Biquandle Virtual Brackets

2.5 MB·

by  Sam Nelson

Biquandle Virtual Brackets
Sam Nelson∗ Kanako Oshiro† Ayaka Shimizu‡ Yoshiro Yaguchi§
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0 Abstract
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We introduce an infinite family of quantum enhancements of the biquandle counting invariant we

n call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot
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and link diagrams with values in a commutative ring R using virtual crossings as smoothings, these
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invariants take the form of multisets of elements of R and can be written in a “polynomial” form for
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convenience. The family of invariants defined herein includes as special cases all quandle and biquandle
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2-cocycleinvariants,allclassicalskeininvariants(Alexander-Conway,Jones,HOMFLYPTandKauffman

polynomials) and all biquandle bracket invariants defined in [11] as well as new invariants defined using
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T virtual crossings in a fundamental way, without an obvious purely classical definition.
G Keywords: Quantum enhancements, biquandles, biquandle counting invariants, virtual knots and
. links
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t 2010 MSC: 57M27, 57M25
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1 Introduction
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1 In [12] the notion was introduced of quantum enhancements of the biquandle counting invariant by means
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of quantum invariants of biquandle-colored knots and links. More precisely, if β is a quantum invariant of
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biquandle colored knots and links, then the multiset of β-values over the set of biquandle colorings of a knot
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9 orlinkK isaninvariantwhosecardinalityrecoversthebiquandlecountinginvariant. In[11],afamilyofsuch
3 quantum enhancements including both classical skein invariants and biquandle cocycle invariants as special
0 cases, known as biquandle brackets, was introduced. These invariants can be understood as skein invariants
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1 of biquandle colored oriented knots and links with skein coefficients depending on the biquandle colors of
0 the semiarcs involved in the crossing being smoothed. The fact that a biquandle coloring is “broken” by
7 smoothings means it is simpler to think of the invariant in terms of the state-sum definition in which all
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smoothings are done at once rather than smoothing crossings one at a time.
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v In this paper we generalize the biquandle bracket idea to use a different set of skein relations in which
Xi a virtual crossing is considered as a type of a smoothing rather than a kind of crossing. We use the term
“biquandle virtual bracket” to distinguish this case from “virtual biquandle brackets” which are brackets
r
a withcoefficientsinavirtualbiquandle. Thisnewinfinitefamilyofinvariantsoforientedclassicalandvirtual
knots and links contains classical biquandle brackets (and hence classical skein invariants and biquandle
2-cocycle invariants) as special cases and uses virtual knot theory in a fundamental way, joining the finite
type invariants of [8] as a family of invariants of classical knots without an obvious definition in term of
purely classical knot and link diagrams.
The paper is organized as follows. In Section 2 we briefly review biquandles, biquandle colorings of
orientedclassicalandvirtualknotsandlinks,andthebiquandlecountinginvariant. InSection3weintroduce
biquandle virtual brackets. In Section 4 we provide some applications and examples. In Section 5 we close
with some questions for future research.
∗Email: [email protected]. PartiallysupportedbySimonsFoundationcollaborationgrant316709
†Email: [email protected]. PartiallysupportedbyJSPSKAKENHIGrantNumber16K17600.
‡Email: [email protected]. PartiallysupportedbyGrantforBasicScienceResearchProjectsfromTheSumitomo
Foundation(160154).
§Email: [email protected]
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2 Biquandles and Virtual Links
We begin with a definition (see [5–7]).
Definition 1. A biquandle is a set X with operations (cid:46), (cid:46) :X →X satisfying for all x,y,z ∈X
(i) x(cid:46)x=x(cid:46)x,
(ii) The maps α ,β :X →X and S :X×X →X×X given by
x x
α (y)=y(cid:46)x, β (y)=y(cid:46)x and S(x,y)=(y(cid:46)x,x(cid:46)y)
x x
are invertible, and
(iii) We have the exchange laws
(x(cid:46)y)(cid:46)(z(cid:46)y) = (x(cid:46)z)(cid:46)(y(cid:46)z)
(x(cid:46)y)(cid:46)(z(cid:46)y) = (x(cid:46)z)(cid:46)(y(cid:46)z)
(x(cid:46)y)(cid:46)(z(cid:46)y) = (x(cid:46)z)(cid:46)(y(cid:46)z).
It is sometimes convenient to write x(cid:46)y as xy and x(cid:46)y as x for the sake of space.
