Knots-quivers correspondence, lattice paths, and rational knots

Transcription

1 Knots-quivers correspondence, lattice paths, and rational knots 1 CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Portugal 2 Mathematical Institute SANU, Belgrade, Serbia Warszawa, September 2018

2 Ingredient 1: Knots Colored HOMFLY PT polynomials: Symmetric (S r )-colored HOMFLY PT polynomials are 2-variable invariants of knots: P r (K)(a, q). For a = q N they are (sl(n), S r ) quantum polynomial invariants: P(a = q N, q) = P sl(n),sr (q).

3 Ingredient 1: Knots Colored HOMFLY PT polynomials: Symmetric (S r )-colored HOMFLY PT polynomials are 2-variable invariants of knots: P r (K)(a, q). For a = q N they are (sl(n), S r ) quantum polynomial invariants: P(a = q N, q) = P sl(n),sr (q). Already interesting is the bottom row : the coefficient of the lowest nonzero power of a appearing in P r (a, q) P r (q) = lim a 0 a P r (a, q)

4 LMOV conjecture Generating function of all symmetric-colored HOMFLY-PT polynomials of a given knot K is: P(x, a, q) := P r (a, q)x r = exp 1 n f r (a n, q n )x rn, r 0 n,r 1 f r (a, q) = i,j N r,i,j a i q j q q 1.

5 LMOV conjecture Generating function of all symmetric-colored HOMFLY-PT polynomials of a given knot K is: P(x, a, q) := P r (a, q)x r = exp 1 n f r (a n, q n )x rn, r 0 n,r 1 f r (a, q) = i,j N r,i,j a i q j q q 1. One can easily get: N r,i,j Q

6 LMOV conjecture Generating function of all symmetric-colored HOMFLY-PT polynomials of a given knot K is: P(x, a, q) := P r (a, q)x r = exp 1 n f r (a n, q n )x rn, r 0 n,r 1 f r (a, q) = i,j N r,i,j a i q j q q 1. One can easily get: N r,i,j Q LMOV conjecture: N r,i,j Z!

7 LMOV conjecture Generating function of all symmetric-colored HOMFLY-PT polynomials of a given knot K is: P(x, a, q) := P r (a, q)x r = exp 1 n f r (a n, q n )x rn, r 0 n,r 1 f r (a, q) = i,j N r,i,j a i q j q q 1. One can easily get: N r,i,j Q LMOV conjecture: N r,i,j Z! N r,i,j are BPS numbers. They represent (super)-dimensions of certain homological groups. Physicaly, they count particles of certain type (therefore are integers).

8 Ingredient 2: Quivers (and their representations) Quivers are oriented graphs, possibly with loops and multiple edges. Q 0 = {1,..., m} set of vertices. Q 1 the set of edges {α : i j}.

9 Ingredient 2: Quivers (and their representations) Quivers are oriented graphs, possibly with loops and multiple edges. Q 0 = {1,..., m} set of vertices. Q 1 the set of edges {α : i j}. Let d = (d 1,..., d m ) N m be a dimension vector. We are interested in moduli space of representations of Q with the dimension vector d: } M d = {R(α) : C d i C d j for all α : i j Q 1 //G, where G = i GL(d i, C).

10 Quivers and motivic generating functions C is a matrix of a quiver with m vertices. P C (x 1,..., x m ) := d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm q-pochhamer symbol (q 2 ; q 2 ) n := n i=1 (1 q2i ).

11 Quivers and motivic generating functions C is a matrix of a quiver with m vertices. P C (x 1,..., x m ) := d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm q-pochhamer symbol (q 2 ; q 2 ) n := n i=1 (1 q2i ). Motivic (quantum) Donaldson-Thomas invariants Ω d1,...,d m;j of a symmetric quiver Q: P C = (d 1,...,d m) 0 j Z k 0 ( 1 ( x d 1 1 x ) ) ( 1) m dm q j+2k+1 j+1 Ω d1,...,dm;j.

12 Quivers and motivic generating functions C is a matrix of a quiver with m vertices. P C (x 1,..., x m ) := d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm q-pochhamer symbol (q 2 ; q 2 ) n := n i=1 (1 q2i ). Motivic (quantum) Donaldson-Thomas invariants Ω d1,...,d m;j of a symmetric quiver Q: P C = (d 1,...,d m) 0 j Z k 0 ( 1 ( x d 1 1 x ) ) ( 1) m dm q j+2k+1 j+1 Ω d1,...,dm;j. Theorem (Kontsevich-Soibelman, Efimov) Ω d1,...,d m;j are nonnegative integers.

