JHEP12(2020)095 Springer July 17, 2020 : October 29, 2020 : December 15, 2020 : Received , Accepted Published b,c Published for SISSA by https://doi.org/10.1007/JHEP12(2020)095 [email protected] and Piotr Sułkowski , b invariants for some knot complements, from an appro- b Z . 3 Jakub Jankowski a 2007.00579v2 The Authors. c Chern-Simons Theories, Topological Field Theories, Topological Strings

Alexander polynomial arises in the leading term of a semi-classical Melvin- , [email protected] [email protected] Faculty of Physics, Universityul. of Warsaw, Pasteura 5, 02-093Walter Warsaw, Burke Poland Institute forPasadena, Theoretical Physics, CA California 91125, Institute U.S.A. ofE-mail: Technology, Department of Mathematics, UniversitätWeyertal zu 86-90, Köln, D-50931, Cologne, Germany b c a Open Access Article funded by SCOAP ArXiv ePrint: sions, which are eitherand cyclotomic, the or knots-quivers encode correspondence. certain features of HOMFLY-PT homology Keywords: priate rewriting, quantization and deformationwe of rederive Alexander conjectural polynomial. expressions Alongobtained for this recently, the route thereby above provingtheorem, mentioned invariants their and for derive consistency various newgiven with knots formulae knot, the for depending Melvin-Morton-Rozansky colored on superpolynomials certain unknown choices, before. our reconstruction For leads a to equivalent expres- Abstract: Morton-Rozansky expansion of coloredposite knot direction, polynomials. we propose In howpolynomials, this to and work, reconstruct newly introduced following colored the HOMFLY-PT polynomials, op- super- Sibasish Banerjee, Revisiting the Melvin-Morton-Rozansky expansion,there or and back again JHEP12(2020)095 r 23 24 15 18 S (1.1) const, = r q = x for ], and later generalized 1 )[ = 2 , k N ~ ) x ) ( 1 14 − k x, N ( +2 colored by symmetric representations k 9 N ) R ∆ , q N =1 – 1 – ∞ q 14 X k ( r + P 8 1 10 . This conjecture was originally formulated by Melvin − 12 x N ) 1 5 x invariants 8 21 4 ) ,... ,... ∆( 1 2 1 4 , the first polynomial knot invariant introduced almost a hun- 8 7 4 ' ) , , x 1 2 ) x, a, q ~ 6 5 ( e , , 1 knot invariants ∆( 1 1 K 3 4 3 ) = F torus knots N ( , q 1 32 N are polynomials in 30 20 26 28 q SU ) + 1) ( r p knot P 2 knot knot knot knot , x, N 4 19 1 2 3 ( 6 6 7 8 5 (2 k →∞ ,r R 0 lim → 4.7 4.8 4.9 4.10 4.1 Trefoil knot, 4.2 Figure-eight knot, 4.3 4.4 4.5 Twist knots 4.6 Twist knots 3.1 The “homological”3.2 approach The “cyclotomic”3.3 approach Relation to 2.1 Alexander polynomial2.2 HOMFLY-PT homology 2.3 Cyclotomic expansions 2.4 Knots-quivers correspondence ~ where and Morton in the case of colored Jones polynomial (i.e. for dred years ago,knot plays theory a and prominent its rolethe Melvin-Morton-Rozansky physical in conjecture, incarnations. subsequently both turned older into Itsthat a quantum essential and theorem, role which most states is recenthave revealed, the developments following for in semi-classical example, expansion by 1 Introduction Alexander polynomial 4 Reconstructing colored (super)polynomials 3 Back again from Alexander polynomial Contents 1 Introduction 2 An unexpected party JHEP12(2020)095 , – 12 14 [ ) x, a, q ( K F or ) ), as a series of some x, q ( 1.1 K ). Interestingly, to fix this F theory on a complement of a 1.1 0) , ], and an analytic version has been (2 5 ]. It has an important interpretation in 4 – 2 – 2 – ]. The Melvin-Morton-Rozansky expansion, which 13 -deform various terms in such a series. Depending on ]. In particular, structural properties of this latter – t 9 8 , 7 ]. However, we stress that in those cases our derivation in the subleading terms in ( . It is thus worth revisiting the Melvin-Morton-Rozansky 3 ) 21 S , ]. 6 12 x, N – ( k ]. Making contact with cyclotomic expansions and knots-quivers 10 R 20 , 19 specialization of colored HOMFLY-PT polynomials and Alexander polyno- -deform, and possibly invariants for knots complements, also denoted a N b Z q -deformation, involves a little arbitrariness, which can be fixed by comparison theory arising from compactification of 6d t = a = 2 ] — which at the same time confirms the existence of cyclotomic expansions for colored - and Note that some of our results reproduce conjectured expressions for colored knot poly- The main goal of this note is to understand in detail how the Melvin-Morton-Rozansky These days an interest in the Melvin-Morton-Rozansky expansion resurfaces. For ex- -deform, q N 18 q , ]. These invariants are also inherently related to physics, as they count BPS states in a -, nomials found e.g. indoes [ not only provide anpressions equivalent indeed conjectural expressions, satisfy but the also Melvin-Morton-Rozanskyit asserts theorem. is that a This those new is ex- and importantthose independent because previously check found that these expressions expressions for are colored correct. polynomials Moreover, were recall derived that taking advantage a with known HOMFLY-PT polynomialsfirst few for polynomials smallarbitrariness colors, in or all examples (at thatwith least we polynomials consider in colored in by principle) this the paper, with fundamental it and is the sufficient second to symmetric compare representation. just 