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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Combinatorics for Knots
J. Ramırez Alfonsın
Universite Montpellier 2
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
1 Basic notions
2 Matroid
3 Knot coloring and the unknotting problem
4 Oriented matroids
5 Spatial graphs
6 Ropes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Reidemeister moves
I
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III !1
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
I
III II
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
III
I
II
I
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
III
I
II
I
I
II
I
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Bracket polynomial
For any link diagram D define a Laurent polynomial < D > in onevariable A which obeys the following three rules where U denotesthe unknot :
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Bracket polynomial
For any link diagram D define a Laurent polynomial < D > in onevariable A which obeys the following three rules where U denotesthe unknot :
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem For any link L the bracket polynomial is independent ofthe order in which rules (i)− (iii) are applied to the crossings.Further, it is invariant under the Reidemeister moves II and III
but it is not invariant under Reidemeister move I ! !
The writhe of an oriented link diagram D is the sum of the signsat the crossings of D (denoted by ω(D)).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem For any link L the bracket polynomial is independent ofthe order in which rules (i)− (iii) are applied to the crossings.Further, it is invariant under the Reidemeister moves II and III
but it is not invariant under Reidemeister move I ! !
The writhe of an oriented link diagram D is the sum of the signsat the crossings of D (denoted by ω(D)).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
(D)=1
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!
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem For any link L define the Laurent polynomial
fD(A) = (−A3)ω(D) < L >
Then, fD(A) is an invariant of ambient isotopy.
Now, define for any link L
VL(t) = fD(t−1/4)
where D is any diagram representing L. Then VL(t) is the Jonespolynomial of the oriented link L.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem For any link L define the Laurent polynomial
fD(A) = (−A3)ω(D) < L >
Then, fD(A) is an invariant of ambient isotopy.
Now, define for any link L
VL(t) = fD(t−1/4)
where D is any diagram representing L. Then VL(t) is the Jonespolynomial of the oriented link L.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Matroids
Let E a finite set. A matroid is a family B of subsets of Everifying certain axioms (the family B is called the bases of thematroid)
There is a natural way to obtain a matroid M from a graph G (theset of bases on M is given by the family of all spanning trees of G ).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Matroids
Let E a finite set. A matroid is a family B of subsets of Everifying certain axioms (the family B is called the bases of thematroid)
There is a natural way to obtain a matroid M from a graph G (theset of bases on M is given by the family of all spanning trees of G ).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
3
1
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4
1
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
The Tutte polynomial of a matroid M(E ) is defined to be the2-variable polynomial
T (M; x , y) =∑A⊆E
(x − 1)r(E)−r(A)(y − 1)|A|−r(A)
where r is the rank function of M.
Remark :The evaluation of the Tutte polynomial in certain pointsmay count something.
Example : T (1, 1) counts the number of bases in the matroid.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
The Tutte polynomial of a matroid M(E ) is defined to be the2-variable polynomial
T (M; x , y) =∑A⊆E
(x − 1)r(E)−r(A)(y − 1)|A|−r(A)
where r is the rank function of M.
Remark :The evaluation of the Tutte polynomial in certain pointsmay count something.
Example : T (1, 1) counts the number of bases in the matroid.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
The Tutte polynomial of a matroid M(E ) is defined to be the2-variable polynomial
T (M; x , y) =∑A⊆E
(x − 1)r(E)−r(A)(y − 1)|A|−r(A)
where r is the rank function of M.
Remark :The evaluation of the Tutte polynomial in certain pointsmay count something.
Example : T (1, 1) counts the number of bases in the matroid.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
CA
B
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
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A
B
C
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A link diagram is alternating if the crossings alternateunder-over-under-over ... as the link is traversed.
A link is alternating if there is an alternating link diagramrepresenting L.
Theorem (Thistlethwaite 1987) If D is an oriented alternating linkdiagram and G denotes its associated unsigned ‘blackface’ graphthen
VL(t) = (t−1/4)3ω(D)−2T (M(G );−t,−t−1)
where M(G ) is the matroid associated to G .