13 Knots quivers correspondence [P. Kucharski, M. Reineke, P. Sulkowski, M.S., Phys. Rev. D 2017] New relationship between HOMFLY PT / BPS invariants of knots and motivic Donaldson-Thomas invariants for quivers Figure: Trefoil knot and the corresponding quiver. The generating series of HOMFLY-PT invariants of a knot matches the motivic generating series of a quiver, after setting x i x.

14 Details of the correspondence Knots Generators of HOMFLY homology Homological degrees, framing Colored HOMFLY-PT LMOV invariants Classical LMOV invariants Algebra of BPS states Quivers Number of vertices Number of loops Motivic generating series Motivic DT-invariants Numerical DT-invariants Cohom. Hall Algebra

15 Application 1 LMOV conjecture BPS/LMOV invariants of knots are refined through motivic DT invariants of a corresponding quiver, and so Theorem For all knots for which there exists a corresponding quiver, the LMOV conjecture holds.

16 Application 2 Lattice paths counting y = 1 4 x Figure: A lattice path under the line y = 1 4x, and a shaded area between the path and the line. x y P (x) = x k = c k (1)x k, k=0 π k-paths k=0 y qp (x) = q area(π) x k = c k (q)x k. k=0 π k-paths k=0

17 Counting lattice paths equivalent formulation Figure: Counting of paths under the line y = 1 2x is equivalent to counting paths in the upper half plane, made of elem. steps (1, 1) and (1, 2).

18 Counting (rational) lattice paths Surprisingly, colored HOMFLY PT polynomials are closely related to the purely combinatorial problem of counting lattice paths under lines with rational slope.

19 Counting (rational) lattice paths Surprisingly, colored HOMFLY PT polynomials are closely related to the purely combinatorial problem of counting lattice paths under lines with rational slope. Again generating function of the bottom row of colored HOMFLY PT polynomial: P(K)(q; x) = 1 + P r (q)x r r=1

20 Counting (rational) lattice paths Surprisingly, colored HOMFLY PT polynomials are closely related to the purely combinatorial problem of counting lattice paths under lines with rational slope. Again generating function of the bottom row of colored HOMFLY PT polynomial: P(K)(q; x) = 1 + P r (q)x r r=1 Observe quotient: P(K)(q; q 2 x) P(K)(q; x)

21 Counting (rational) lattice paths Surprisingly, colored HOMFLY PT polynomials are closely related to the purely combinatorial problem of counting lattice paths under lines with rational slope. Again generating function of the bottom row of colored HOMFLY PT polynomial: Observe quotient: P(K)(q; x) = 1 + P r (q)x r r=1 P(K)(q; q 2 x) P(K)(q; x) Finally, take q 1 limit ( classical limit): P(K)(q; q 2 x) y(x) = lim = 1 + q 1 P(K)(q; x) a n x n. n=1

22 Counting (rational) lattice paths [M. Panfil, P. Sulkowski, M.S., 2018] Proposition Let r and s be mutually prime. Let K = Tr,s f = rs be the (rs)-framed (r, s)-torus knot. Then the corresponding coefficients a n are equal to the number of directed lattice path from (0, 0) to (sn, rn) under the line y = (r/s)x.

23 Knots and quivers results This relationship naturally suggests a particular refinement of the numbers a n.

24 Knots and quivers results This relationship naturally suggests a particular refinement of the numbers a n. For example, for 2/3 slope, after computing the relevant invariants (for the bottom row) of the T 2,3 knot, and applying all the machinery, we get that the matrix of a quiver corresponding to the knot is: [ ].

25 Knots and quivers results This relationship naturally suggests a particular refinement of the numbers a n. For example, for 2/3 slope, after computing the relevant invariants (for the bottom row) of the T 2,3 knot, and applying all the machinery, we get that the matrix of a quiver corresponding to the knot is: [ a (2/3) n = = i+j=n n i=0 1 7i + 5j n + i + 1 (rediscovered Duchon formula) ]. ( ) ( ) 7i + 5j + 1 5i + 5j + 1 i j ( ) ( ) 5n + 2i 5n + 1. i n i

26 Paths under the line with slope 2/3 a (2/3) n = i+j=n 1 7i + 5j + 1 ( ) ( 7i + 5j + 1 5i + 5j + 1 a (2/3) n : 1, 2, 23, 377,... j i j ) i

27 Knots and quivers results New results: For 2/5 slope, i.e. T 2,5 the corresponding quiver matrix is: a (2/5) n = 1 11i+9j+7k+1( 11i+9j+7k+1 i i+j+k=n ) ( ) 9i+9j+7k+1 j ( 7i+7j+7k+1 k )

28 Consequence 1 binomial identities All this also rediscovers some binomial identities, like e.g. ( ) 5n = 2n n i=0 5n 5n + 2i ( 5n + 2i i ) ( ) 5n n i Comes from counting of paths under line with slope 2/3.