17 HOMFLY-PT polynomials and superpolynomials —certain or features take of equivalent HOMFLY-PT form homologycorrespondence that and [ encodes is relevant incorrespondence the is context also of an knots-quivers important aspect of this work. The second step, which captures find that this reconstruction processof can be Alexander split polynomial, into twoparticular i.e. steps. form, the First, essentially we first using writeto term the the inverse inverse in binomial the theorem.some expansion choices Second, made ( in we the propose first step, how the expressions that we obtain are either cyclotomic [ expansion and, among others, understanding its role in theseexpansion new arises from developments. various recently— found expressions taking for colored the knotHOMFLY-PT opposite polynomials, polynomials and and direction superpolynomials — starting from to Alexander polynomial. provide We prescription how to reconstruct colored level. On the otherintroduced hand, Melvin-Morton-Rozansky expansion enters a16 definition of newly 3d knot under consideration in knot Floer homology —ored) which HOMFLY-PT categorifies homology Alexander polynomial [ homology theory — recently and enabled a topolynomials putative derive and conjectural (col- superpolynomials formulae [ for coloredrelates HOMFLY-PT mial, should therefore be one manifestation of such a general theory on a decategorified physics, namely the above limitfrom captures an contributions to abelian Chern-Simons flat partition connection.formulated function and Its proven proof in is [ given in [ ample, it is believed that a general knot homology theory exists, which encompasses both by Rozansky to more general knot polynomials [ JHEP12(2020)095 X and 2 , such 6 3 S ) = 1 + torus knots in (labeled by ] in the case x ( 1 14 +1 K 3 p , differ only by 2 ,... , ∆ 2 1 2 ) 8 x T , are closely related x , knots differ only by − 1 ), and to satisfy the 3 (1 6 pX , 6 1 − 4 = = 2 and . Rewriting the right hand N X r 2 ) ) = 1 6 x , t ), whose Alexander polynomials ]. Colored polynomials for twist 1 ∆( (with . It would be interesting to unveil ]. Formulae for 10 a, knot, which is one of the first knots 2 ( ,... 0 4 3 X 11 , 7 =1 , ]. Colored superpolynomials for r 2 + → , P 10 10 ~ X = 1 ) = p , t ], i.e. the statement that colored superpolyno- – 3 – 1 13 a, ) = 1 + , etc. We focus on derivation of colored superpoly- – ( 4 x 9 r 7 ( P 3 , 6 , and they were originally introduced in [ 3 , are found in [ ) 2 7 ∆ ) in the limit 7 x, q knot, are given in [ 1.1 ( and 1 K and 7 (also labeled by 2 F 1 6 ]. Therefore the first knots for which colored polynomials have X 12 ,... − 2 7 . Similarly, Alexander polynomials for X , . More generally, twist knots in the series 2 twist knot — it is therefore instructive to see how our reconstruction − 5 pX X , satisfy the relation 2 1 9 3 − ) as an illustrative example, among others because it has the same Alexander ), whose Alexander polynomials take form 4 ) = 1 7 x ( = 1 , whose Alexander polynomials can be written respectively as ,... a, q, t ) = 1 + 2 1 ( 3 6 x r 4 1 , so in particular for , 4 p 2 ∆ for knot complements might be found. For a complement of a knot P , ∆( ∆ Apart from reconstructing formulae for colored polynomials and understanding their In the course of our analysis we also find an interesting relation between colored To illustrate the power of our formalism, we also derive new formulae for colored poly- b Z knots are found in [ and = 1 1 3 for invariants are also denoted corresponding to Jones polynomial.to In have this the original setting, samesame these expansion quantum invariants A-polynomial ( are equation supposed as colored Jones polynomials. A generalization of to twist knots read signs, other consequences of this correspondence. structure, another important motivation for this work is that along similar lines expressions which differ only byof certain our reconstruction signs process, and3 as deformation mentioned terms above. thatand arise The simplest in pair thep of second such step knots are HOMFLY-PT polynomials or superpolynomialslogical for diagrams differ various only pairsby by of a one knots, diamond element: and whose a ader homo- zig-zag trivial polynomials zig-zag of of length for length 3signs. these 1 for for knots, one the In knot other written turn, is knot. as this replaced In polynomials leads consequence, Alexan- to in very similar cyclotomic expansions for such pairs of knots, nomials for polynomial as process differs for those twocolored knots. superpolynomials Following for our other prescription, knots one with can 7 analogously and derive more crossings. polynomials for knots upknots, to so 5 in crossings particular for werefor given all in [ 6 not been written down before are Melvin-Morton-Rozansky theorem that weone take might advantage expect of that our is techniquesthat proven enable do for to not get all satisfy colored knots, polynomials the and also exponential for growth. these knots nomials, unknown before. In particularfor we which focus explicit colored on polynomials have not been found to date. Indeed, colored super- mials side of this relation inthat terms we of obtain, binomial expansion anderty leads ultimately