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A link diagram is alternating if the crossings alternateunder-over-under-over ... as the link is traversed.
A link is alternating if there is an alternating link diagramrepresenting L.
Theorem (Thistlethwaite 1987) If D is an oriented alternating linkdiagram and G denotes its associated unsigned ‘blackface’ graphthen
VL(t) = (t−1/4)3ω(D)−2T (M(G );−t,−t−1)
where M(G ) is the matroid associated to G .
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A link diagram is alternating if the crossings alternateunder-over-under-over ... as the link is traversed.
A link is alternating if there is an alternating link diagramrepresenting L.
Theorem (Thistlethwaite 1987) If D is an oriented alternating linkdiagram and G denotes its associated unsigned ‘blackface’ graphthen
VL(t) = (t−1/4)3ω(D)−2T (M(G );−t,−t−1)
where M(G ) is the matroid associated to G .
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knot coloring
A knot diagram K is called colorable if each arc can be drawnusing one of three colors (say, red, blue, green) in such a way that1) at least two of the colors are used and2) at any crossing at which two colors appear, all three appear.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knot coloring
A knot diagram K is called colorable if each arc can be drawnusing one of three colors (say, red, blue, green) in such a way that1) at least two of the colors are used and2) at any crossing at which two colors appear, all three appear.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knot coloring
A knot diagram K is called colorable if each arc can be drawnusing one of three colors (say, red, blue, green) in such a way that1) at least two of the colors are used and2) at any crossing at which two colors appear, all three appear.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A knot diagram K is called colorable mod n if each arc can belabeled with an integer from 0 to p − 1 such a way that1) at least two labels are distinct and2) at each crossing the relation 2x − y − z = 0 mod p holds wherex is the label on the overcrossing and y and z the other two labels.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A knot diagram K is called colorable mod n if each arc can belabeled with an integer from 0 to p − 1 such a way that1) at least two labels are distinct and2) at each crossing the relation 2x − y − z = 0 mod p holds wherex is the label on the overcrossing and y and z the other two labels.
2
0 1
6
4
66
3
6
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem If a diagram of a knot K is colorable mod n then everydiagram of K is colorable mod n.
Corollary There exist non-trivial knots.
Corollary If a link is splittable then it is colorable mod 3.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem If a diagram of a knot K is colorable mod n then everydiagram of K is colorable mod n.
Corollary There exist non-trivial knots.
Corollary If a link is splittable then it is colorable mod 3.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem If a diagram of a knot K is colorable mod n then everydiagram of K is colorable mod n.
Corollary There exist non-trivial knots.
Corollary If a link is splittable then it is colorable mod 3.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
[UP] Unknotting problem : Given a knot diagram K , decidewhether K is trivial.
Tait 1877 Knot classification.Papakyriakopoulos 1957 A knot is trivial if and only if thefundamental group of its complement is abelian.
Haken 1961 [UP] is decidible.Hass, Lagarias and Pippenger 1999 [UP] is NP.
Is there a good method to simplify diagrams ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
[UP] Unknotting problem : Given a knot diagram K , decidewhether K is trivial.
Tait 1877 Knot classification.Papakyriakopoulos 1957 A knot is trivial if and only if thefundamental group of its complement is abelian.
Haken 1961 [UP] is decidible.Hass, Lagarias and Pippenger 1999 [UP] is NP.
Is there a good method to simplify diagrams ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
[UP] Unknotting problem : Given a knot diagram K , decidewhether K is trivial.
Tait 1877 Knot classification.Papakyriakopoulos 1957 A knot is trivial if and only if thefundamental group of its complement is abelian.
Haken 1961 [UP] is decidible.Hass, Lagarias and Pippenger 1999 [UP] is NP.
Is there a good method to simplify diagrams ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
[UP] Unknotting problem : Given a knot diagram K , decidewhether K is trivial.
Tait 1877 Knot classification.Papakyriakopoulos 1957 A knot is trivial if and only if thefundamental group of its complement is abelian.
