question: Why are knots in wood so difficult to cut and/or drill through?
comment: Wood is made up of long fibers. The wood's "grain" is determined by the direction that these fibers point in. The direction of the grain affects how the wood feels to cut - cutting *across* the grain chops the fibers in half, whilst cutting *with* the grain separates them from each other. You need different tools to make these cuts - [crosscut saws]( URL_1 ) and [rip saws]( URL_2 ), respectively. A knot is formed when the trunk of a tree grows over where a branch used to be before falling off. The grain of the wood changes significantly around a knot - up to [90 degrees]( URL_0 ). So when you cut into a knot, you wind up encountering all sorts of different grain directions, and your saw is going to behave poorly *somewhere* in that cut.
passages: When a tree is very young it is covered with limbs almost, if not entirely, to the ground, but as it grows older some or all of them will eventually die and are either broken off or fall off. Subsequent growth of wood may completely conceal the stubs which will however remain as knots. No matter how smooth and clear a log is on the outside, it is more or less knotty near the middle. Consequently, the sapwood of an old tree, and particularly of a forest-grown tree, will be freer from knots than the inner heartwood. Since in most uses of wood, knots are defects that weaken the timber and interfere with its ease of working and other properties, it follows that a given piece of sapwood, because of its position in the tree, may well be stronger than a piece of heartwood from the same tree.
   The types of wood warping include:
  
  BULLET::::- bow: a warp along the length of the face of the wood
  
  BULLET::::- crook: (also called wain) a warp along the length of the edge of the wood
  
  BULLET::::- kink: a localized crook, often due to a knot
  
  BULLET::::- cup: a warp across the width of the face, in which the edges are higher or lower than the center of the wood
  
  BULLET::::- twist or wind: a distortion in which the two ends do not lie on the same plane. Winding sticks assist in viewing this defect.
   In some decorative applications, wood with knots may be desirable to add visual interest. In applications where wood is painted, such as skirting boards, fascia boards, door frames and furniture, resins present in the timber may continue to 'bleed' through to the surface of a knot for months or even years after manufacture and show as a yellow or brownish stain. A knot primer paint or solution (knotting), correctly applied during preparation, may do much to reduce this problem but it is difficult to control completely, especially when using mass-produced kiln-dried timber stocks.
  
  Section::::Physical properties.:Heartwood and sapwood.
   In the tree a knot is either the base of a side branch or a dormant bud. A knot (when the base of a side branch) is conical in shape (hence the roughly circular cross-section) with the inner tip at the point in stem diameter at which the plant's vascular cambium was located when the branch formed as a bud.
   In grading lumber and structural timber, knots are classified according to their form, size, soundness, and the firmness with which they are held in place. This firmness is affected by, among other factors, the length of time for which the branch was dead while the attaching stem continued to grow.
   Tools are sometimes employed in the finishing or untying of a knot, such as a fid, a tapered piece of wood that is often used in splicing. With the advent of wire rope, many other tools are used in the tying of "knots." However, for cordage and other non-metallic appliances, the tools used are generally limited to sharp edges or blades such as a sheepsfoot blade, occasionally a fine needle for proper whipping of laid rope, a hot cutter for nylon and other synthetic fibers, and (for larger ropes) a shoe for smoothing out large knots by rolling them on the ground.
   BULLET::::- Freeing – Working a badly fallen tree that may have become trapped in other trees. Working out maximum tension locations to decide the safest way to release tension, and a winch may be needed in complicated situations. To avoid cutting straight through a tree in tension, one or two cuts at the tension point of sufficient depth to reduce tension may be necessary. After tension releases, cuts are made outside the bend.
  
  BULLET::::- Limbing – Cutting the branches off the log. The operator must be able to properly reach the cut to avoid kickback.
   As a tree grows, lower branches often die, and their bases may become overgrown and enclosed by subsequent layers of trunk wood, forming a type of imperfection known as a knot. The dead branch may not be attached to the trunk wood except at its base, and can drop out after the tree has been sawn into boards. Knots affect the technical properties of the wood, usually reducing the local strength and increasing the tendency for splitting along the wood grain, but may be exploited for visual effect. In a longitudinally sawn plank, a knot will appear as a roughly circular "solid" (usually darker) piece of wood around which the grain of the rest of the wood "flows" (parts and rejoins). Within a knot, the direction of the wood (grain direction) is up to 90 degrees different from the grain direction of the regular wood.
   Knots do not necessarily influence the stiffness of structural timber, this will depend on the size and location. Stiffness and elastic strength are more dependent upon the sound wood than upon localized defects. The breaking strength is very susceptible to defects. Sound knots do not weaken wood when subject to compression parallel to the grain.
   Knot (papermaking)
  
  Knots are unwanted, large, dark aggregates of wood fibres when making chemical pulp.
   One or more of the rope characteristics (strength, durability, and flexibility) is often altered, depending upon the ultimate use of the rope, at the expense of the other properties. For example, rope used in caving is generally exposed to increased abrasion, so the mantle is woven more tightly than rope used in climbing or rappelling. However, the resulting rope is cumbersome and difficult to tie knots in.
   It is impossible to cut knurling "Like extremely coarse pitch threads" both because lathe gear trains will not support such longitudinal speeds and because reasonable cutting speeds would be impossible to achieve.
  
  Section::::Types.
  
  BULLET::::- Annular rings: Frequently used when the mating part is plastic. Rings allow for easy mating but ridges make it difficult to pull the components apart.
  