holds to the only analogous same expressions for to results. ato those restricted determine However, family the colored of exponential knots polynomials growth (in only prop- particular for the thin this knots), restricted so class. it enables On the other hand, the of the exponential growth condition [ JHEP12(2020)095 . . ~ K 4 e ]). F we 1 16 = , as in p q invariants or invariants, ) ) ~ ]. Yet another x, q we present our 16 ( x, a, q ( 3 K K F F ]. Furthermore, it would ). There are also different ], and upon specialization torus knots, for all 20 23 , 1.1 . Apart from the dependence – , which plays a crucial role in +1 19 p 2 ) 2 21 , -deformed knot. x 2 x a − 4 T -deformed quantum A-polynomial 7 (1 a = ). Therefore one might hope that ∆(1) = 1 ], and we concisely summarize various aspects -expansion ( X 1.1 2 15 ~ invariants is discussed in [ ) ), and our reconstruction prescription, are 1.1 x, q ( – 4 – K F we illustrate how this procedure works in various 4 can be defined in various ways: by considering a knot diagram in terms of a variable ) case was proposed in [ and we use normalization ) x ]; it is thus important to understand how our results fit into this ) ) x 1 N 14 ]. The latter invariants satisfy ∆( − ( ∆( x are closely related to colored polynomials (as also explained in [ 16 ) SU , from the leading terms of the q , associated to other flat connections, in particular the one that arises in the ) = ∆( ~ x, a, q x ( K ∆( F have the same semi-classical limit as in ( N , we also express q The plan of the paper is as follows. In section Finally, let us mention a few other problems worth pursuing, related to the results of x = relations, etc. These arelist standard definitions, explicitly so Alexander instead polynomials ofNote for providing that details, various in knots table thaton we will considercyclotomic in expansions section that we discuss in what follows. notation that we use in the rest of2.1 the paper. Alexander polynomial Alexander polynomial and associating weights to its crossings, in terms of a Seifert matrix, by Conway’s skein we derive expressions for colored superpolynomials for 2 An unexpected party In this sectionphysical we concepts, briefly that summarize play a various role developments in in what knot follows. theory, At the and same related time, we also set up the then reveal. of knot invariants thatreconstruction play procedure. a role In inexamples, section which the involve rest knots of up to the 8 paper. crossings and In some section infinite series, and in particular constrained by the underlying integralcaptured by structure LMOV of invariants), or colored the HOMFLY-PT integralseries polynomials structure (captured of (as by corresponding motivic quiver generating Donaldson-Thomasbe invariants) [ rewarding to lift ourwhat reconstruction relations between program knot to Floer a homology homological and level, colored and HOMFLY-PT homology to it understand would volume conjecture. All theseto expansions some should extent be e.g. related inframework. by [ the In a resurgence, similar as vein, discussed modularity, it whose is role worth in uncovering the how contextinteresting our of results question are is related to how quantum the expansion ( this case this paper. TheIn first this interesting work issue we iswhich postulate an depend how on interplay to between reconstruct expansionsexpansions colored in in polynomials or were introduced in [ equations, the same as coloreda HOMFLY-PT polynomials [ invariants could be reconstructedshow following that similar this steps is indeed as the we case present at in least for this a work. family of We these invariants to JHEP12(2020)095 (2.3) (2.1) (2.2) 3 X i + 3 X 2 2
and in terms of i X 2 X i X + 2 2 x 2 p + X X 2 + 5 X + 2 − + X =1 + X p i X X pX X pX X P + 5 − − X X X X X 2 X X X − − = 1 = 1 = 1 + 6 = 1 + = 1 + 5 = 1 + 2 = 1 + . . 3 . 2 2 = 1 + 3 ) x ) px x = 1 x x s px = 1 s 2 = 1 + 2 = 1 + 3 = 1 + 4 = 1 + 4 +1 ( = 1 + ( − ) x − x p + x i,j,k = 1 + 5 x d + d 2 x x x x 2 x ) x x 2 3 + 2 x ) + H 1+ + p x − x x − p 3 x ∆( 1+ x 2 3 − that categorifies HOMFLY-PT poly- 1 + + [ p 3 + 2 5 + 3 7 + 4 7 + 4 − HFK 1 − 3 + 3 x + 3 − dim 2 \ x − + 5 HFK − − − − − x 1 k \ 1 − HFK 1 = − 1 1 1 1 i,j,k + 5 − + 1 1 − − dim 1) + (1 + 2 i 1 + + 3 x − − − − 1 x + (1 H x dim − 1 s x 1 x x x x 1 − 2 − − i 1 − q x s − ( 2 3 4 4 − − x + d + 1 − q − j x 1 p t 3 x d q 2 px − 3 i + 3 – 5 – − px 1) − x − 1) a − 2 2 d,s − x X − 2 − + − ( − 2 − ( x p − x 2 − X x i,j,k − 3 − − x ) = d,s = − x 2 X for various knots, written as a function of p i ), which relates Alexander polynomial to colored x ) ) = − q, t P x 3 1.1 ) = − q a, q ∆( x ( ∆( P HFK( ,... ,... 1 2 +1 p 8 7 2 , , , 1 2 2 6 5 T ] , , 1 1 4 3 1 1 1 2 1 2 3 1 2 3 4 2 -deformation of Alexander polynomial, and can be also obtained as a 24 19 [ t 3 4 5 5 6 6 6 7 7 7 7 9 8 Knot , so that ) \ . HFK a, q 2 ( Torus knots ) x . Alexander polynomials P x Twist knots Twist knots − (1 Furthermore, Alexander polynomial arises as an Euler characteristic of knot Floer = While knot Floer homologyexists categorifies a triply-graded Alexander HOMFLY-PT polynomial, homology nomial it is expected that there This provides a specialization of a superpolynomial, i.e.briefly Poincaré characteristic discussed of in HOMFLY-PT the homology, next section. 