Haken 1961 [UP] is decidible.Hass, Lagarias and Pippenger 1999 [UP] is NP.
Is there a good method to simplify diagrams ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Question (Welsh) Is there a function f such that any diagram ofthe trivial knot with n crossngs can be transformed to a a diagramwithout crossing by using at most f (n) Reidemeister moves ?
Theorem (Hass and Lagarias 2001) f (n) = 2cn where c = 1011.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Question (Welsh) Is there a function f such that any diagram ofthe trivial knot with n crossngs can be transformed to a a diagramwithout crossing by using at most f (n) Reidemeister moves ?
Theorem (Hass and Lagarias 2001) f (n) = 2cn where c = 1011.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Oriented matroids
Let E a finite set. An oriented matroid is a family C of signedsubsets of E verifying certain axioms (the family C is called thecircuits of the oriented matroid).
There is a natural way to obtain an oriented matroid from aconfiguration of points in IRd .
If C ∈ C conv(pos. elements C ) ∩ conv(neg. elements C ) 6= ∅.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Oriented matroids
Let E a finite set. An oriented matroid is a family C of signedsubsets of E verifying certain axioms (the family C is called thecircuits of the oriented matroid).
There is a natural way to obtain an oriented matroid from aconfiguration of points in IRd .
If C ∈ C conv(pos. elements C ) ∩ conv(neg. elements C ) 6= ∅.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Oriented matroids
Let E a finite set. An oriented matroid is a family C of signedsubsets of E verifying certain axioms (the family C is called thecircuits of the oriented matroid).
There is a natural way to obtain an oriented matroid from aconfiguration of points in IRd .
If C ∈ C conv(pos. elements C ) ∩ conv(neg. elements C ) 6= ∅.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Oriented matroids
Let E a finite set. An oriented matroid is a family C of signedsubsets of E verifying certain axioms (the family C is called thecircuits of the oriented matroid).
There is a natural way to obtain an oriented matroid from aconfiguration of points in IRd .
If C ∈ C conv(pos. elements C ) ∩ conv(neg. elements C ) 6= ∅.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Example : d = 3.
1 3
2
541
2
3
4
5
(+,+,+,+,!)(+,+,+,!,!)
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Cyclic Polytope
Let t1, . . . , tn ∈ IR. The cyclic polytope of dimension d with nvertices is defined as
Cd(t1, . . . , tn) := conv(x(t1), . . . , x(tn))
where x(ti ) = (ti , t2i , . . . , td
i ) are points of the moment curve
Cd(t1, . . . , tn) → Cd(n)
Upper bound theorem (McMullen 1970) The number of j-faces ofa d-dimensional polytope with n vertices is maximal for Cd(n).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Cyclic Polytope
Let t1, . . . , tn ∈ IR. The cyclic polytope of dimension d with nvertices is defined as
Cd(t1, . . . , tn) := conv(x(t1), . . . , x(tn))
where x(ti ) = (ti , t2i , . . . , td
i ) are points of the moment curve
Cd(t1, . . . , tn) → Cd(n)
Upper bound theorem (McMullen 1970) The number of j-faces ofa d-dimensional polytope with n vertices is maximal for Cd(n).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Cyclic Polytope
Let t1, . . . , tn ∈ IR. The cyclic polytope of dimension d with nvertices is defined as
Cd(t1, . . . , tn) := conv(x(t1), . . . , x(tn))
where x(ti ) = (ti , t2i , . . . , td
i ) are points of the moment curve
Cd(t1, . . . , tn) → Cd(n)
Upper bound theorem (McMullen 1970) The number of j-faces ofa d-dimensional polytope with n vertices is maximal for Cd(n).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Cyclic Polytope
Let t1, . . . , tn ∈ IR. The cyclic polytope of dimension d with nvertices is defined as
Cd(t1, . . . , tn) := conv(x(t1), . . . , x(tn))
where x(ti ) = (ti , t2i , . . . , td
i ) are points of the moment curve
Cd(t1, . . . , tn) → Cd(n)
Upper bound theorem (McMullen 1970) The number of j-faces ofa d-dimensional polytope with n vertices is maximal for Cd(n).