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Definition 2. A map f :X →Y between biquandles is a biquandle homomorphism if
f(x(cid:46)y)=f(x)(cid:46)f(y) and f(x(cid:46)y)=f(x)(cid:46)f(y)
for all x,y ∈ X. A subset S ⊂ X of a biquandle X is a subbiquandle if S is closed under the biquandle
operations of X, including the inverses of the maps in axiom (ii).
Example 1. A biquandle X in which x(cid:46)y =x for all y ∈X is a quandle. Examples of quandles include
• n-fold conjugation quandles: Every group G (or union of conjugacy classes in a group) is a quandle
under x(cid:46)y =y−nxyn for n∈Z,
• Alexander quandles: Every module X over Z[t±1] is a quandle under x(cid:46)y =tx+(1−t)y,
• Symplectic quandles: Every vector space V over a field F of characteristic (cid:54)=2 with a symplectic form
(cid:104),(cid:105):V ×V →F is a quandle under x(cid:46)y =x+(cid:104)x,y(cid:105)y.
Example 2. Examples of non-quandle biquandles include
• Constant action biquandles: Every set X is a biquandle under x(cid:46)y = x(cid:46)y = σ(x) for a bijection
σ :X →X,
• Alexander biquandles: Every module X over the ring Z[t±1,s±1] is a biquandle under the operations
x(cid:46)y =tx+(s−t)y and x(cid:46)y =sx.
• Fundamental biquandle of an oriented link: The set B(L) of equivalence classes of biquandle words
in generators corresponding to semiarcs in a diagram of an oriented link L modulo the congruence
generated by the crossing relations
and the biquandle axioms is a strong invariant of virtual links, conjectured to be complete up to
reflection for virtual knots and known to be complete up to reflection for classical knots (see [5,6,9]
for more).
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Given a finite set X ={x ,...,x } we can define a biquandle structure on X by listing operation tables
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of (cid:46) and (cid:46) such that the axioms are satisfied – the diagonals of the tables must agree, the columns must
be permutations, etc. We can conveniently encode these tables as an n×2n block matrix by dropping the
“x”s and listing only the subscripts, e.g.
(cid:46) x1 x2 x3 (cid:46) x1 x2 x3  1 3 2 1 1 1 
x x x x x x x x
1 1 3 2 1 1 1 1 → 3 2 1 2 2 2 
x x x x x x x x
2 3 2 1 2 2 2 2 2 1 3 3 3 3
x x x x x x x x
3 2 1 3 3 3 3 3
Next, we recall the definition of virtual knots and links (see [10] for more). A virtual knot diagram has
classical and virtual crossings as depicted:
The virtual crossings are interpreted as not really crossings, but artifacts of drawing a non-planar knot
diagram on planar paper – we can regard a virtual crossing as indicating genus in the supporting surface
on which the knot diagram is drawn. In this way, a virtual crossing may be considered a kind of smoothing
rather than a kind of crossing.
Two virtual link diagrams are equivalent if they are related by the oriented virtual Reidemeister moves:
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Note that there are more oriented III, vIII and v moves, but in the presence of the other pictured moves
they can be recovered from the pictured moves. In particular, the above constitute one generating set of
orientedvirtualReidemeistermoves. Classicalknotsandlinksformasubsetofvirtualknotsandlinks, with
two classical knots or link equivalent as virtual knots or links if and only if they are equivalent as classical
knots or links; see [10] for more.
Given a biquandle X and an oriented virtual knot or link L represented by a diagram D, we can color
D with X by assigning elements of X to the semiarcs in D, i.e. the portions of D between crossing points,
such that we locally have the following pictures:
It is then straightforward to verify that for any X-coloring of D before a Reidemeister move, there is a
unique corresponding coloring after the move. In particular, if X is finite, the the number of X-colorings of
D is finite. In terms of the fundamental biquandle, a coloring is a homomorphism f : B(L) → X, and the
set of colorings is Hom(B(L),X); then the counting invariant is ΦZ(L)=|Hom(B(L),X)|.
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Hence, we have
Theorem 1. The number of X-colorings of a virtual knot or link K by a biquandle X, denoted ΦZ(K) and
X
known as the biquandle counting invariant, is an invariant of oriented classical and virtual links.
