Knot theory via Seifert surfaces. Christopher William Davis Joint with Grant Roth (UWEC) 1/14 A link is an embedding of many disjoint circles into 3-dimensional space. It is often important to distinguish different components of the same link. The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other. =? =? Knots and Links A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. 2/14 It is often important to distinguish different components of the same link. The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other. =? =? Knots and Links A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. A link is an embedding of many disjoint circles into 3-dimensional space. 2/14 The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other. =? =? Knots and Links A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. A link is an embedding of many disjoint circles into 3-dimensional space. It is often important to distinguish different components of the same link. 2/14 Knots and Links A knot is an embedded circle in 3-dimensional space. A knot has a directionality, encoded with an arrow. A link is an embedding of many disjoint circles into 3-dimensional space. It is often important to distinguish different components of the same link. The central problem in all of knot and link theory is determining when two knots (or links) might be isotoped (deformed without cutting or sliding strands through each other) into each other. =? =? 2/14 Fact The linking number is not changed by isotopy. + lk(L1; L2) = +1 lk(L1; L2) = 0 No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? Answer: 3/14 Fact The linking number is not changed by isotopy. + lk(L1; L2) = +1 lk(L1; L2) = 0 Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? Answer: No. 3/14 Fact The linking number is not changed by isotopy. + lk(L1; L2) = +1 lk(L1; L2) = 0 The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. 3/14 Fact The linking number is not changed by isotopy. + lk(L1; L2) = +1 lk(L1; L2) = 0 For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. 3/14 + lk(L1; L2) = +1 lk(L1; L2) = 0 Fact The linking number is not changed by isotopy. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. 3/14 lk(L1; L2) = +1 lk(L1; L2) = 0 Fact The linking number is not changed by isotopy. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? + Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. 3/14 Fact The linking number is not changed by isotopy. Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? + lk(L1; L2) = +1 lk(L1; L2) = 0 Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. 3/14 Linking numbers Here are two links: The Hopf link and the unlink. Can the Hopf link be isotoped to the unlink? =? + lk(L1; L2) = +1 lk(L1; L2) = 0 Answer: No. Informally, Imagine a disk bounded by the red disk. There is no way of sliding the blue curve off of the red disk without crossing through the red curve. The linking number is a formalization of this idea. For a 2-component link L1 [ L2 the linking number lk(L1; L2) is given by counting how many times L1 passes \positively" over L2 and subtracting the number of \negative" times. Fact The linking number is not changed by isotopy. 3/14 − − + A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K. + lk(L1; L2) = 0 If F is ANY Seifert surface for L1 then lk(L1; L2) is given by counting up the number of times L2 passes through F in the positive direction and the subtracting the number of times is passes through in the negative direction. Seifert surfaces The definition previous is completely diagrammatic. Let's reformulate the linking number in a more geometric way. − − + + lk(L1; L2) = 0 4/14 − − + + lk(L1; L2) = 0 If F is ANY Seifert surface for L1 then lk(L1; L2) is given by counting up the number of times L2 passes through F in the positive direction and the subtracting the number of times is passes through in the negative direction. Seifert surfaces The definition previous is completely diagrammatic. Let's reformulate the linking number in a more geometric way. A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K. − − + + lk(L1; L2) = 0 4/14 − − + + lk(L1; L2) = 0 Seifert surfaces The definition previous is completely diagrammatic. Let's reformulate the linking number in a more geometric way. A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K. − − + + lk(L1; L2) = 0 If F is ANY Seifert surface for L1 then lk(L1; L2) is given by counting up the number of times L2 passes through F in the positive direction and the subtracting the number of times is passes through in the negative direction. 4/14 − + + lk(L1; L2) = 0 Seifert surfaces The definition previous is completely diagrammatic. Let's reformulate the linking number in a more geometric way. A Seifert surface for a knot K is an oriented surface (2-dimensional shape) bounded by K. − − + + − lk(L1; L2) = 0 If F is ANY Seifert surface for L1 then lk(L1; L2) is given by counting up the number of times L2 passes through F in the positive direction and the subtracting the number of times is passes through in the negative direction. 4/14 + + lk(L1; L2) = 0 Seifert surfaces The definition previous is completely diagrammatic.