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (R.A. 2009) Let D(K ) be a diagram of a knot K on ncrossings. Then, there exists a cycle in C3(m) isotopic to K wherem ≤ 7n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (R.A. 2009) Let D(K ) be a diagram of a knot K on ncrossings. Then, there exists a cycle in C3(m) isotopic to K wherem ≤ 7n.
9
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
1
2
3
4
5
6
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J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Geometric algorithm
Given a diagram of a knot K .
Construct a cycle C in C3(m) isotopic to K .
Detect useless edges in C by using the circuits of the orientedmatroid associated to C3(m).
Theorem (R.A. 2010) This method detect if a diagram with ncrossing is trivial and its order is O(2cn) where c is a constant.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Geometric algorithm
Given a diagram of a knot K .
Construct a cycle C in C3(m) isotopic to K .
Detect useless edges in C by using the circuits of the orientedmatroid associated to C3(m).
Theorem (R.A. 2010) This method detect if a diagram with ncrossing is trivial and its order is O(2cn) where c is a constant.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 57: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/57.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Geometric algorithm
Given a diagram of a knot K .
Construct a cycle C in C3(m) isotopic to K .
Detect useless edges in C by using the circuits of the orientedmatroid associated to C3(m).
Theorem (R.A. 2010) This method detect if a diagram with ncrossing is trivial and its order is O(2cn) where c is a constant.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 58: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/58.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Geometric algorithm
Given a diagram of a knot K .
Construct a cycle C in C3(m) isotopic to K .
Detect useless edges in C by using the circuits of the orientedmatroid associated to C3(m).
Theorem (R.A. 2010) This method detect if a diagram with ncrossing is trivial and its order is O(2cn) where c is a constant.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 59: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/59.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Geometric algorithm
Given a diagram of a knot K .
Construct a cycle C in C3(m) isotopic to K .
Detect useless edges in C by using the circuits of the orientedmatroid associated to C3(m).
Theorem (R.A. 2010) This method detect if a diagram with ncrossing is trivial and its order is O(2cn) where c is a constant.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 60: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/60.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Spatial graphs
A spatial representation of a graph G is an embedding of G in IR3
where the vertices of G are points and edges are represented bysimple Jordan curves.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Spatial graphs
A spatial representation of a graph G is an embedding of G in IR3
where the vertices of G are points and edges are represented bysimple Jordan curves.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Spatial graphs
A spatial representation of a graph G is an embedding of G in IR3
where the vertices of G are points and edges are represented bysimple Jordan curves.Spatial representation of K5.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Let m(L) be the smallest integer such that any spatialrepresentation of Kn with n ≥ m(L) contains cycles isotopic to L.Theorem (Conway and Gordon 1983)• For any spatial representation of K6, it holds∑
(λ1,λ2)
lk(λ1, λ2) ≡ 1 mod 2
where (λ1, λ2) is a 2-component link contained in K6
and lk denotes the linking number.• For any spatial representation of K7, it holds∑
λ
Arf (λ) ≡ 1 mod 2
where λ is a 7-cycle of K7 and Arf denotes the Arf invariant.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Let m(L) be the smallest integer such that any spatialrepresentation of Kn with n ≥ m(L) contains cycles isotopic to L.Theorem (Conway and Gordon 1983)• For any spatial representation of K6, it holds∑
(λ1,λ2)
lk(λ1, λ2) ≡ 1 mod 2
where (λ1, λ2) is a 2-component link contained in K6
and lk denotes the linking number.• For any spatial representation of K7, it holds∑
λ
Arf (λ) ≡ 1 mod 2
where λ is a 7-cycle of K7 and Arf denotes the Arf invariant.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 65: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/65.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Let m(L) be the smallest integer such that any spatialrepresentation of Kn with n ≥ m(L) contains cycles isotopic to L.Theorem (Conway and Gordon 1983)• For any spatial representation of K6, it holds∑
(λ1,λ2)
lk(λ1, λ2) ≡ 1 mod 2
where (λ1, λ2) is a 2-component link contained in K6
and lk denotes the linking number.• For any spatial representation of K7, it holds∑
λ
Arf (λ) ≡ 1 mod 2
where λ is a 7-cycle of K7 and Arf denotes the Arf invariant.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A spatial representation is linear if the curves are line segments.
