Classical Knot Theory

This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.

A classical knot is a (smooth or piecewise-linear locally-flat) embedding of a circle S 1 = {z ∈ C : 47 |z| = 1} into 3-dimensional space. Two such knots are said to be equivalent if one can be continuously 48 deformed into the other without breaking or cutting. More precisely, f 1 : S 1 → R 3 and f 2 : are equivalent if and only if there is an orientation preserving homeomorphism of pairs (R 3 , f 1 (S 1 )) →

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(R 3 , f 2 (S 1 )). We often replace R 3 with S 3 = {(x, y, z, w) : x 2 + y 2 + z 2 + w 2 = 1} which is the 51 one-point compactification of 3-space.  Since 3-colorability is defined in terms of a representative diagram, we must show that it is invariant 74 under the Reidemeister moves. Observe that for a type I move (top of Fig. 5) all three colors must or they have the same color. In the latter case, the short arc that is introduced has the same color as these 77 two arcs do. In the former case, the short arc has the third color.

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For the type III move there are five cases to consider:      From these illustrations, we can see that if a diagram is 3-colorable, then any diagram of the knot is 92 3-colorable in a way that uses three distinct colors. Moreover, any coloring of the unknot is monochro-93 matic. Thus the unknot is distinct from the trefoil. Any attempt to 3-color the figure-8 knot with three 94 distinct colors will fail. We leave that as an exercise for the reader. To continue, we define a set X that has a binary operation : X × X → X (written in in-fix notation) 103 a quandle if the following three axioms hold: 104 I for any a ∈ X, we have a a = a; 105 II for each a, b ∈ X, there is a unique c ∈ X such that c b = a; 106 III for each a, b, c ∈ X, we have (a b) c = (a c) (b c).

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As indicated in Fig. 10, the axioms correspond to the Reidemeister moves in the sense that if a knot 108 diagram is colored by elements of a quandle X (in such a way that the under-arc towards which the hu-109 munculus's left hand points receives a product), then any diagram related by a sequence of Reidemeister 110 moves will also be colored. The tetrahedral quandle is defined similarly as rotations of the tetrahedron through a vertex. In general, 132 a (connected) quandle exhibits aspects of symmetry since it can be defined in terms of a binary operation 133 on a set of cosets of a group of automorphisms of the quandle. This description appears elsewhere [2,7,9] 134 and would take us far away from the current purposes of the paper. sphere S 3 = {(x, y, z, w) ∈ R 4 : x 2 + y 2 + z 2 + w 2 = 1}. For brevity, we write the image of the 138 embedding as K, and we speak of the knot K. The smooth or PL-locally-flat condition suffices to provide 139 a tubular neighborhood of the knot. This is a smooth embedding of a solid torus N : that is a tubular neighborhood of the knot. That is, letting D 2 = {(x, y) ∈ R 2 : x 2 + y 2 ≤ 1} denote the   First, a homotopy between paths γ 0 and γ 1 is a map H : We say that two paths are homotopic if there is a homotopy between them. This induces an equivalence relation on the set of paths; we call an equivalence class a loop in space. If α, β : are a pair of paths, then they can be multiplied by the rule This means that a particle traveling along the composition first travels along α at double speed and then 149 along β at double speed. 150 We define a group structure on the set of homotopy classes of loops by declaring the composition to 151 be induced by path multiplication, the identity element is the equivalence class of the constant path, and 152 the inverse of α is to traverse α backwards (specifically α −1 (s) = α(1 − s)). The illustrations in Fig. 12 indicate the geometric notions, and outline the homotopies that are needed to demonstrate that α · α −1 154 is homotopic to the constant map, and that · induces an associative product on equivalence classes. In 155 this figure, the exterior of the trefoil knot is shown as the space that is interior to the torus. It is not easy 156 to make the intuitive leap from the complement of the knot as depicted in a non-compact 3-dimensional 157 space to the concise picture given. Following the discussion about handles in the next section, we will demonstrate that the fundamental 159 group can be generated by the set of loops that encircle each of the arcs that appear in the diagram. We

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For any non-negative integer k, let sphere which is the boundary of the disk. In case k = 0, the k-disk is a point and its boundary is empty.

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A k-handle is a subset of a topological space that is homeomorphic to D k × D n−k . An explicit 179 identification between the subset and this product of disks is assumed throughout. There are several 180 important subsets of the k-handle.

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• The core disk is the subset D k × {0}.

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• The co-core disk is the disk {0} × D n−k .

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• The attaching region is S k−1 × D n−k .

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• The belt region is D k × S n−k−1 .

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Of course, a k-handle is also an (n − k)-handle. When we consider it as such, we are turning the   Boy's surface (Fig. 17) is an immersion of the projective plane that is immersed in 3-dimensional boundary. We will use the knot diagram to construct a handle decomposition of the knot complement.

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In this decomposition, the arcs correspond to 1-handles and the crossings (under arcs) correspond to 229 2-handles.

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In order to help develop this description in general, we begin with a decomposition of the unknot 231 (Fig. 19). A 0-handle appears at the top of the diagram. A 1-handle is attached in such a way that its    The arcs of the diagram correspond to holes that have been drilled from the space above the diagram 249 as indicated in Fig. 20. Such "holes removed" can be reinterpreted as 1-handles that are attached to a 250 0-handle that is envisioned as lying above the plane in which the diagram is drawn. The crossings -251 representing arcs that go under the plane of the diagram -define 2-handles.

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The most simple case that can be depicted and that includes a crossing is that of the unknot with one  The intersections between 1-handles and 2-handles can be indicated completely within a planar dia-260 gram built from the knot diagram. Figure 22 demonstrates one such diagram associated to the trefoil.

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Over arcs correspond to the segments of the co-cores of the 1-handles that intersect the plane of the 262 diagram. The attaching spheres for the 2-handles encircle the pair of segments of the under crossing.
Any path in space is homotopic to a path that misses the 3-handle. Thus any loop in space can be moved into the union of the 0, 1, and 2-handles. We realize, therefore that the intersection sequences inside.
Attach 3-handle A,B, and C define a presentation for the fundamental group of the trefoil. Specifically, the fundamental group is given: x, y, z : Just as in the case of surfaces, handles in the decompositions of knot exteriors (or any 3-dimensional       In the initial stage of this decomposition, the complement is built out from the boundary torus. The 306 "upside-down" decomposition of the torus that is depicted on the lower right side of Fig. 15 is thickened 307 to a decomposition of the torus times an interval. The 0-handle envelopes the torus. The 1-handles are 308 spatial regular neighborhoods of the letter C, with the co-core disks being planar neighborhoods. These 309 1-handles correspond to the longitude and meridian of the knot, but be aware that since the cores are 310 short arcs, that which is apparently a longitude is a segment of the co-core disk of the meridian and vice 311 versa. Figure 25 contains the details.

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At each crossing a 1-handle is attached. It is a pillar erected between neighborhoods of arcs at the 313 crossing. The core disk runs from the lower arc to the upper arc, and the belt sphere is oriented in a 314 counter clockwise fashion. The attaching sphere lies on the 0-handle that envelopes the boundary torus.    We close this discussion with the remark that the dot notation in [1] can be chosen to coincide with 340 the incidence of the meridional co-core and the regions with some crossing conventions. Thus there 341 is a calculus to go from this presentation of the fundamental group to a presentation of the Alexander 342 module. This topic and more will be the subject of a subsequent paper.