2.2 HOMFLY-PT homology responding homology theories shouldnatural exist. to consider From the the Poincaré characteristic homological of perspective, it is also Knot Floer homology is expectedHOMFLY-PT to homology arise that through the weMorton-Rozansky action briefly of expansion discuss a certainHOMLFY-PT in ( differential polynomials, the from indicates next on section. decategorified level that Also a the relation Melvin- between cor- X homology Table 1 JHEP12(2020)095 , 3 are , (2.7) (2.6) (2.4) (2.5) ) 2 i , , t . Knots 1 i , q i differentials a ( const N d = , which categori- i δ R i,j,k = H δ , i -degree of generators re- t . Superpolynomials for t a i 1) q q − . i ]. In particular it follows that a . ]. For a large class of knots, 7 a R 8 Khovanov-Rozansky homology, , q, R i,j,k r -grading, is not known, however it is ex- i S i,j,k (1 N δ H X H sl P = i,j,k . -degree and dim i q t H dim k ) = i,j,k k i,j,k − q 1) t i j H H − q q ( i ∆( j a + q dim i i – 6 – a a k t X and i,j,k j ’th generator as q X = 2 i i i,j,k 1) a i ) = − δ ) = X i,j,k . a, q, t a, q, 2 , have been formulated in [ a, q ( ( ’th generator. Conjecturally, the action of ( r r ) = i P R S P P = a, q, t ) = ( R reduces the HOMFLY-PT homology to the knot Floer homology. P 0 ], defined for the a, q d ( 7 P ]. These properties impose constraints that enable to determine e.g. a 7 [ Z reduces the HOMFLY-PT homology to the ∈ -grading [ 0 δ in the summation in the expression on the right labels homology generators, so N i N > Conjecturally, there exists also colored HOMFLY-PT homology One aspect of the HOMFLY-PT homology that we take advantage of, are relations It is useful to present the structure of a superpolynomial in a so-called homological An important quantity that characterizes generators of HOMFLY-PT homology is the for N The structure and propertiesmetric of representations various differentials inthese such structural homology properties theories, enabledefined for to as sym- determine corresponding colored superpolynomials, fies HOMFLY-PT polynomials colored by a representation between its generators imposed by the cancelinggenerators differentials of [ the HOMFLY-PTwhich homology can we be refer assembled tohomological into as diagrams two presented a types in zig-zag of whatzig-zag, and structures, follows. which a A consists given diamond. of diagram an necessarily We odd contains denote one number of zig-zags dots, and and diamonds a in number of all diamonds. diagram, whose horizontal and verticalspectively. axes encode Each dot in ain diagram the represents one superpolynomial. generator oretc. Examples the corresponding of monomial homological diagrams are shown in figures A knot is calledthat thin do if not all satisfy its this generators condition have are the called same thick. In consequence, HOMFLY-PT andsuperpolynomial, Alexander polynomials arisevarious as knots specializations are of listed the in table so-called where that each monomial inreferred the to superpolynomial as represents degreeswith one of generator, the and while the action of pected to satisfy a numberd of properties, which include thesuperpolynomial, existence of i.e. certain Poincaré differentials characteristic of An explicit construction of the homology theory JHEP12(2020)095 2 2 q a 6 (2.8) (2.9) t + is the (2.10) 3 2 q t 2 s 3 6 3 q a a qt (where we 6 3 + + qt a ) 2 3 7 3 t t q t t ) 2 + q a ( 2 + ) 2 2 3 a 9 4 q ), superpolyno- q  t 4 − + 4 aq − 5 p P + + p qt a 6 4 a 2 2.3 2 t 2 t + 3 3 t 3 t 1 qt a + + ≡ q 2 4 3 , q 2 − 2 7 3 a . ) 7 3 − a p aqt ) qt t p
a + 2 2 a qt q 4 ) a r + + aqt 4 ( + a aqt a for various knots. 2 p + S i,j,k + a 4 5 2 2 + ) P t t ) + 7 + qt + ) = + q 8 a, q, t 4 H 2 t + aqt 4 ( a 2 5 qt 2 q ... aqt + qt r qt t ... ≡ 4 ( 2 q qt 4 + 3 4 a + + 2 + q Q qt a ) aqt a, q, t a + at 7 − qt a dim ) a 3 ( 2 t a, q, t q 7 + 4 t 3 ( 4 t a k at ( + + 1  t + 2 + + t (1 2 4 5 1 + 1 a 2 + 2 a 6 r 3 . q P + 1 + 2 2 4 a 1) + + 5 a P r 3 qt Q t ) + 2 qt ) t 8 1 3 ) aqt + at qt + at − aq ≡ 3 + 3 t 2 aqt + t − 5 3 ( a − 3 a 5 ). Recall that some knots satisfy the 4 , t ) t a t 1 a 5 t j a + + t 2 t t 1 3 t a 1 2 + 3 q 2 + − 3 )( 3 + 2 + a with positive coefficients, and i − a at + a a, q, t t 1 3 2 a + a + 2 a, 3 a ( 4 at 2 or t aq +2 5 t t + 3 + ( 5 2 + a + a, q, t t p  )( 2 + + 2 t qt ( 2  4 at + (1 + a aqt 2 a 3 a P 4 2 3 ) 1 3 3 a + X at i,j,k P P at t a a t qt + 1 + rs + 2 + qt 2 + 2 q a 3 aqt and + 2 ( 6 a a 3 + + a + 1 + + 2 t t – 7 – + 2 − 2 1 q q at 4 4 2 3 )(1 + + (1 + t ) = t − at 1 q t q a 2 qt + a 3 at 1 s 1) = t + 2 3 0 + 2 1 + a q 2 , t 2 a, q − 5 t + a aqt r − + )(1 + a 1 + 2 + 2 qt at t 1 at p 1 + rs − t + q q 3 4 ) 2 2 2 + at atq 1 2 t q q − a, qt + a q a a 3 q + q 3 ( 5 t 2 t a q rs rs a + 2 4 r + a/q q − a, q, q 4 4 + + ( + a 1 t atq ( − − 4 a q P 3 1 r t + 3 7 qt + q a a a q a 3 + 1 P qt 2 2 1 + a t + (1 + 4 aqt 1 3 3 q = + 2 qt 2 a q a 1 t a + ) = q 2 3 + t − ) = + 2 a + q t 2 t + 2 a t t 1 + (1 + 2 4 q 1 − 2 2 t a q a + a q q 2 a, q 2 + 1 q 3 a, q, t ( t 2 2 ( r q a 4 q + a r a P q a P are polynomials in + + 6 r t 3 3 2 q a . (Uncolored) superpolynomials Q 1 q 3 , corresponding knot polynomials (Jones, HOMFLY-PT ( a and = 1 ), we denote HOMFLY-PT polynomials colored by symmetric representations )) are referred to as uncolored, and denoted ,... ,... 