Let m(L) be the smallest integer such that any spatial linearrepresentation of Kn with n ≥ m(L) contains cycles isotopic to L.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A spatial representation is linear if the curves are line segments.
Let m(L) be the smallest integer such that any spatial linearrepresentation of Kn with n ≥ m(L) contains cycles isotopic to L.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
!"#$%&!&"#'(
)*+,-."/0
1
F8
23
32
2
( T )
T(5,2)
4%&,*".-
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Negami 1991) m(L) exists and it is finite for any link L.
Theorem (R.A. 1998) m(T or T ∗) = 7.
Theorem (R.A. 2000) m(421) > 7.
Theorem (R.A. 2009) m(F8),m(T (5, 2)) > 8.
Theorem (R.A. 2009) Let D(L) be a diagram of link L with ncrossings. Then,
m(L) ≤ 227n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Negami 1991) m(L) exists and it is finite for any link L.
Theorem (R.A. 1998) m(T or T ∗) = 7.
Theorem (R.A. 2000) m(421) > 7.
Theorem (R.A. 2009) m(F8),m(T (5, 2)) > 8.
Theorem (R.A. 2009) Let D(L) be a diagram of link L with ncrossings. Then,
m(L) ≤ 227n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Negami 1991) m(L) exists and it is finite for any link L.
Theorem (R.A. 1998) m(T or T ∗) = 7.
Theorem (R.A. 2000) m(421) > 7.
Theorem (R.A. 2009) m(F8),m(T (5, 2)) > 8.
Theorem (R.A. 2009) Let D(L) be a diagram of link L with ncrossings. Then,
m(L) ≤ 227n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Negami 1991) m(L) exists and it is finite for any link L.
Theorem (R.A. 1998) m(T or T ∗) = 7.
Theorem (R.A. 2000) m(421) > 7.
Theorem (R.A. 2009) m(F8),m(T (5, 2)) > 8.
Theorem (R.A. 2009) Let D(L) be a diagram of link L with ncrossings. Then,
m(L) ≤ 227n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Negami 1991) m(L) exists and it is finite for any link L.
Theorem (R.A. 1998) m(T or T ∗) = 7.
Theorem (R.A. 2000) m(421) > 7.
Theorem (R.A. 2009) m(F8),m(T (5, 2)) > 8.
Theorem (R.A. 2009) Let D(L) be a diagram of link L with ncrossings. Then,
m(L) ≤ 227n.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knots physical models
For a given diameter, one needs certain minimum length of rope inorder to tie a (nontrivial) knot.
Moreover, the more complicated the knot you want to tie, themore rope you need.
Question (Siebenmenn 1985) Can you tie a knot in a 30 cm lengthof 3 cm rope ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knots physical models
For a given diameter, one needs certain minimum length of rope inorder to tie a (nontrivial) knot.
Moreover, the more complicated the knot you want to tie, themore rope you need.
Question (Siebenmenn 1985) Can you tie a knot in a 30 cm lengthof 3 cm rope ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Knots physical models
For a given diameter, one needs certain minimum length of rope inorder to tie a (nontrivial) knot.
Moreover, the more complicated the knot you want to tie, themore rope you need.
Question (Siebenmenn 1985) Can you tie a knot in a 30 cm lengthof 3 cm rope ?
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A number r > 1 is nice if for any distinct points x and y on K wehave D(x , r) ∩ D(y , r) = ∅. The disk thickness of K is defined tobe t(K ) = sup{r |r is nice}.A thick realization K0 of K is a knot of unit thickness which is ofthe same type as K .