29 Consequence 1 binomial identities All this also rediscovers some binomial identities, like e.g. ( ) 5n = 2n n i=0 5n 5n + 2i ( 5n + 2i i ) ( ) 5n n i Comes from counting of paths under line with slope 2/3. One can obtain such identities precisely for the quivers that correspond to (torus) knots.

30 Counting of weighted lattice paths Proposition The generating function y qp (x) of lattice paths under the line of the slope r/s, weighted by the area between this line and a given path, is equal to y qp (x) = k=0 π k-paths q area(π) x k = P C (q 2 x 1,..., q 2 x m ) P C (x 1,..., x m ) xi =xq 1. For the line of the slope r/s, the quiver in question is defined by the matrix C that encodes extremal invariants of left-handed (r, s) torus knot in framing rs.

31 Counting of weighted lattice paths Example: line of slope 2/3 y qp (x) = 1 + (q 4 + q 6 )x + (q 8 + 3q q q q q q 20 + q 22 + q 24 )x (q q q q q q q q q q q q q q q q q q q q 50 + q 52 + q 54 )x q x + 23x x

32 Formulae for classical DT quiver invariants P C (x 1,..., x m ) = d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm

33 Formulae for classical DT quiver invariants P C (x 1,..., x m ) = d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm P C (q 2 x 1,..., q 2 x m ) y(x 1,..., x m ) = lim = b l1,...,l q 1 P C (x 1,..., x m ) m x l xm lm. l 1,...,l m

34 Formulae for classical DT quiver invariants P C (x 1,..., x m ) = d 1,...,d m m ( q) i,j=1 C i,j d i d j (q 2 ; q 2 ) d1 (q 2 ; q 2 x d 1 1 ) x m dm. dm P C (q 2 x 1,..., q 2 x m ) y(x 1,..., x m ) = lim = b l1,...,l q 1 P C (x 1,..., x m ) m x l xm lm. l 1,...,l m Coefficients b l1,...,l m b l1,...,l m = A(l 1,..., l m ) where take form m j=1 m 1 A(l 1,..., l m ) = 1 + ( 1) (C i,i +1)l i 1 + m i=1 C i,jl i k=1 admissible Σ k ( 1 + m i=1 C ) i,jl i l j (i u,j u) Σ k C iu,j u l iu.

35 Expression for the coefficient A(l 1,..., l m ) C is an m m matrix. Definition Let k {1,..., m}. For a set of k pairs {(i 1, j 1 ),..., (i k, j k )}, with 1 i u, j u m, we say that it is admissible, if it satisfies the following two conditions: (1) there are no two equal among j 1,..., j k (2) there is no cycle of any length: for any l, 1 l k, there is no subset of l pairs (i ul, j ul ), l = 1,..., l, such that j ul = i ul+1, l = 1,..., l 1, and j ul = i u1.

36 Alternative, recursive definition We observe the set of polynomials in m variables p(c)(x 1,..., x m ), whose coefficients depend are functions of the entries of matrix C. Define actions of the symmetric group S m on m m matrices by: [σ C] i,j := C σi,σ j, i, j = 1,..., m, and on polynomials p(c)(x 1,..., x m ) by σ p(c)(x 1,..., x m ) := p(σ C)(x σ1,..., x σm ).

37 Alternative, recursive definition Then, for a given m m matrix C we define the polynomial P m (C)(x 1,..., x m ) by the following: σ P m (C)(x 1,..., x m ) = P m (C)(x 1,..., x m ), σ S m P 1 (C)(x) = 1, ( P m (C)(x 1,..., x m 1, 0) = P m 1 (C )(x 1,..., x m 1 ) 1 + m 1 where C denotes the submatrix of C formed by its first m 1 rows and columns. i=1 C i,m x i ).

38 Alternative, recursive definition Then, for a given m m matrix C we define the polynomial P m (C)(x 1,..., x m ) by the following: σ P m (C)(x 1,..., x m ) = P m (C)(x 1,..., x m ), σ S m P 1 (C)(x) = 1, ( P m (C)(x 1,..., x m 1, 0) = P m 1 (C )(x 1,..., x m 1 ) 1 + m 1 where C denotes the submatrix of C formed by its first m 1 rows and columns. Then: A(l 1,..., l m ) = P m (C)(l 1,..., l m ). i=1 C i,m x i ).

