%This is a smaller-sized version (slightly edited) of my 1998 ICM talk.
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\centerline{POLYNOMIAL KNOTS}
\centerline{Alan H. Durfee}
\centerline{Mount Holyoke College, Massachusetts}
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Mount Holyoke College summer research program for undergraduates (REU)
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STUDENT PARTICIPANTS 1998:
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Eva Curry (University of Maine)
Eli Lebow (Harvard)
Bryant Mathews (Simon's Rock/Harvard)
Sang Pahk (Williams)
Melanie Pivarski (Carnegie Mellon)
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Also a group working on coding theory with Giuliana Davidoff.
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DETAILS ON OUR REU PROGRAM:
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Started in 1988
Criteria for research topics
Undergraduates doing research
Supported by the National Science Foundation
Two months (June and July)
Daily schedule
More: http://www.mtholyoke.edu/acad/math/reu
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A {\it polynomial knot} is a polynomial map $K : \real \to \real^3$
which is a smooth embedding.
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($K(t) = (x(t), y(t), z(t))$ where $x(t),$ $y(t)$,$z(t)$ are real
polynomials)
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Example: The trefoil (picture)
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BACKGROUND:
(1) Shastri (1992): Question of Abhyankar: Equivalence of embeddings of
$\real \to \real^3$ and $\complex \to \complex^3$.
Examples of polynomial knots:
\ \ \ Trefoil
$$x(t) = t^3 - 3t$$
$$y(t) = t^4 - 4t^2$$
$$z(t) = t^5 - 10t$$
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\ \ \ Figure-eight (picture):
$$x(t) = t^3 - 3t$$
$$y(t) = t(t^2-1)(t^4-4)$$
$$z(t) = t^7 -42 t$$
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The {\it degree} $deg(K)$ of a polynomial knot $K$ is the maximum of the degrees of
$x(t),$ $y(t)$,$z(t)$.
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(2) Vassiliev (1990): Approximate the (infinite dimensional)
space ${\cal K}$ of all
knots by the (finite dimensional) space of
polynomial knots of degree $d$.
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More precisely, let ${\cal K}^V_d$ be the space of polynomial knots of the
form
$$x(t) = t^d + a_1t^{d-1} + a_2t^{d-2} + \dots a_{d-1}t$$
$$y(t) = t^d + b_1t^{d-1} + b_2t^{d-2} + \dots b_{d-1}t$$
$$z(t) = t^d + c_1t^{d-1} + c_2t^{d-2} + \dots c_{d-1}t$$
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(3) Vassiliev, On spaces of polynomial knots (1996):
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\ \ \ (a) ${\cal K}^V_3$ is contractible.
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\ \ \ (b) ${\cal K}^V_4$ is a homology $S^1$.
\ \ \ In particular, $d \leq 4$ implies that $K$ is unknotted.
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\ \ \ (c) For $d$ even, ${\cal K}^V_c = X \times S^1$ for some space $X$.
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PROBLEM: FIND MORE EQUATIONS FOR KNOTS
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Shastri (1992): Trefoil, Figure-eight
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Auerbach (REU 1995): Torus knots of type (5,2), (7,2), and (9,2)
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Shastri: Any knot has a polynomial approximation.
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Proof: First find $x(t), y(t)$ (the diagram of the knot projection to
the $xy$-plane). (Details?)
Then find $z(t)$ so that the underpasses and overpasses are OK.
QED
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1998: Equations for many more knots
\ \ \ Picture of torus knot of type (5,2)
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Problem: The equations are complicated!
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MINIMAL DEGREE
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Problem: Given a knot, what is the minimal degree in which it can be
realized?
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Relate polynomial knots to usual knots:
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Choose identification $\real^3 \approx \sphere^3 - \{N\}$.
(for example, by stereographic projection)
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If $K \subset \real^3$ is a polynomial knot, let $\hat K \subset
\sphere^3$ be the closure of its image in the sphere.
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$\hat K$ is an oriented, tame knot. (May have cusp at the north pole)
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We say that oriented tame knots in $\sphere^3$ are {\it equivalent} if they are
isotopic.
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Proposition: Let $K$ be a polynomial knot, and let $T: \real^3 \to
\real^3$ be an orientation preserving linear transformation (det
$\neq 0$). Then $\hat K$ and $\hat{ TK}$ are equivalent.
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Proof: The set of such $T$ is connected. (The kink at the north
pole moves around.)
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Thus can assume that
$$deg \ x(t) > deg \ y(t) > deg \ z(t) $$
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CROSSING NUMBER
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The {\it crossing number} $c(L)$ of $L \subset \sphere^3$ is the least number
of crossings in any projection of any $L'$ equivalent to $L$.
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Proposition (L. Rudolph, 1995): If $K$ is a polynomial knot, then
$$c(\hat K) \leq (1/2)(deg(K) -2)(deg(K) -3)$$
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Proof: Project $K$ to the $xy$-plane.
Suppose $x(t)$ has degree $p$ and $y(t)$ has degree $q$.
Crossings are numbers $s \neq t$ such that $x(s) = x(t)$ and $y(s) =
y(t)$.