  BULLET::::- Linear knurl: Used with mating plastic pieces, the Linear Knurl allows greater torsion between components.
   Withington's original binder used wire to tie the bundles. There were problems with using wire and it was not long before William Deering invented a binder that successfully used twine and a knotter (invented in 1858 by John Appleby).
   In most situations forming loops and bends with conventional knots is far more practical than using rope splices, even though the latter can maintain nearly the rope's full strength. Prudent users allow for a large safety margin in the strength of rope chosen for a task due to the weakening effects of knots, aging, damage, shock loading, etc. The working load limit of a rope is generally specified with a significant safety factor, up to 15:1 for critical applications. For life-threatening applications, other factors come into play.
  
  Section::::Properties.:Security.
   Knots are incompletely cooked wood chips coming out of the digester. Their origin is often dense parts of branches, such as compression wood or timber knots – hence the name. Knots can also stem from large/oversized wood chips due to insufficient impregnation with cooking liquors. The content of lignin is very high. In kraft pulping knots are typically 0.5–3.0 % of the digester throughput. The knots are screened from the pulp, because if left in the pulp they may damage washing equipment and consume large amounts of bleaching chemicals. They are normally sent back to the digester and re-cooked so that their fibres are not wasted.
   The lower bound (big Omega) is shown with two families (("k", "k"−1) torus knots and "k"-Hopf links) that realize this bound. A former upper bound of O(Cr("C")) has been shown using Hamiltonian cycles in graphs embedded in a cubic integer lattice. The current best near-linear upper bound was established with a divide-and-conquer argument to show that minimum projections of knots can be embedded as planar graphs in the cubic lattice. However, no one has yet observed a knot family with super-linear length dependence "L"("C")  O(Cr("CK")) and it is conjectured that the upper bound is in fact linear.
  
  Section::::Ropelength as a knot invariant.
   Hand-tying of knots is a very labour-intensive task. An average weaver can tie almost 10,000 knots per day. More difficult patterns with an above-average knot density can only be woven by a skillful weaver, thus increasing the production costs even more. An average weaver may tie 360 knots per hour (1/10 seconds), while 1200 knots approaches the maximum a skilful weaver can tie per hour.
   Once screw turning machines were in common use, most commercially available wood screws were produced with this method. These cut wood screws are almost invariably tapered, and even when the tapered shank is not obvious, they can be discerned because the threads do not extend past the diameter of the shank. Such screws are best installed after drilling a pilot hole with a tapered drill bit. The majority of modern wood screws, except for those made of brass, are formed on thread rolling machines. These screws have a constant diameter, threads with a larger diameter than the shank, and are stronger because the rolling process does not cut the grain of the metal.
   In 2000, William R. Taylor identified a deep knot in acetohydroxy acid isomeroreductase (PDB ID: 1YVE), by using an algorithm that smooths protein chains and makes knots more visible. The algorithm keeps both termini fixed, and iteratively assigns to the coordinates of each residue the average of the coordinates of the neighboring residues. It has to be made sure that the chains do not pass through each other, otherwise the crossings and therefore the knot might get destroyed. If there is no knot, the algorithm eventually produces a straight line that joins both termini.
   A deep knot is preserved even though the removal of a considerable number of residues from either end does not destroy the knot. The higher the number of residues that can be removed without destroying the knot, the deeper the knot.
  
  Section::::Formation of knots.
   It is important for safety reasons to mention that the rope used for the mechanics should be smaller in diameter than the tension rope. This allows for movement but when a large amount of tension is applied to the mechanical rope, friction is created between the two ropes allowing the friction knot to lock on to the tension rope. If two identical ropes of the same diameter are used for the friction knot and tension rope, the knot may move freely like a slip knot (lasso or noose) and not lock into place. When encircling any cylindrical object, most ropes can only be tightened to a diameter slightly greater than the ropes own diameter(USMC Assault Climber Course).
   As a rule, reaction wood is undesirable in any structural application, primarily as its mechanical properties are different from normal wood: it alters the uniform structural properties of timber. Reaction wood can twist, cup or warp dramatically during machining. This movement can occur during the milling process, making it occasionally dangerous to perform certain operations without appropriate safety controls in place. For instance, ripping a piece of reaction wood on a table saw without a splitter or riving knife installed can lead to kick back of the stock. Reaction wood also responds to moisture differently from normal wood.
  
  Section::::See also.
   A complete algorithmic solution to this problem exists, which has unknown complexity. In practice, knots are often distinguished by using a "knot invariant", a "quantity" which is the same when computed from different descriptions of a knot. Important invariants include knot polynomials, knot groups, and hyperbolic invariants.
  
  The original motivation for the founders of knot theory was to create a table of knots and links, which are knots of several components entangled with each other. More than six billion knots and links have been tabulated since the beginnings of knot theory in the 19th century.
   Tait subsequently began listing unique knots in the belief that he was creating a table of elements. He formulated what are now known as the Tait conjectures on alternating knots. (The conjectures were proved in the 1990s.) Tait's knot tables were subsequently improved upon by C. N. Little and Thomas Kirkman.
   Traditionally, knots have been catalogued in terms of crossing number. Knot tables generally include only prime knots, and only one entry for a knot and its mirror image (even if they are different) . The number of nontrivial knots of a given crossing number increases rapidly, making tabulation computationally difficult . Tabulation efforts have succeeded in enumerating over 6 billion knots and links . The sequence of the number of prime knots of a given crossing number, up to crossing number 16, is 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988, , , ... . While exponential upper and lower bounds for this sequence are known, it has not been proven that this sequence is strictly increasing .
  