The rope length L(K ) of K is the infimum of the length of K0
taken over all thick realizations of K .
Theorem (Cantarella, Kusner and Sullivan 2002) L(K ) exists.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A number r > 1 is nice if for any distinct points x and y on K wehave D(x , r) ∩ D(y , r) = ∅. The disk thickness of K is defined tobe t(K ) = sup{r |r is nice}.A thick realization K0 of K is a knot of unit thickness which is ofthe same type as K .
The rope length L(K ) of K is the infimum of the length of K0
taken over all thick realizations of K .
Theorem (Cantarella, Kusner and Sullivan 2002) L(K ) exists.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 84: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/84.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
A number r > 1 is nice if for any distinct points x and y on K wehave D(x , r) ∩ D(y , r) = ∅. The disk thickness of K is defined tobe t(K ) = sup{r |r is nice}.A thick realization K0 of K is a knot of unit thickness which is ofthe same type as K .
The rope length L(K ) of K is the infimum of the length of K0
taken over all thick realizations of K .
Theorem (Cantarella, Kusner and Sullivan 2002) L(K ) exists.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Diao, Ernst and Yu 2004) There exists a constant csuch that for any knot K
L(K ) ≤ c · (Cr(K ))3/2
where Cr(K ) is the crossing number of K .
The cubic lattice consists of all points in IR3 with integralcoordinates and all unit line segments joining these points.
A cubic lattice knot is a polygonal knot represented in the cubiclattice.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 86: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/86.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Diao, Ernst and Yu 2004) There exists a constant csuch that for any knot K
L(K ) ≤ c · (Cr(K ))3/2
where Cr(K ) is the crossing number of K .
The cubic lattice consists of all points in IR3 with integralcoordinates and all unit line segments joining these points.
A cubic lattice knot is a polygonal knot represented in the cubiclattice.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 87: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/87.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Diao, Ernst and Yu 2004) There exists a constant csuch that for any knot K
L(K ) ≤ c · (Cr(K ))3/2
where Cr(K ) is the crossing number of K .
The cubic lattice consists of all points in IR3 with integralcoordinates and all unit line segments joining these points.
A cubic lattice knot is a polygonal knot represented in the cubiclattice.
J. Ramırez Alfonsın Combinatorics for Knots
![Page 88: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/88.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
The trefoil represented in the cubic lattice.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (Diao, Ernst and Yu 2004) Let K be a knot. Then, K canbe embedded into the cubic lattice with length at most
136 (Cr(K ))3/2 + 84Cr(K ) + 22√
Cr(K ) + 11.
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
13
!1
2
3 4 56
78
9
10
11
12
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
13
!21 3 5 6 7 8 9 10 12 13
114
!1
2
3 4 56
78
9
10
11
12
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
12
!21 3 5 6 7 8 9 10 12 13
114
1 2 3 4 5 6 7 8 9 10 11 13
J. Ramırez Alfonsın Combinatorics for Knots
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Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (R.A. 2010) Let K be a knot. Then, K can be embeddedinto the cubic lattice with length at most O(Cr(K )).
Theorem (R.A. 2010) There exists a constant c such that for anyknot K
L(K ) ≤ c · (Cr(K ))
where Cr(K ) is the crossing number of K .
J. Ramırez Alfonsın Combinatorics for Knots
![Page 95: Combinatorics for Knots](https://reader033.vdocument.in/reader033/viewer/2022060610/6296b5a72d6ef6430a712aec/html5/thumbnails/95.jpg)
Basic notionsMatroid
Knot coloring and the unknotting problemOriented matroids
Spatial graphsRopes and thickness
Theorem (R.A. 2010) Let K be a knot. Then, K can be embeddedinto the cubic lattice with length at most O(Cr(K )).
Theorem (R.A. 2010) There exists a constant c such that for anyknot K
L(K ) ≤ c · (Cr(K ))
where Cr(K ) is the crossing number of K .
J. Ramırez Alfonsın Combinatorics for Knots