1 r simply by Table 2 1 2 +1 p 8 7 Q 2.7 1 1 1 2 1 2 3 1 2 3 4 2 19 r 2 2.4 , , 3 4 5 5 6 6 6 7 7 7 7 9 , 8 1 2 2 S Knot For 6 5 T , , = 1 1 4 3 mial ( skip a dependence onexponential other growth variables, condition, such which as is(or the various specializations statement that thereof) their satisfy colored the superpolynomials relation R Rasmussen invariant. These conditions,existence and of other other analogous differentials,follows strongly conditions we constrain arising verify the from whether form the determine colored of for HOMFY-PT superpolynomials. various polynomials knots In oras indeed what superpolynomials satisfy in that these ( we conditions. For brevity, and analogously which impose the following conditions on colored superpolynomials where For example, such colored homology theories are expected to possess canceling differentials, JHEP12(2020)095 ] ]; 12 17 (2.12) (2.13) (2.14) (2.11) arrows ]. It is and does 25 i,j r C , ) , so that ) i , a, q, t ) ( , are expected to have xq r K m e S a, q depends on a choice of a − ( m . ) ) ) K m (1 q q q 1 d ( ( ; − 3 m K m =0 K m t ) m i , d r d q ) ; Q q m aq r (here we skip dependence on other ( ) q = − , so that K m aq ; ( K d ( r m m ) ) ) − rm ], and for HOMFLY-PT polynomial and q rm q q − ; ( ; − r, q r 18 q ( q x m # ( ) # m aq r q c ( m r m ; " " – 8 – rm =0 r for a knot +2 , for X − r =0 r m ) =0 r m q q X q m X )  ( m ( q r m γr ; ) = K =0  r r βr t r q X q m P ( − βr βr q q αr K q r ( ≡ a αr P ) = ) m αr a ) q q a ( ( q ]. According to this correspondence, colored HOMFLY-PT r ) = ; ± K r P ) = J +2 20 , while color-independent factor , r a, q ), colored by symmetric representations ) = ( q t K r, q ]. For colored HOMFLY-PT polynomial, the cyclotomic expansion 19 ( K r 12 or m P . c a, q, t ) = ( a γ ( . A generalization of this statement to superpolynomials takes form [ K β r r, q is a universal factor that captures the whole dependence on color ( P and ) m c and r, q α, β α ( m . In case of Jones polynomial the existence of such an expansion is proven in [ c K In addition to the above 3 gradings, it is conjectured that also a closely related spondence, found inhomology, [ and thus allresentation knot theory. polynomials Namely, mentioned it abovegiven turns too, knot, out are which that related in generators to stringstates, of theory quiver HOMFLY-PT can realization rep- homology are be for identified also a with associated certain fundamental to BPS nodes of some particular quiver. There are for some 2.4 Knots-quivers correspondence The last ingredient playing an important role in our analysis is the knots-quivers corre- for some Subsequently, analogous conjectures werefor formulated refined for (or other generalized) Jones coloredsuperpolynomial polynomial knot in in polynomials: [ [ is characterized by knot in this case where not depend on a knot 2.3 Cyclotomic expansions Colored knot polynomials variables, such as cyclotomic expansions, which is the statement that they can be written in the form expected to hold for the knot under consideration. quadruply-graded homology theory andnot corresponding hard to superpolynomials generalize existthis all [ task our for results brevity. to such a quadruply-graded theory, however we skip We also verify whether the expressions that we find satisfy this condition, if only it is JHEP12(2020)095 . ) ], 2 27 . To , i (2.16) (2.15) x x, a, q 26 ( K invariants F ) . ) ]. Geometric j d 30 ]; more recently . =1 x, a, q i j ( m d m 28 P K , x F +1 20 i ··· α ( 1 1 ), upon the identification i d 1 − β =1 x k i ) in various monomials in the r 2.9 q t ) P ; q q q i ; ) ) in the form of an analogous ( i d i q ) α ( and α q j 2.7 ) − d ; 2 i q i q ; α d ( ( a, q q ], and its generalization in the context ( i,j =1 =1 C k i m i ] and arborescent knots [ =1 32 k i , i,j , where P Q 29 ) 1 2 Q 31 P m q q i ) specializing to ( α r – 9 – = =1 k i m , and its advantage is an immediate connection to 2.15 d , . . . , x are powers of P + 1 ) with the above mentioned identification of i ) X we discuss the relation of these results to t x ... 3.2 a ( + r − 1 2.15 ( ]. P d 3.3 and i ). This explains the relation of quivers to HOMFLY-PT d 33 i =0 ]. = ∞ r ) = i , q 2.4 ,...,k i β 20 m P [ X a + =1 i ’th node in this quiver, which encode interactions between these i j α i,j ), and making contact with various concepts presented in section ) = ; it relies on the structure of superpolynomials and HOMFLY-PT C = m , . . . , x 1.1 ], this generating series is equal to the generating function of col- 1 k 3.1 d , where x i i ( + t d r 20 ... ) , 1) P , . . . , x q + 1 ; 2 − 19 d ’th and the x ( q i i ( ( + a 1 P a d =1 i k i ) t q − Q ; i q a ( q Quivers corresponding to a large class of knots were found in [ In what follows we also identify quivers corresponding to various knots, by rewriting = i homology and leads tofrom the expressions perspective for of colored“cyclotomic”, knots-quivers is polynomials presented correspondence. which in section arecyclotomic Another expansions. most rewriting, In