39 Paths under the line with slope 3/4 C (3,4) = paths = A (3,4) (l 1,l 2,l 3,l 4,l 5 ) l 1 + +l 5 =n 1 7l 1 +7l 2 +7l 3 +7l 4 +7l 5 +1( 7l 1 +7l 2 +7l 3 +7l 4 +7l 5 +1 l 1 ) 1 7l 1 +9l 2 +8l 3 +9l 4 +9l 5 +1( 7l 1 +9l 2 +8l 3 +9l 4 +9l 5 +1 l 2 ) 1 7l 1 +8l 2 +9l 3 +9l 4 +10l 5 +1( 7l 1 +8l 2 +9l 3 +9l 4 +10l 5 +1 l 3 ) 1 7l 1 +9l 2 +9l 3 +11l 4 +11l 5 +1( 7l 1 +9l 2 +9l 3 +11l 4 +11l 5 +1 l 4 ) 1 7l 1 +9l 2 +10l 3 +11l 4 +13l 5 +1( 7l 1 +9l 2 +10l 3 +11l 4 +13l 5 +1 l 5 ).

40 A (3,4) (l 1,l 2,l 3,l 4,l 5 )= 1+28 l l l l l l 1 l l2 1 l l3 1 l l l 1 l l 2 1 l l l 1 l l l l 1 l l2 1 l l3 1 l l 2 l l 1 l 2 l l 2 1 l 2 l l2 2 l l 1 l2 2 l l3 2 l l l 1 l l2 1 l l 2 l l 1 l 2 l l 2 2 l l l 1 l l 2 l l l l 1 l l2 1 l l3 1 l l 2 l l 1 l 2 l l 2 1 l 2 l l2 2 l l 1 l2 2 l l3 2 l l3 3 l l l 1 l l2 1 l l 3 l l 1 l 3 l l 2 1 l 3 l l 2 l 3 l l 1 l 2 l 3 l l2 2 l 3 l l2 3 l l 1 l2 3 l l 2 l2 3 l l 2 l l 1 l 2 l l2 2 l l 3 l l 1 l 3 l l 3 l l l 2 l 3 l l l 1 l l 2 3 l l l 1 l l 2 l l2 1 l l3 1 l l 2 l l 1 l 2 l l2 1 l 2 l l 2 2 l l 1 l2 2 l l3 2 l l 2 l 3 l l2 3 l l 4 l l 1 l 4 l l2 4 l l 2 l l 3 l l 1 l 3 l l 2 1 l 3 l l 2 l 3 l l 1 l 2 l 3 l l2 2 l 3 l l2 3 l l l 1 l l 1 l 2 3 l l 2 l2 3 l l3 3 l l 4 l l 1 l 4 l l2 1 l 4 l l 2 l 4 l l 2 l 4 l l 3 l 4 l l 1 l 2 l 4 l l 2 2 l 4 l l 3 l 4 l l 1 l 3 l 4 l l 2 l 3 l 4 l l2 3 l 4 l l 3 l l 2 4 l l 1 l2 4 l l 2 l2 4 l l 3 l2 4 l l3 4 l l l 1 l l 4 l l 2 1 l l 2 l l 1 l 2 l l2 2 l l 3 l l 1 l 3 l l4 5.

41 Schröder paths <latexit sha1_base64="d1fgbxu9z8ueplpbludsgokzc2c=">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</latexit> x Figure: An example of a Schröder path of length 6.

42 Schröder paths and full colored HOMFLY-PT Quiver corresponding to the full colored HOMFLY-PT invariants of knots in framing f = 1 [ ] 2 1 C = 1 1 This corresponds to counting paths under the diagonal line y = x.

43 Schröder paths and full colored HOMFLY-PT Quiver corresponding to the full colored HOMFLY-PT invariants of knots in framing f = 1 [ ] 2 1 C = 1 1 This corresponds to counting paths under the diagonal line y = x. In this case we take specializations: x 1 x, x 2 ax

44 Schröder paths and full colored HOMFLY-PT Quiver corresponding to the full colored HOMFLY-PT invariants of knots in framing f = 1 [ ] 2 1 C = 1 1 This corresponds to counting paths under the diagonal line y = x. In this case we take specializations: x 1 x, x 2 ax Then from the quiver generating function of C we get y(x, a, q) = 1 + (q + a)x + ( q 2 + q 4 + (2q + q 3 )a + a 2) x with the height of a path measured by the power of x and the number of diagonal steps measured by the power of a.

