Tunnel Numbers of Knots
byKanji Morimoto
Department of IS and Mathematics, Konan University
Okamoto 8-9-1, Higashi-Nada, Kobe 658-8501, Japan
morimoto@konan-u.ac.jp
Abstract
Tunnel number of a knot is a geometric invariant of a knot in the 3-sphere,
and has interesting properties related to the connected sum of knots. In thisarticle, we show the existence of infinitely many pairs of knots each tunnel
number of which goes up under tne connected sum, and show the existence of
infinitely many pairs of knots each tunnel number of which goes down under theconnected sum. In addition, we study the degeneration ratio of tunnel numbers
under the connected sum.

1. Introduction
In the present article, we introduce the tunnel numbers of knots in the 3-sphere S 3 ,and calculate the tunnel numbers of several examples. Further, we study the relation
between the tunnel numbers and the connected sum of knots.
This geometric invariant, the tunnel number, is closely related to the Heegaard
genus of the 3-manifolds as the knot exteriors. Therefore, in studying tunnel numbersof knots, we need not only knot theoretical technique but also 3-manifold topology
argument. For knot theory we refer to Rolfsen’s book [Rs] and for 3-manifold topology
we refer to Hempel’s book [He].

2. Definitions and examples
To define the tunnel number, we need :Fact 2.1 For any knot K in the 3-sphere S 3 , there is an arc system {γ1 , γ2 , · · · , γt }in S 3 with (γ1 ∪ γ2 ∪ · · · ∪ γt ) ∩ K = ∂(γ1 ∪ γ2 ∪ · · · ∪ γt ) such that the exteriorof the union of K and the arcs is homeomorphic to a genus t + 1 handlebody, i.e.,cl(S 3 − N (K ∪ γ1 ∪ γ2 ∪ · · · ∪ γt ) ∼= a genus t + 1 handlebody, where N (·) denotes aregular neighborhood.
Definition2.2 We call the arc system in Fact 2.1 an unknotting tunnel system of K.

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In particular, if the system consists of a single arc, then we call the arc an unknotting
tunnel of K.
Definition2.3 We define the tunnel number of K, denoted by t(K), as the minimal
number of the arcs among all unknotting tunnel systems of K.
Proof of Fact 2.1. Consider the projection of K. Then we can take a small arc
γi (i = 1, 2, · · · , c) at each crossing point, where c is the crossing number of theprojection.
γ2γ1γ3γca standard genus c + 1 handlebody

Figure 1: an unknotting tunnel system
Then, by the deformation in Figure 1, N (K ∪ γ1 ∪ γ2 ∪ · · · ∪ γc ) is isotopic to a
standard genus c + 1 handlebody in S 3 . This means that the exterior of N (K ∪ γ1 ∪γ2 ∪ · · · ∪ γc ) is also a genus c + 1 handlebody, and completes the proof.
Examples(1) The trivial knot has tunnel number 0.
(2) The trefoil knot has tunnel number 1.
(3) The knot 816 in Rolfsen’s table ([Rs]) has tunnel number 2.
(4) Every 2-bridge knot has tunnel number 1.
(5) Every torus knot has tunnel number 1.
(6) Let pi be an odd integer with |pi | > 1 (i = 1, 2, 3), and let K be the pretzel knotof type (p1 , p2 , p3 ). Then K has tunnel number 2.
Proof. (1) The exterior of the trivial knot is homeomorphic to the solid torus, i.e., agenus 1 handlebody. Thus the trivial knot has tunnel number 0.
(2) By the deformation in Figure 2, we see that the arc γ is an unknottin tunnel ofthe trefoil knot. Thus the trefoil knot has tunnel number 1 because the trefoil knot
is a non-trivial knot.
γFigure 2: an unknotting tunnel2(3) The knot illustrated in Figure 3 is 816 in Rolfsen’s table ([Rs]). Then, by a littledeformation, we see that the graph K ∪ γ1 ∪ γ2 becomes a trivial graph as in Figure
3. This means that the arc system {γ1 , γ2 } is an unknotting tunnel system of 816 ,and hence t(816 ) ≤ 2, On the other hand, by the deformation in Figure 4, we seethat 816 has a 2-string essential tangle decomposition, where the definition of tangle
decomposition will be given after Theore 3.5. Thus, by [Sm], t(816 ) ≥ 2. Thus t(816 )has tunnel number 2.

γ1γ2Figure 3: an unknotting tunnel system of 816

Figure 4: a 2-string essential tangle decomposition of 816(4) Every 2-bridge knot K has a 2-string trivial tangle decomposition as in Figure 5.
Then, the arc γ is an unknotting tunnel, and hence t(K) = 1.