appropriate which section weinvariants. call for some knot complements, fromRozansky Alexander expansion ( polynomial, by meansThe of the first Melvin-Morton- stepon in the this inverse process binomialvarious is forms theorem. of appropriate final expressions. rewriting Dependingdiscussed The of in on first section Alexander rewriting, a that polynomial, choice we refer of based to such as “homological”, rewriting, is we obtain 3 Back again fromIn Alexander this polynomial section wecolored present HOMFLY-PT the polynomials main and superpolynomials, idea as of well this as work, i.e. we explain how to reconstruct colored polynomials in thethis form end ( we take advantage, among the others, of the following relation they were also identifiedinterpretation for of all knots-quivers rational is knots discussedof in [ topological [ string theory in [ multiplicities of composite BPS states,tal which BPS are made states. ofthe the A above generating mentioned simple fundamen- function generalizationquiver of of generating the colored series, above superpolynomials determined expressionthe by ( numbers enables the of also same arrows to quiver, present whose structure is captured by As found inored [ HOMFLY-PT polynomials, withx ( uncolored superpolynomial ( homology. Furthermore, factorization of the above series determines LMOV invariants, i.e. BPS states. To such adefined quiver as one can assign the so-called quiver generating series [ between the JHEP12(2020)095 t of − t (3.1) − = 1 = , a   = 1 = ’s would have j i 2 t a x we demonstrate 1 − =0 X j p ) 4.10 x − for a knot that is thin, so . i (1 t t x 1) i , q (i.e. for each term in the above we identified terms associated − 1 q i i , ) 1 a , for each knot these generators x a (1   i  p ∆( , so clearly monomials in Alexander P − 2.2 P has the same parity for all generators x 1) i . In the first, “homological” approach, t − = ). Suppose that the zig-zag has length ) ) = i knot, which is thick. x p − = x i 19 q ∆( (because contributions from diamonds cancel 8 + a, q, t . Let us show first, that such cancellations in ( 1 , q, t 1 – 10 –  − p − P x = 1 ’s would also have different parity — so powers of i = , while the terms that correspond to diamonds we q a − ) ( t x ∆(1) = 1  ( ... z P ∆ + specialization to Alexander polynomial, their ) = +1 q 1 p − − ∆( x = − t p − takes the same value for each generator x i t − i ) = (for various diamonds labeled by q x ( ) + and do not affect this overall sign). Furthermore, from specialization of a z i x is even). Furthermore, if a pair of generators would cancel due to a minus sign ( a ∆ i i a ∆ 2 = 2 = 1 . It follows from specialization of a superpolynomial that it must have form δ Therefore, suppose that in Alexander polynomial To sum up, for thin knots, monomials in Alexander polynomial can be immediately Thus, consider the superpolynomial x + 1 in those terms would be different, and thus they could not cancel. It follows that for p where interchanging signs arise fromsign properties follows from of the canceling normalization differentials,for while the overall superpolynomial and properties of canceling differentials it follows that terms assembled to a zig-zag, whichdenote we denote 2 upon reduction offor a relatively superpolynomial simple thick topolynomial knots, Alexander into it terms polynomial). is that alsothat form not Nonetheless, our a reconstruction hard zig-zag at procedure and to least works diamonds. determine for how In to section split Alexander monomials in the superpolynomial, which is equal to grouped into aknots zig-zag one and can diamonds, alsoto group analogously this the as end terms some in in additional a Alexander terms polynomial must superpolynomial. be into first such For added patterns, and thick however subtracted (those which cancel different parity. It followsq that their thin knots the numberfrom of superpolynomial monomials to does Alexander not polynomial,polynomial and change therefore upon corresponds each the monomial to in specialization Alexander sum a of particular absolute homology values generator. of coefficients This in also Alexander implies polynomial that is the equal to the number of that summation). It follows that(since the combination that would arise upon of generators. Moreover,the Alexander superpolynomial, polynomial i.e. arises aspolynomial are a related specialization to thosemight in arise superpolynomial, upon possibly up thefact to specialization do some cancellations not that arise for thin knots. Our starting point iswe Alexander write polynomial it inHOMFLY-PT homology. a way As thatare we assembled makes reviewed into manifest in one section themonomials zig-zag in structure and the superpolynomial, several of so diamonds. the generators number of All of such these monomials the is generators equal uncolored correspond to the to number 3.1 The “homological” approach JHEP12(2020)095 (3.8) (3.7) (3.2) (3.3) (3.4) (3.5) (3.6) -, and a -, . q ) 1 − as in (or gen- -Pochahmmer p k q m + ) ... x . ( + 1 m , f k , )
) ) x i ( x +2( ( f yq . m f i m s ) x ) − , we get additional binomial x + ! x p m . 2 1 (1 ) x − 1 − − − m i x p p k − u ( -Pochhammers symbols ) represents the left-most end of , =0 (1 k k Y i q 2 (1 f m ! 1) )