45 Schröder paths and full colored HOMFLY-PT Figure: All 6 Schröder paths of height 2 represented by the quadratic term q 2 + q 4 + (2q + q 3 )a + a 2 of the generating function.

46 Consequence 2 Some divisibilities (integrality) If p is prime, then: p ( 3p 1 p 1 ) 1.

47 Consequence 2 Some divisibilities (integrality) If p is prime, then: p ( 3p 1 p 1 ) 1. p 2 ( 3p 1 p 1 ) 1.

48 Consequence 2 Some divisibilities (integrality) If p is prime, then: p ( 3p 1 p 1 ) 1. p 2 ( ) 3p 1 p 1 1. ( (3p 1 p 3 ) ) 2 1. p 1

49 Consequence 2 Some divisibilities (integrality) If p is prime, then: p ( 3p 1 p 1 ) 1. p 2 ( ) 3p 1 p 1 1. ( (3p 1 p 3 ) ) 2 1. p 1 If r N, then r 2 µ ( ) ( r 3d 1 ) d d 1. d r { ( 1) µ(n) = k, n = p 1 p 2 p k, 0, p 2 n

50 Consequence 2 Some divisibilities (integrality) If p is prime, then: p ( 3p 1 p 1 ) 1. p 2 ( ) 3p 1 p 1 1. ( (3p 1 p 3 ) ) 2 1. p 1 If r N, then r 2 µ ( ) ( r 3d 1 ) d d 1. d r { ( 1) µ(n) = k, n = p 1 p 2 p k, 0, p 2 n Corresponds to the fact that DT invariants are non-negative integers (in this case of the quiver of the framed unknot one vertex, m-loop quiver)

51 Consequence 2 Integrality of DT invariants Let C be a 2-vertex quiver: [ ] α β C = β γ with α, β, γ N. Then for every r, s N we have β Ω r,s = ( 1) (α+1)r/d+(γ+1)s/d µ(d) (αr + βs)(βr + γs) d gcd(r,s) ( ) ( ) αr/d + βs/d βr/d + γs/d N r/d s/d

52 Rational knots (joint with P. Wedrich) p/q = [a 1,..., a r ] = a r + Rational tangle encoded by [2, 3, 1] 1 a r a r

53 Rational knots T ( ) :=, R( ) :=

54 Rational knots T ( ) :=, R( ) := UP :, OP :, RI :

55 Skein theory l k k l = l k l h=0 ( q)h l k l k h l l k k l = l k k h=0 ( q)h k k l k h l

56 Basic webs and twist rules UP[j, k] = j j k k j j, OP[j, k] = j j k k j j, RI [j, k] = j j k j j 1 TUP[j, k] = j [ h=k ( q)h j q k2 h ] k UP[j, h] + 2 RUP[j, k] = k h=0 ( q)h j a h j [ q 2kh+k2 +j 2 j h ] k h OP[j, h] + 3 TOP[j, k] = j h=k ( q)h a k q k2 2jk [ ] h k RI [j, h] + 4 ROP[j, k] = k h=0 ( q)h j a k j [ q 2h(j k)+(k j)2 j h ] UP[j, h] 5 TRI [j, k] = j h=k ( q)h a h q k2 2jh [ h k 6 RRI [j, k] = k h=0 ( q)h q h(2j 2k)+k2 j 2 [ j h k h ] + k h + OP[j, h] ] RI [j, h] +

57 Rational knots: Knots-quivers corresponds works! Theorem Let K be a rational knot and let Q K be the corresponding quiver. Then, the vertices of Q K are in bijection with generators of the reduced HOMFLY-PT homology of K, such that the (a, q, t)-trigrading of the i th generator is given by (a i, Q i,i q i, Q i,i ) where Q i,i denotes the number of loops at the i th vertex of Q K.

58 Open questions work in progress Find quivers for other larger classes of knots How to find a quiver for a given knot directly (geometrically, topologically...)? Other, better definition? The (non)uniqueness of a quiver. More proofs... Closed formulas for q-version of paths. Further integralities and combinatorial identities, Rogers-Ramanujan identities,... Links and quivers... Extend to all representations, not necessarily symmetric. What is so special for quivers that correspond to knots? Combinatorial identities for binomial coefficients, and extended integrality/divisibility hold precisely for them.

59 Thank you for your attention!