not intersect the remaining part of the projections.

Front 45

(T/F)
All of the compositions of knots J and K where the orientations match will yield multiple composite knots.

Back 45

False

All of the compositions of the two knots in this scenario will yield the same composite knot.

Front 46

(T/F)
All of the compositions of knots J and K where the orientations match will yield a single connected sum.

Back 46

True

Front 47

Ambient Isotopy

Back 47

The rearranging/movement of the string through three-dimensional space with out letting it pass through itself.
(Not allowed to shrink a part of the knot down to a point)

An ambient isotopy of R^3 is a continuous family of homeomorphisms of R^3.

Front 48

Planar Isotopy

Back 48

When a deformation of a knot projection deforms the projection plane as is if it were made of rubber with the projection drawn upon it.

A P.I. of a knot projection is a continuous deformation of the projection plane.

Front 49

Reidemeister Move

Back 49

One of three ways to change a projection of the knot that will change the relation between the crossings.

Front 50

First Reidemeister Move

Back 50

Allows one to put in or take out a twist in the know.

Front 51

Second Reidemeister Move

Back 51

Allows one to either add two crossings or remove two crossings.

Front 52

Third Reidemeister Move

Back 52

Allows one to slide a strand of the know from one side of a crossing to the other side of the crossing.

Front 53

Link

Back 53

A set of knotted loops all tangled up together.
Two links are considered to be the same if we can deform the one link to the other link without ever having any one of the loops intersect itself or any of the other loops in the process.

Front 54

Whitehead Link

Back 54

Two linked circles with more than two crossings.
It would seem the simplest has 4 component crossings, with two crossing onto itself by one of the components.

Front 55

(T/F)
A link is made up of at least three components.

Back 55

False.
Two Components.

Pg. 17

Front 56

Borromean Rings

Back 56

The well-known link with three components. If you remove any one component, the other two are unlinked.

Front 57

Unlink (Trivial Link)

Back 57

A link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.

Front 58

Linking Number

Back 58

A method for measuring numerically how linked up two components are.
If you have to move the under crossing clockwise to get the orientations of two DIFFERENT components to align in the least amount of space: +1.
If you must go counterclockwise: -1.
Or, point with your thumb in the direction of the over crossing, with your palm facing your eyes, If your fingers point in the same direction as the under crossing, +1.

Add them all up, divide by two.

Front 59

We say that the linking number is an INVARIANT of the oriented link, that is, once the orientations are chosen on the two components of the link, the linking number is unchanged by ambient isotopy.

Back 59

True.
Basically, it should have the same linking number (absolute value) regardless of orientation.

Pg. 21

Once orientations are chosen, two equivalent links have the same linking number.

Front 60

A link is called BRUNNIAN if the link itself is nontrivial, but the removal of any one of the components leaves us with a set of trivial unlinked circles.

Back 60

True.

Pg 22.

Front 61

Dowker Notation

Back 61

A sequence of even integers that tabulates notes. Assigning a negative symbol to an even digit implies that it is the strand point forming an undercrossing. Leaving it positive implies an overcrossing.

Front 62

What's one way to tell if two knots are composite based off of the given sequence in Dowker Notation?

Back 62

Take the sequence 4 6 2 10 12 8,

It is a shuffling of the numbers 2, four times. 4, 6 and then a shuffling of the three numbers 8, 10, and 12.

When the permutation of the even sequence can be broken into two separate subpermutation, the resulting knot are composite (assuming nontriviality)

Front 63

Conway's Notation

Back 63

Two rational tangles are equivalent iff the continued fractions associated with these tangles yield the same number.

Notation example: 2 3 2