γ

K =Figure 5: an unknotting tunnel of a 2-bridge knot
(5) For every torus knot K, take an arc γ indicated in Figure 6, i.e., the arc connecting

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adjascent parallel two strings, where the knot illustrated in Figure 6 is the torus knotof type (5, 4). Then, by [BRZ], the arc γ is an unknotting tunnel, and hence t(K) = 1.

γK =Figure 6: an unknotting tunnel of a torus knot
(6) For a pretzel knot K of type (p1 , p2 , p3 ), the arc system {γ1 , γ2 } as in Figure 7 isan unknotting tunnel system of K, where the pretzel knot illustrated in Figure 7 isof type (3, 5, 7). Then by [MSY2], we have t(K) = 2.
γ1γ2

K =Figure 7: an unknotting tunnel system of a pretzel knot

3. Connected sum
Let K1 and K2 be two knots in S 3 . Then the connected sum of K1 and K2 , denotedby K1 #K2 , is defined as in Figure 8.
The behavior of geometric invariants under connected sum is a very interesting
problem in knot theory. For example, for the Seifert genus g(K), we have in [Rs]:g(K1 #K2 ) = g(K1 ) + g(K2 )

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K1

K2

K1 #K2Figure 8: the connented sum
This means that the Seifert genus is additive under connected sum. For the bridge
index b(K), by [Sb], we have:b(K1 #K2 ) = b(K1 ) + b(K2 ) − 1
In this case, b(K) − 1 is additive under connected sum. On the other hand, for
the unknotting number u(K) or th crossing number c(K), it is still unknown if thoseinvariants are additive or not.
Now, for the tunnel numbers, the most basic fact is:Fact 3.1 t(K1 #K2 ) ≤ t(K1 ) + t(K2 ) + 1
Proof. Let {γ1 , γ2 , · · · , γt } be the unknotting tunnel system of K1 and {δ1 , δ2 , · · · , δs }be the unknotting tunnel system of K2 . Consider the arc system {γ1 , γ2 , · · · , γt , δ1 , δ2 ,· · · , δs , ρ}, obtained from the union of the two unknotting systems by adding an extraarc ρ indicated in Figure 9. Then we can see that it is an unknotting tunnel systemof K1 #K2 . It should be noted that, in general, we need the extra arc ρ to obtain anunknotting tunnel system of the connected sum.
δ1γ1

ργtδsFigure 9: an unknotting tunnel system of the connented sum
In the early years of research of tunnel numbers, there were very few families ofknots whose tunnel numbers were identified. In fact, we had only 2-bridge knots
and torus knots. In addition, for any 2-bridge knots or any torus knots K1 , K2 , theadditivity t(K1 #K2 ) = t(K1 ) + t(K2 ) holds. Therefore, those days, the following twoquestions had puzzled knot theorists.
Q1 : Are there knots K1 , K2 such that t(K1 #K2 ) = t(K1 ) + t(K2 ) + 1 ?
Q2 : Are there knots K1 , K2 such that t(K1 #K2 ) < t(K1 ) + t(K2 ) ?

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The first result on the additivity problem of tunnel numbers is :Theorem 3.2 ([Nw], [Sm]) Tunnel number one knots are prime.
This theorem says that there are no knots K1 , K2 as in Q2 with t(K1 #K2 ) = 1.
However, in 1990’s, such knots as in Q1 and Q2 were found as follows :Theorem 3.3 ([MSY1]) Let Km be the knot as in Figure 10. Then t(Km ) =t(Km′ ) = 1 and t(Km #Km′ ) = 3 for any integers m and m′ , i.e., “1 + 1 = 3”.
10m - 4 crossings(7, 17) - torus braid

Figure 10: the knot KmTheorem 3.4 ([M2]) Let Kn be the knot as in Figure 11 for any integer n ̸= 0, −1.
Then t(Kn ) = 2 and t(Kn #K ′ ) = 2 for any 2-bridge knot K ′ , i.e., “2 + 1 = 2”.

(4, 3) torus braid

2n + 1crossings

Figure 11: the knot Kn
Soon after that, we succeeded to characterize those knot types with “2 + 1 = 2” asfollows :Theorem 3.5 ([M3]) Let K be a tunnel number two knot. Then, the tunnel numberof K#K ′ is two again for any 2-bridge knot K ′ if and only if K has a 2-string

6essential free tangle decomposition (S 3 , K) = (B1 , t11 ∪ t21 ) ∪ (B2 , t12 ∪ t22 ) such that oneof the two tangles has an unknotted component.
In this theorem, a “2-string essential free tangle decomposition” means a decomposition of a knot into two 2-string essential free tangles, where a 2-string essential freetangle is a pair (B, t1 ∪ t2 ) of a 3-ball B and a pair of arcs t1 ∪ t2 properly embeddedin B, such that t1 and t2 cannot be separated in B and the exterior of t1 ∪ t2 in B isa genus two handlebody.