1 ! − 3.3 x − = 2 ( invariants, by appropriate 1 − ··· − -Pochhammers, and allow introducing ) m i − p ) q ! X m m that are linear in summation variables, − , i.e. we have only one zig-zag and no (1 q 1 2 ) m i j x ; m t k k s + 2 x y x + x ( n x, a, q i − 1 ( k ! ) =

N 1 − and =0 K x 1) (1 p j m k (

F – 11 – =0 i a − ∞

P − X m ∆ =0 ∞ j m x X , 2 m = ≤ ) ’th diamond. Altogether, it follows that i x i i 1 X ) = ( 1) k 1 n x ) − − ≤ − =0 ( ) = X j u i p N ... k ) + x ( 1 ( x ( ), in case they are known independently. As a general strat- ≤ p ∆ x − 1 in the large bracket in ( 2 ( f x that are quadratic in summation variables. In what follows -binomials, replace Pochhammer symbols by − − z for the =0 X 1 q p 1.1 Y (1 i ), with additional expansion of the term , we write the leading term in the Melvin-Morton-Rozansky q m i = ) = k ) s x ≤ x 1 = ( 3.6 1 in ( ( − − f f ) ) = ∆ p m ) N k ) and x ) 1 x ≤ k i x 0 ( k ∆( x, N − ( , and comparing either with several first colored polynomials, or several ∆( = k 2 R m ) as a series  = (1 ), is the expression that we wish to promote to colored HOMFLY-PT poly- j 2 u 1.1 x 3.7 1 − =0 X j p x  Therefore the series ( Note that the first term -deformation. Such deformations can be implemented by invoking various features pre- in the summand extraand overall extra powers overall of powers of we use the following notations for the Pochhammer and t sented in section first coefficients egy, we replace binomials by symbols, replace powers of certain expressions by If in addition to ainvolve a zig-zag number there of are binomial some coefficients. diamonds, we still geteralizing) similar ( expressions, which nomials, superpolynomials, or hopfully to coefficients. For example,diamonds, we if get expansion ( Moreover, from the subsequent multinomial expansion of a zig-zag. We can now use the inverse binomial theorem Identifying where for some particular into diamonds have the form JHEP12(2020)095 - ). = q (3.9) m 3.12 ) (3.13) (3.12) (3.10) (3.11) x − ) or anal- (1 3.7 = ), and (when
gets deformed 2 m − 2.8 ) m m 1 ) m + − x , q N m ; − m ) m ) x q ) . ( ; (1 q specialization of ( m 1 . m ; ) ) − q . 1 q m q ( ) ) ; ; − m aq t , a, q, t 1 1 ) ( X r 1 − ( − q # q − q = ; g ( ; r q q m aq r t aq ( ( f − " q − ( m 1 raised to a power that is at most ( ) = = # q = ≡ q ; m r t m ) i 1 m 2 " 1 ) − − − , such that binomials are replaced by are raised to a linear power is summation x r i q ) ). m aq ). To get HOMFLY-PT polynomials, it is ; x

t r ) − x ) − x ( q ( q, q ). ( 3.6 m (1 ; ( 3.11 # ) m f i – 12 – q and ) m r x ( a m m q ), and subsequent remarks, we predict that colored ) t a 2.10 ) " . To get superpolynomials, the above deformation ; − s , in the form q 1 =1 N i X ; 2 q, q − =0 r ) q q 3.11 (1 ]. Similarly, we replace all binomials in ( ( X x N ( indeed arises in various expressions for colored super- m − x ! q − = 12 = 2 ( – rp rp (1 m a ) # q a − q 10 r = ) and ( ; m ) = 1

t m " x m 1 and ) = X ). We also verify that resulting expressions are consistent with 3.6 − m + r ∆( ) q x aq 1.1 N -binomials. Furthermore, another factor of − = a, q, t

− is a deformation of q ( (  in ( r x r (1 m m P ) )  that often accompanies ( ! that arise universally in ( 2 m m ) x, N − ) q ( ; x , a, q, t m R 3 m r t − q r -binomials are defined as + ( q q aq (1 f N − m

( To sum up, in view of ( The above general strategy concerns in particular the terms ! 1) m − N ( equivalent to those found ascyclotomic in form. the To previous this section,written aim, as however the now a main they polynomial observation are in is written that in Alexander the polynomial can be differentials in HOMFLY-PT homologiesrelevant) and with satisfy the conditions exponential such growth as ( ( 3.2 The “cyclotomic”Let approach us present now another expansion that we consider. It leads to expressions that are quadratic in summation variables,variables. and Colored HOMFLY-PTQuadratic polynomials and linear arise powers mentioned as colored above superpolynomials, can be or HOMFLY-PT fixed polynomials, by orcoefficients comparing (in with principle) the with first the few first few superpolynomials have the structure where binomials, and the summand in addition involves The combination polynomials identified before [ ogous expressions by into where we identified is further modified by a single factor of natural to deform them as follows and the JHEP12(2020)095 ! k , and (3.17) (3.15) (3.16) (3.14) ) at each X ( x h can be written ) x , while the form on ∆( . . ) now yields k 2 by comparing with the ) , and most importantly X = 1 3.5 s x k 1) a c x p that enables the expansion − − is coupled to some other , 1 N =0 ) vanishes, so the remaining ∞ ( k X k (1 ). Moreover, we still have the k p ) ) q x for some polynomial by comparing with the next-to- = ) for x ; 3.13 , ) 3 1 3.10 k t m − ) − 3.1 r X ) s ( q = 1 + in ( a ; (1 X h aq . This enables us, after appropriate
3 ( i ) t − ). Examples of Alexander polynomials g r X ( x ! rk 2 aq − clearly can be written as a combination of 3.13 − ). Therefore the whole − ) − q ( . (1 ) = 1 + x # x x 1 3.2 r m k . Altogether these terms combine to ( m ∆( " – 13 – z −

kr + 1 ∆ − , then adjust k q ), in the second step we can deform it. Regarding ) N

