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Classical Knot Invariants Classical knot invariants Jennifer Schultens Jennifer Schultens Classical knot invariants Knots Definition 3 A knot K in S is a smooth isotopy class of smooth embeddings of 1 3 S into S . Jennifer Schultens Classical knot invariants Knots Figure: The unknot Jennifer Schultens Classical knot invariants Knots Figure: The unknot Jennifer Schultens Classical knot invariants Knots Figure: A knot Jennifer Schultens Classical knot invariants Knots Figure: Two knots Jennifer Schultens Classical knot invariants Knots Figure: The connected sum of two knots Jennifer Schultens Classical knot invariants Crossing numbers of knots Crossing number Jennifer Schultens Classical knot invariants Example 1: The unknot has crossing number 0. Crossing numbers of knots Definition The crossing number of a knot K is the least number of crossings in a diagram of K. Jennifer Schultens Classical knot invariants Crossing numbers of knots Definition The crossing number of a knot K is the least number of crossings in a diagram of K. Example 1: The unknot has crossing number 0. Jennifer Schultens Classical knot invariants This can be seen by checking all diagrams with crossing number 0; 1; 2 and noting that they do not give the trefoil. (Except that you need to know that the trefoil is truly knotted.) Theorem A reduced alternating diagram of a knot K realizes the crossing number of K. Crossing numbers of knots Example 2: The trefoil has crossing number 3. Jennifer Schultens Classical knot invariants Theorem A reduced alternating diagram of a knot K realizes the crossing number of K. Crossing numbers of knots Example 2: The trefoil has crossing number 3. This can be seen by checking all diagrams with crossing number 0; 1; 2 and noting that they do not give the trefoil. (Except that you need to know that the trefoil is truly knotted.) Jennifer Schultens Classical knot invariants Crossing numbers of knots Example 2: The trefoil has crossing number 3. This can be seen by checking all diagrams with crossing number 0; 1; 2 and noting that they do not give the trefoil. (Except that you need to know that the trefoil is truly knotted.) Theorem A reduced alternating diagram of a knot K realizes the crossing number of K. Jennifer Schultens Classical knot invariants Fun interlude Figure: T(2, 3), also known as the trefoill Jennifer Schultens Classical knot invariants Theorem: c(T (p; q)) = minfq(p − 1); p(q − 1)g Fun interlude Question 1: What is the crossing number of a torus knot, T (p; q)? Jennifer Schultens Classical knot invariants Fun interlude Question 1: What is the crossing number of a torus knot, T (p; q)? Theorem: c(T (p; q)) = minfq(p − 1); p(q − 1)g Jennifer Schultens Classical knot invariants Nothing! Fun interlude Question 2: What is known about the crossing number of a satellite knot? Jennifer Schultens Classical knot invariants Fun interlude Question 2: What is known about the crossing number of a satellite knot? Nothing! Jennifer Schultens Classical knot invariants Bridge numbers of knots Bridge number Jennifer Schultens Classical knot invariants Definition Let K be a knot. The bridge number, b(K), of K, is the least possible number of maxima of K with respect to a height function. Bridge numbers of knots Definition 3 3 A height function on R (or S ) is a smooth function 3 h : R ! R 3 (or h : S ! R) without critical points. Jennifer Schultens Classical knot invariants Bridge numbers of knots Definition 3 3 A height function on R (or S ) is a smooth function 3 h : R ! R 3 (or h : S ! R) without critical points. Definition Let K be a knot. The bridge number, b(K), of K, is the least possible number of maxima of K with respect to a height function. Jennifer Schultens Classical knot invariants In fact, the unknot is the only knot with bridge number 1. Example 2: The trefoil (a nontrivial knot) has bridge number 2. Theorem (Schubert): b(K1#K2) = b(K1) + b(K2) − 1 Bridge numbers of knots Example 1: The unknot has bridge number 1. Jennifer Schultens Classical knot invariants Example 2: The trefoil (a nontrivial knot) has bridge number 2. Theorem (Schubert): b(K1#K2) = b(K1) + b(K2) − 1 Bridge numbers of knots Example 1: The unknot has bridge number 1. In fact, the unknot is the only knot with bridge number 1. Jennifer Schultens Classical knot invariants Theorem (Schubert): b(K1#K2) = b(K1) + b(K2) − 1 Bridge numbers of knots Example 1: The unknot has bridge number 1. In fact, the unknot is the only knot with bridge number 1. Example 2: The trefoil (a nontrivial knot) has bridge number 2. Jennifer Schultens Classical knot invariants Bridge numbers of knots Example 1: The unknot has bridge number 1. In fact, the unknot is the only knot with bridge number 1. Example 2: The trefoil (a nontrivial knot) has bridge number 2. Theorem (Schubert): b(K1#K2) = b(K1) + b(K2) − 1 Jennifer Schultens Classical knot invariants Bridge numbers of knots K1 K2 Figure: Schematic for bridge number inequality Jennifer Schultens Classical knot invariants Fun interlude Figure: T(2, 3), also known as the trefoill Jennifer