Concordance Crosscap Numbers of Knots and the Alexander Polynomial

0.09 MB·English

by  Charles Livingston

CONCORDANCE CROSSCAP NUMBERS OF KNOTS AND THE
ALEXANDER POLYNOMIAL
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a
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9 Abstract. ForaknotK theconcordancecrosscapnumber,c(K),isthemin-
imumcrosscapnumberamongallknotsconcordanttoK. Buildingonworkof

] G.Zhangwhichstudiedthedeterminantsofknotswithc(K)<2,weapplythe
T
Alexander polynomial to construct new algebraic obstructions to c(K) < 2.
G With the exception of low crossing number knots previously known to have
. c(K) < 2, the obstruction applies to all but four prime knots of 11 or fewer
h crossings.
t
a
m
[ Every knot K ⊂S3 bounds an embedded surface F ⊂S3 with F ∼=#nP2−B2
forsomen≥0,whereP2denotestherealprojectiveplane. Thecrosscapnumberof
2
K, γ(K), is defined to be the minimum suchn. The careful study of this invariant
v
9 began with the work of Clark in [Cl]; other references include [HT, MY1]. The
0 study of the 4–dimensionalcrosscapnumber, γ (K), defined similarly but in terms
4
1 of F ⊂B4, appears in such articles as [MY2, Vi, Ya].
3 Gengyu Zhang [Zh] recently introduced a new knot invariant, the concordance
0
crosscap number, γ (K). This is defined to be the minimum crosscap number
7 c
0 of any knot concordant to K. This invariant is the nonorientable version of the
/ concordancegenus,originallystudiedbyNakanishi[Na]andCasson[Ca],andlater
h
investigated in [Li].
t
a In [Zh], Zhang presented an obstruction to γ (K) ≤ 1 based on the homology
c
m
of the 2–fold branched cover of the knot, or equivalently, det(K). Inspired by her
: work,inthis notewewillobservethatthe obstructionfoundin[Zh]extends toone
v
i based on the Alexander polynomial of K, ∆K(t), and the signature of K, σ(K).
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r Theorem 1. Suppose γ (K)=1 and set q =|σ(K)|+1. For all odd prime power
a c
divisors p of q, the 2p–cyclotomic polynomial φ (t) has odd exponent in ∆ (t).
2p K
Furthermore, every other symmetric irreducible polynomial δ(t) with odd exponent
in ∆ (t) satisfies δ(−1)=±1.
K
Proof. Any knot K′ with γ(K′) = 1 bounds a Mobius band, and is thus a (2,r)–
cable of some knot J for some odd r. If K is concordant to K′, then σ(K) =
σ(K′) = ±(|r|−1); the signature σ(K′) is given by a formula of Shinohara [Sh]
for the signature of 2–stranded cable knots. It follows that |σ(K)| = |r|−1, so
|r|=|σ(K)|+1=q.
AccordingthearesultofSeifert[Se],theAlexanderpolynomialofK′ isgivenby
∆ (t)∆ (t2), where ∆ (t) is the Alexander polynomial of the (2,q)–torus knot.
2,q J 2,q
Astandardresultstates that∆ (t)= (t2q−1)(t−1) = tq+1. This canbe writtenas
2,q (tq−1)(t2−1) t+1
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2 CHARLESLIVINGSTON
the product of cyclotomic polynomials,
∆2,q(t)= Y Φ2p(t).
p|q , p>1
SinceK isconcordanttoK′,K#−K′isslice,andthushasAlexanderpolynomial
of the form g(t)g(t−1). That is, with q =|σ(K)|+1,
∆ (t)∆ (t2)∆ (t)=g(t)g(t−1).
K J 2,q
We now make two observations: (1) Any symmetric irreducible polynomial has
even exponent in g(t)g(t−1), and thus even exponent in ∆ (t)∆ (t2)∆ (t); (2)
K J (2,q)
since ∆ (t) is an Alexander polynomial, ∆ (1)=±1, and thus ∆ (t2)| =±1.
J J J t=−1
By Lemma 2, Φ (−1) = p if p is an odd prime power, and Φ (−1) = ±1 if p