Example 3. The Hopf link below has four colorings by the biquandle X =Z with x(cid:46)y =x(cid:46)y =x+1 as
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depicted:
The virtual Hopf link, on the other hand,
x = x+1
y = y+1
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hasnocoloringsbyX sincetherearenosolutionstothesystemofcoloringequations,sowehaveΦZ(Hopf)=
X
4(cid:54)=0=ΦZ(vHopf) and the counting invariant distinguishes these two virtual links.
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3 Biquandle Virtual Brackets
The invariant in example 3 gives a value of
(cid:26) 2c L(cid:54)∈O
Z
Φ (L)=
X 0 L∈O
where O is the set of virtual links in which any component has an odd number of (classical) crossing points
and c is the number of components of L. Thus ΦZ with the choice of biquandle X in example 3 fails
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to distinguish any two classical knots; we would like to strengthen and enhance this and other biquandle
countinginvariantstoobtainstrongerinvariants. IfβisaninvariantofX-coloreddiagrams,thenthemultiset
of β-values over the set of X-colorings of L is a generally stronger invariant from which we can recover ΦZ
X
bytakingthemultiset’scardinality. Inthissectionweintroduceanewinfinitefamilyofsuchenhancements.
Definition 3. Let X be a biquandle and R a commutative ring with identity. Then a biquandle virtual
bracket consists of six maps A,B,C,D,U,V : X ×X → R and two distinguished elements δ ∈ R, w ∈ R×
satisfying the following conditions:
w = δA +B +V (i.i)
x,x x,x x,x
w−1 = δC +D +U (i.ii),
x,x x,x x,x
1 = A C +V U (ii.i)
x,y x,y x,y x,y
1 = B D +V U (ii.ii)
x,y x,y x,y x,y
0 = A U +V C (ii.iii)
x,y x,y x,y x,y
0 = B U +V D (ii.iv)
x,y x,y x,y x,y
0 = δB D +A D +B C (ii.v)
x,y x,y x,y x,y x,y x,y
0 = δA C +A D +B C (ii.vi),
x,y x,y x,y x,y x,y x,y
A A A +V A V = A A A +V A V (iii.i)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
A A B +B A A
x,y xy,zy y,z x,y xy,zy y,z
+δB A B +B A V
x,y xy,zy y,z x,y xy,zy y,z
+B B B +B V B
x,y xy,zy y,z x,y xy,zy y,z
+V A B = A B A (iii.ii)
x,y xy,zy y,z yx,zx x,z xz,yz
A B A = A A B +B A A
x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
+δB A B +B A V
yx,zx x,z xz,yz yx,zx x,z xz,yz
+B B B +B V B +V A B (iii.iii)
yx,zx x,z xz,yz yx,zx x,z xz,yz yx,zx x,z xz,yz
A V A = A A V +V A A (iii.iv)
x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
A A V +V A A = A V A (iii.v)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz
B B A +B V V = A B B +V V B (iii.vi)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
A B B +V V B = B B A +B V V (iii.vii)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
B B V +B V A = A B V (iii.viii)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz
A B V = B B V +B V A (iii.ix)
x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
V B A = A V B +V B B (iii.x)
x,y xy,zy y,z yx,zx x,z xz,yz yx,zx x,z xz,yz
A V B +V B B = V B A (iii.xi)
x,y xy,zy y,z x,y xy,zy y,z yx,zx x,z xz,yz
V V A = A V V (iii.xii)
x,y xy,zy y,z yx,zx x,z xz,yz
A V V = V V A (iii.xiii)
x,y xy,zy y,z yx,zx x,z xz,yz
V B V = V B V (iii.xiv)
x,y xy,zy y,z yx,zx x,z xz,yz
V V V = V V V (iii.xv)
x,y xy,zy y,z yx,zx x,z xz,yz
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Note that in the case that A ,B ,C and D are all invertible for a pair (x,y), conditions (ii.iii) and
x,y x,y x,y x,y
(ii.iv) reduce to
δ = −A B−1 −C D−1 (ii.iii(cid:48))
x,y x,y x,y x,y
δ = −A−1B −C−1D (ii.iv(cid:48)).
x,y x,y x,y x,y
The biquandle virtual bracket axioms are chosen so that the skein relations
with writhe correction factor w and smoothed component value δ
define an invariant of X-colored Reidemeister moves.