Figure 12: a 2-string essential free tangle with an unkntted component
Hence, by Figure 4, 816 has a 2-string essential free tangle decomposition such that
one of the two tangles has an unknotted component. This means that t(816 ) = 2 andt(816 #K ′ ) = 2 for any 2-bridge knot K ′ . Since the knots up to 7 crossings ara all
2-bridge knot, we see that 816 is the first tunnel number two knot with tunnel number
degeneration.
By the way, we here introduce a concept of “meridionally primitive”. Let K be aknot with the tunnel number t. Then there is a genus t + 1 Heegaard splitting (V1 , V2 )of S 3 such that V1 contains K as a central loop of a handle.
Definition 3.6 We say that K is meridionall primitive if there is a genus t + 1Heegaard splitting (V1 , V2 ) as above such that there is a meridian disk D1 of V1 and ameridian disk D2 of V2 with D1 ∩ K = 1 point and D1 ∩ D2 = ∂D1 ∩ ∂D2 = 1 point.
Then, concerning Q1, we had gotten the following :Theorem 3.7 ([M1, M4]) Let K1 and K2 be two knots with t(K1 ) = t(K2 ) = 1.
Then t(K1 #K2 ) = 3 if and only if none of K1 and K2 is meridionally primitive.
We proved Theorem 3.3 by using this theorem. In fact, we proved that the knot
Km is not a meridionally primitive via Quantum ivariant formula due to Yokota ([Y]).
As a generalyzation of Theorem 3.7, we have ;Theomre 3.8 ([M4]) Let K1 and K2 be two knots in S 1 and suppose both K1 andK2 are meridionally small. Then t(K1 #K2 ) = t(K1 ) + t(K2 ) + 1 if and only if noneof K1 and K2 is meridionally primitive.

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In this theorem, we say that a knot K is meridionally small if the exterior E(K) =cl(S 3 − N (K)) contains no properly embedded essential surface F such that eachcomponent of ∂F is a meridian of K. By using the concept “meridionall small”, wegot the following :Theorem 3.9 ([M5]) Let K1 , K2 , · · · , Kn be all meridionally small knots in S 3 .
Then we have t(K1 #K2 # · · · #Kn ) ≥ t(K1 ) + t(K2 ) + · · · + t(Kn ).
Even if we drop the assumption of “meridionally small”, Scharlemann and Schultens
got the following :Theorem 3.10 ([SS1]) Let K1 , K2 , · · · , Kn be knots in S 3 . Then we havet(K1 #K2 # · · · #Kn ) ≥ n.
By the way, in studying of 3-manifolds, it is very important if a given 3-manifold
contains an essential torus or not. As such studying for knot exteriors, we havecharacterized the knot types of tunnel number one knots containing essential tori in[MS].

4. Degeneration ratio
In the studying of the degeneration of tunnel numers, it seems that the ratio oft(k1 #K2 ) to t(K1 ) + t(K2 ) is more important than the diﬀerence between t(k1 #K2 )and t(K1 ) + t(K2 ). Therefore Schrlemann and Schultens introduced the degeneration
ratio d(K1 , K2 ) for any two knots K1 and K2 as follows :d(K1 , K2 ) = 1 −t(K1 #K2 )t(K1 ) + t(K2 )Our first example “2+1=2” has the degeneration ration 13 as follows :d(K1 , K2 ) = 1 −2
1=
2+13Then we can ask :Q3 : What is the upper limit of the degeneration ratio ?
Concerning this question, Scharlemann and Schultens got :Thoerem 4.1 ([SS2])] For any two prime knots K1 and K2 , we have d(K1 , K2 ) ≤3.
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Recently, as the next step to “2+1=2”, Nogueira got :Theorem 4.2 ([Ng]) There are infinitely many pairs of knot K1 and K2 such that

8t(K1 ) = 3, t(K2 ) = 2 and t(K1 #K2 ) = 3, i.e., “3 + 2 = 3′′ .
In this therem, Nogueira have gotton the concrete examples by using the knot Kmin Figure 10. Then the degeneration ratio is 52 as follows :d(K1 , K2 ) = 1 −

3
2
=
3+2
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For the time being, this example is the biggest degeneration ratio. Thus, as thesequence of the series, we can ask :Q4 : Are there infinitely many pairs of knot K1 , K2 such that “4+3=4”, “5+4=5”,“6+5=6”, “7+6=7”, · · · ?
5If there is such a sequence, we have the sequence of degeneration ratio 37 , 94 , 11,71613 , 15 , · · · → 2 .

Concerning the degeneration ratio, we have several examples and results in [M6].

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