Look at
$$ {x(s) - x(t) \over s-t} = 0$$
This is a polynomial equation of degree $p-1$
$${ y(s) - y(t) \over s-t} = 0$$
This is a polynomial equation of degree $q-1$
Bezout's theorem implies that they have at most $(p-1)(q-1)$ intersections.
Since $(s,t)$ is the same as $(t,s)$, there are at most
$(1/2)(p-1)(q-1)$ crossings. QED
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Corollary:
\ \ \ (a) If $deg(k) \leq 4$ then $\hat K$ is unknotted. (ie, partially
recover Vassiliev's result)
\ \ \ (b) If $deg(K) = 5$ and $\hat K$ is nontrivial, then $\hat K$ is
the trefoil.
\ \ \ (c) If $deg(K) = 6$ then $c(K) \leq 6$ and ?
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BRIDGE NUMBER
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The {\it bridge number} $bridge(L)$ of $L \subset \sphere^3$ is the
least number of bridges in any projection of any $L'$ equivalent to $L$.
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Proposition (Schubert, 1954) The bridge number of a torus knot of type
$(p,q)$ is $p$.
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Proposition (L. Rudolph, 1995) If $K$ is a polynomial knot, then
$$deg(K) \geq 2(bridge(\hat K) ) + 1$$
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Proof: Also
$bridge(L)$ is equal to
minimum over ${L' \sim L}$ of
minimum over directions $v$ of
the number of local maxima of $L$ in the direction $v$.
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Picture of $K$:
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In all cases,
$$bridge(\hat K) \leq (1/2)(deg(K) +1)$$
QED
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Corollary (again): If $degree(K) \leq 4$ then $\hat K \subset \sphere^3$ is
unknotted.
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SUPERBRIDGE NUMBER (Kuiper, 1987)
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The {\it superbridge number} $sbridge(L)$ of $L \subset \sphere^3$ is
equal to
minimum over ${L' \sim L}$ of
maximum over directions $v$ of
the number of local maxima of $L$ in the direction $v$.
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%Denote the torus knot of type (p,q) by $T_{p,q}$.
Proposition (Kuiper): The superbridge number of a torus knot of type
$(p,q$) is $q$ for $p < q < 2p$, and $2p$ if $2p < q$.
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Proposition (1998): If $K$ is a polynomial knot, then
$$deg(K) \geq 2(sbridge(\hat K)) -1$$
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Proof: Look at the number of local maxima in the $x$ direction.
Recall that $deg(x(t)) = deg(K)$
Have $sbridge(\hat K) \leq (1/2)(deg(x(t)) + 1)$. QED
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POLYNOMIAL KNOTS IN MINIMAL DEGREE
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If $deg(K) \leq 4$ then $\hat K$ is unknotted.
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If $deg(K) = 5$ then $\hat K$ is a trefoil
(or unknotted).
This is the only possibility in degree 5.
(Rudolph, 1995: crossing number).
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If $deg(K) = 6$ then $\hat K$ is a figure-eight (or one of the above).
These are the only possibilities.
(1998: construction; $\hat K$ is a two-bridge knot.)
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$deg(K) = 7$:
\ \ \ $5_1$ exists, and this is its minimal degree
(1998: construction; superbridge number)
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\ \ \ $8_{19}$ (the torus knot of type (3,4) exists, and this is its
minimal degree
(1998: construction; bridge number)
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CONNECTED SUMS
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Method of construction (1998):
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$deg(K_1 \# K_2) \approx deg(K_1) \ deg(K_2)$
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The connected sum of $n$ trefoils can be realized in degree $2n+3$, and
this is the minimal degree in which this can be done.
First example of an infinite class of knots.
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The torus knot of type $(m, m-1)$ for $m \leq 7$ can be realized in degree $2m-1$.
This is the minimal degree in which this can be done (bridge number).
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OTHER
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``New'' proof of $sbridge(L) > bridge(L)$
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State of the REU in its sixth week
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POLYGONAL KNOTS
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A {\it polygonal knot} is a polygon (whose sides are possibly unequal)
embedded in $\real^3$.
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Randell (1994), Adams (1994), Millett (1994), Adams REU (1997),
Meissen (1998), . . .
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Let $edge(K)$ denote the number of edges of a polygonal knot $K$.
If $edge(K) \leq 5$ then $K$ is unknotted.
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If $edge(K) = 6$ then $K$ is the trefoil or
the unknot.
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If $edge(K) = 7$ then $K$ is the figure-eight or one of the above.
These are the only possibilities
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If $edge(K) = 8$: All knots of 5 and 6 crossings
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SIMILARITIES OF POLYNOMIAL AND POLYGONAL KNOTS
Remove an edge from a polygonal knot and it looks like a polynomial knot:
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Also:
Connected sums of trefoils
Torus knot of type $(m,m-1)$
Estimates using crossing number, bridge number, superbridge number
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BUT
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Connected sums appear different:
\ \ \ For polynomial knots, in product of degrees.
\ \ \ For polygonal knots, in sum of degrees.
\ \ \ Also picture.
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