To accomplish this, we start with the idea that for a given X-coloring f oriented link diagram L with c
classicalcrossings, thereisasetof3c completelysmootheddiagramsknownasstates inwhicheachclassical
crossing is smoothed either with the orientation, against the orientation, or made virtual. Each such state
S has an associated value β ∈ R defined as the product of the coefficients associated to the choices of
S
smoothingstimesδk wherek isthenumberofclosedcurvecomponentsofthestate(whichmayhavevirtual
crossings) times wn−p where n is the number of negative classical crossings and p is the number of positive
classical crossings of L. We then sum these β values over the complete set of states obtained from f to
S
obtain the state sum value β for the given coloring of the link, and the multiset of β values over the set of
f
all X-colorings f of L is then an invariant of oriented links.
We have the following critical observation:
Observation 1. A move on a tangle which does not change connectivity on the boundary (i.e., boundary
pointsconnectedornotconnectedinthepre-movetanglearestillconnectedornotconnectedinthepost-move
tangle) and for which the coefficient weight sums are equal before and after the move does not change the
overall contribution to the state sum.
Using this observation, we obtain the biquandle virtual bracket axioms from the oriented virtual Reide-
meister moves.
The Reidemeister I moves require that w =δA +B +V and w−1 =δC +D +U :
x,x x,x x,x x,x x,x x,x
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Comparing the skein expansions before and after the direct and reverse oriented Reidemeister II moves
yields the conditions
1 = A C +V U
x,y x,y x,y x,y
1 = B D +V U
x,y x,y x,y x,y
0 = A U +V C
x,y x,y x,y x,y
0 = B U +V D
x,y x,y x,y x,y
0 = δB D +A D +B C
x,y x,y x,y x,y x,y x,y
0 = δA C +A D +B C
x,y x,y x,y x,y x,y x,y
The two sides of the third Reidemeister move, with each smoothed state labeled with its coefficient
product,
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are left-hand side
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and right-hand side
.
Comparingcoefficientsumsofequivalenttanglesusingobservation1yieldsthefifteenaxioms(iii.i)through
(iii.xv).
The pure virtual moves do not affect the state sum value in any way, and the mixed virtual move yields
the same sum on both sides up to virtual moves, and thus does not change the state sum:
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Hence, we have
Theorem 2. Let X be a biquandle and β a biquandle virtual bracket. Then for any X-coloring f of an
oriented (classical or) virtual link L, the state sum
(cid:88)
β = β
f S
S State
is an invariant of X-colored virtual Reidemeister moves.
Example 4. Treating an oriented virtual link L as colored by its fundamental biquandle B(L) with f
assigning to each semiarc its generator, the state sum β is the fundamental biquandle virtual bracket of L.
f
While it is difficult to compare values of the fundamental virtual bracket directly (since doing so requires
comparingfundamentalbiquandlesoflinksdirectly),wefinditusefultousethefundamentalvirtualbracket
expressionasastepincomputingstatesumsforvirtualbracketsoverfinitebiquandlesbysubstitutingimage
values of generators under the coloring maps f ∈Hom(B(L),X); see Section 4.
Corollary 3. For any biquandle X, biquandle virtual bracket β and oriented virtual link L, the multiset of
state sum values
Φβ,M(L)={β | f ∈Hom(B(L),X)}
X f
is an invariant of classical and virtual links.
Definition 4. We call the multiset in corollary 3 the biquandle virtual bracket multiset invariant of L with
respect to the biquandle X and the biquandle virtual bracket β.
If R is a number ring, it is common practice (see [3,5] etc.) to write the multiset in a “polynomial”
form vaguely analogous to a generating function by making the multiplicities coefficients and the elements
exponents, e.g. encoding {−2,1,1,1,3,3} as u−2 +3u+2u3. When written this way, we will refer to the
invariant as the biquandle virtual bracket polynomial of L, denoted Φβ (L).
X
Example 5. LetX beabiquandleandβ abiquandlevirtualbracketoveracommutativeringwithidentity
R. If V =U =0 for all x,y ∈X, then β is a biquandle bracket as defined in [11]. We will refer to such
x,y x,y
virtual brackets as biquandle classical brackets or just biquandle brackets.
Example 6. LetX beabiquandleandβ abiquandlevirtualbracketoveracommutativeringwithidentity
R. If A =B =C =D =0 for all x,y ∈X and w =δ =1, then the virtual bracket axioms reduce
x,y x,y x,y x,y
to
U =V−1 and V V V =V V V
x,y x,y x,y xy,zy y,z yx,zx x,z xz,yz
for all x,y,z ∈ X, making V a biquandle 2-cocycle and ΦB exactly the biquandle 2-cocycle invariant with
X
Boltzmann weights written multiplicatively (see [2,4,5]).
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