2p 2p
is an odd composite. Thus, for p an odd prime power divisor of q, Φ (t) has odd
2p
exponent in ∆ (t) and does not divide ∆ (t2), so has odd exponent in ∆ (t).
2,q J K
Any other irreducible factor of ∆ (t) with odd exponent is either a factor δ(t) of
K
∆ (t),andthusoftheformΦ (t)withpanoddcomposite(andsoδ(−1)=±1),
2,q 2p
orelseisnotafactorof∆ (t)andsohasoddexponentin∆ (t2),andagainmust
2,q J
satisfy δ(−1)=±1. This completes the argument. (cid:3)
Lemma 2. The cyclotomic polynomial Φ (t) satisfies Φ (−1)=p if p is an odd
2p 2p
prime power and Φ (−1)=±1 if p is an odd composite.
2p
Proof. For an odd r, h (t) = tr+1 satisfies h (−1) = r by L’Hospital’s rule. We
r t+1 r
have that h (t) is the product
r
hr(t)= Y Φ2p(t).
p|r , p>1
For p a prime power, sn, Φ (t) = tsn+1 , and so, again by L’Hospital’s rule,
2p tsn−1+1
Φ2sn(−1)=s. Thus, the product
Y Φ2p(−1)=r.
p|r , p>1, paprimepower
It followsthat allthe other terms in the productexpansionof h (t) must equal±1
r
when evaluated at t=−1, as desired. (cid:3)
Examples
Theorem1is quite effective inruling outγ (K)=1. Forinstance, there are801
c
prime knots with 11 or fewer crossings. Of these, 51 are known to be topologically
slice,and 23areknownto be concordantto a (2,q)-torus knotfor some q andthus
haveγ =1. Ofthe remaining727knots,allbutfourcanbe shownto haveγ ≥2.
c c
These four are 11n and 11n , both of which are possibly slice, and 9 and
45 145 40
11n ,bothofwhicharepossibleconcordantto the trefoil. Ofthecollectionof727
66
knots,Yasuhara’sresult[Ya]appliesto showthat207ofthemhave4–ballcrosscap
number γ (K)≥2. The 4–ball crosscap numbers of the rest are unknown.
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As a second set of examples, consider knots K with ∆ (t) of degree 2. It
K
follows immediately from Theorem 1 that there are only two possibilities: either
σ(K) = 0 and ∆ is reducible (an irreducible symmetric quadratic f(t) cannot
K
satisfy f(1)=±1 and f(−1)=±1), or σ(K)=±2 and ∆ (t)=t2−t+1.
K
CONCORDANCE CROSSCAP NUMBERS 3
We conclude with the further special case consisting of (p,q,r)–pretzel knots,
P(p,q,r), with p,q, and r odd; some of these were studied in [Zh]. If we let
D =D(p,q,r)=pq+qr+rp, then
D+1 D−1 D+1
∆ (t)= t2− t+ ,
P(p,q,r) 4 2 4
which has discriminant −D. Thus, by the previous argument we have:
Corollary3. Ifγ (P(p,q,r))=1theneitherσ(P(p,q,r))=0andD(p,q,r)=−l2
c
for some integer l or σ(P(p,q,r))=±2 and D(p,q,r)=3.
These pretzel knots include some shown by Zhang [Zh] to have 4-dimensional
crosscapnumber γ (K)=1.
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References
[Ca] A.Casson,unpublished.
[Cl] B.Clark,Crosscapsandknots,Internat.J.Math.Math.Sci.1(1978),no.1,113–123.
[HT] M. Hirasawa and M. Teragaito, Crosscap numbers of 2-bridge knots, Topology 45
(2006), no.3,513–530.
[Li] C.Livingston,Theconcordance genus ofknots,Alg.andGeom.Top.4(2004), 1–22.
[MY1] H. Murakami and A. Yasuhara, Crosscap number of a knot, Pacific J. Math. 171
(1995), no.1,261–273.
[MY2] H. Murakami and A. Yasuhara, Four-genus and four-dimensional clasp number of a
knot, Proc.Amer.Math.Soc.128(2000), no.12,3693–3699.
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[Ya] A. Yasuhara, Connecting lemmas and representing homology classes of simply con-
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maticalSociety39,755-761.