Schultens Classical knot invariants Answer: min(p; q). One direction is easy, the other is a theorem of Schubert. Question 2: What can we say about the bridge number of a satellite knot? Answer: It's at least the bridge number of the companion knot times the wrapping number of the pattern. Fun interlude Question 1: What is the bridge number of a torus knot T (p; q)? Jennifer Schultens Classical knot invariants One direction is easy, the other is a theorem of Schubert. Question 2: What can we say about the bridge number of a satellite knot? Answer: It's at least the bridge number of the companion knot times the wrapping number of the pattern. Fun interlude Question 1: What is the bridge number of a torus knot T (p; q)? Answer: min(p; q). Jennifer Schultens Classical knot invariants Question 2: What can we say about the bridge number of a satellite knot? Answer: It's at least the bridge number of the companion knot times the wrapping number of the pattern. Fun interlude Question 1: What is the bridge number of a torus knot T (p; q)? Answer: min(p; q). One direction is easy, the other is a theorem of Schubert. Jennifer Schultens Classical knot invariants Answer: It's at least the bridge number of the companion knot times the wrapping number of the pattern. Fun interlude Question 1: What is the bridge number of a torus knot T (p; q)? Answer: min(p; q). One direction is easy, the other is a theorem of Schubert. Question 2: What can we say about the bridge number of a satellite knot? Jennifer Schultens Classical knot invariants Fun interlude Question 1: What is the bridge number of a torus knot T (p; q)? Answer: min(p; q). One direction is easy, the other is a theorem of Schubert. Question 2: What can we say about the bridge number of a satellite knot? Answer: It's at least the bridge number of the companion knot times the wrapping number of the pattern. Jennifer Schultens Classical knot invariants Tunnel numbers of knots Tunnel number Jennifer Schultens Classical knot invariants Preliminaries for tunnel number Figure: A handlebody Jennifer Schultens Classical knot invariants Preliminaries for tunnel number Figure: Another handlebody Jennifer Schultens Classical knot invariants Definition Let K be a knot. A tunnel system for K is a collection of arcs 3 a1;:::; an properly embedded in (S ; K) such that 3 S − η(K [ a1 [···[ an) is a handlebody. Definition The tunnel number of K is the least number of arcs in a tunnel system for K. Tunnel numbers of knots Definition A handlebody is a 3-dimensional regular neighborhood of a graph. Jennifer Schultens Classical knot invariants Definition The tunnel number of K is the least number of arcs in a tunnel system for K. Tunnel numbers of knots Definition A handlebody is a 3-dimensional regular neighborhood of a graph. Definition Let K be a knot. A tunnel system for K is a collection of arcs 3 a1;:::; an properly embedded in (S ; K) such that 3 S − η(K [ a1 [···[ an) is a handlebody. Jennifer Schultens Classical knot invariants Tunnel numbers of knots Definition A handlebody is a 3-dimensional regular neighborhood of a graph. Definition Let K be a knot. A tunnel system for K is a collection of arcs 3 a1;:::; an properly embedded in (S ; K) such that 3 S − η(K [ a1 [···[ an) is a handlebody. Definition The tunnel number of K is the least number of arcs in a tunnel system for K. Jennifer Schultens Classical knot invariants Why? Preliminaries for tunnel number Figure: The complement of the unknot The complement of the unknot is a solid torus, which is a 3-dimensional regular neighborhood of the circle (a graph), hence a handlebody. Jennifer Schultens Classical knot invariants Preliminaries for tunnel number Figure: The complement of the unknot The complement of the unknot is a solid torus, which is a 3-dimensional regular neighborhood of the circle (a graph), hence a handlebody. Why? Jennifer Schultens Classical knot invariants The unknot is isotopic to 1 2 3 f(x; y; p ; 0) 2 T g 2 S 2 2 2 1 1 1 3 η(unknot) ≈ f(x; y; z; w)jx + y = ; z > p ; w < p g 2 S 2 2 2 2 2 3 C(K) = S − η(unknot) 2 2 1 1 1 3 ≈ f(x; y; z; w)jx + y = ; z ≤ p ; w ≥ p g 2 S 2 2 2 2 2 Preliminaries for tunnel number 1 1 2 = f(x; y; z; w) 2 4jx2 + y 2 = ; w 2 + z2 = g T R 2 2 Jennifer Schultens Classical knot invariants 2 2 1 1 1 3 η(unknot) ≈ f(x; y; z; w)jx + y = ; z > p ; w < p g 2 S 2 2 2 2 2 3 C(K) = S − η(unknot) 2 2 1 1 1 3 ≈ f(x; y; z; w)jx + y = ; z ≤ p ; w ≥ p g 2 S 2 2 2 2 2 Preliminaries for tunnel number 1 1 2 = f(x; y; z; w) 2 4jx2 + y 2 = ; w 2 + z2 = g T R 2 2 The unknot is isotopic to 1 2 3 f(x; y; p ; 0) 2 T g 2 S 2 Jennifer Schultens Classical knot invariants Preliminaries for tunnel number 1 1 2 = f(x; y; z; w) 2 4jx2 + y 2 = ; w 2 + z2 = g T R 2 2 The unknot is isotopic to 1 2 3 f(x; y; p ; 0) 2 T g 2 S 2 2 2 1 1 1 3 η(unknot) ≈ f(x; y; z; w)jx + y = ; z > p ; w < p g 2 S 2 2 2 2 2 3 C(K) = S − η(unknot) 2 2 1 1 1 3 ≈ f(x; y; z; w)jx + y = ; z ≤ p ; w ≥ p g 2 S 2 2 2 2 2 Jennifer Schultens Classical knot invariants In fact, the unknot is the only knot tunnel number 0.
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