) 1 , so the sum over x x ) −

, analogously as in ( 1 k x  − 2 ), we make analogous inverse binomial expansion as in 3.15 − ∆( r k =0 ) ∞  x k = = 1 X x m ) there is no overall term (1 . x in ! x k x 3.13 − = k c x 3.6 arises only through 1 . One can adjust first the coefficient (1 x 1 + 2 − ) = ∆( ) x x N − , which universally gets deformed to x ) 1 − k 1 x ) are provided in table ∆( (1 − x , and for x and rearranging summations, to write the above expression as a series ∆( X = , etc. At the end a constant term needs to be fixed. However, we also − , and altogether we obtain ( 1 m x ) (which is still not entirely obvious due to overall powers of 1 ) − (1 s X x ( 2 + , we find that in general one factor of 3.13 )). Nonetheless, in this “cyclotomic” approach, these structural features are g k − 2 ∆(1) = 1 ) ). 3.2 1 x k x in the denominator is simply x − k (1 ). Namely, one can complete a zig-zag to a combination of diamonds, with an extra 3.13 x with certain coefficients Having found the expansion ( Thus, starting now with ( The above statement can be also related to the structure of a zig-zag and diamonds = 3.3 X k while in X in denominator and gives risesecond to factor of Note that in comparisonthe to whole ( dependenceexpansion on of the previous section. Note thatis the different interpretation of — the as term obtained from mentioned filling above, in it a zig-zag. represents a The inverse corner binomial of theorem ( a certain virtual diamond, each zig-zag can be writtenalso in each the diamond form has thein structure the as form in ( ( diamond ( not that essential, and itform is ( simply crucial that we can write Alexander polynomial in the The middle expression abovethe is right can written be inremoved. interpreted the For as longer form zig-zags a such ( diamond an interpretation with works one analogously. corner It then (represented follows by that the first “1”) written in terms of in ( operation that removes one corner ofarises one for diamond. a The zig-zag simplest of example length of 3, such a for process which we can write powers of term at the highesthighest power of power know that constant term is Indeed, we know that JHEP12(2020)095 ) , that (3.19) (3.18) x, a, q x ( K F torus knots, quadratic in q ), and adjusting + 1) . Considering the x ) for HOMFLY-PT p ), while subsequent 3.10 2 . , . invariants, introduced m 3.6 x 2.13 (2 ) ) ]), one might hope to get due to negative values of ) contains explicitly both , 16 k x e c x, a, q k 3.18 x, a, q ( ) ( q q f ; K 3 (as in [ m F t ) r 1 rp follows from ( or − aq − q q x ) ; − ( m ). This is so for x rp , and these are precisely the factors that ) ( a r rk q x, q m q we replace Pochhammers and binomials by ; ( 3.4 − ) q ) are positive — for such knots, q K k q ( = # c ; F ! r k 1 3.4 . This is how a cyclotomic expansion of colored x k – 14 – " − k ) of superpolynomials, or ( c ! r aq =0 under the summation, so (after extending the range in ( k ( X k r 2.14 x in this way. It is an interesting problem whether there q =0 ∞ ) X ) = m = invariants for other knots too. x ) invariants ) = ) a, q, t x, a, q linear in summation variables and powers of ( ( r t K ) may introduce negative powers of x, a, q P ( F x, a, q x, a, q ( K 3.4 ( invariants. In this case the formula ( -binomials, and allow an additional deformation that may involve and F K is a deformation q K ) F ) F a ). For other knots, whose diagrams involve at least one diamond, the -independent. Note however, that this expression is a well-defined series ]. These invariants could be derived analogously as above, if only it would x 3.4 x, a, q 16 a, q, t ( – ( K k e c 14 F appears only in positive powers in ( = x k e c Also, note that the second, “cyclotomic” approach, does not seem to be relevant for , if x . This is indeed the case for other examples that we discuss in this paper, and thus for -Pochhammers and i 4 Reconstructing colored (super)polynomials In this section we illustrate ouras reconstruction well procedure as for for various infinite knots series up of to torus 8 knots crossings, and twist knots. We obtain expressions for colored identify some analytic continuationwould of lead the to above proper expression to positive powers of identifying positive and negative powers of of summation to infinity) we do not obtain a well defined series in second summation in ( s them we cannot identify exist knots for whichinvariants should all be powers given of by the above formula. On the other hand, one might hope to In particular, note thatdeformations the are whole dependencein on whose homological diagram consistssummation in of ( a zig-zag only, which is represented by the first first, “homological” approach, and taking advantagenormalization of by the removing first the line overall of prefactor ( 3.3 Relation to Let us alsorecently discuss in a [ possiblebe relation possible to to write the resulting series as a well-defined expansion in where polynomials arises from this second form of expansion of Alexander polynomial. appear in the cyclotomic expansionpolynomials. ( Furthermore, in coefficients q at most powers of summation variables. Ultimately we get which captures the whole dependence on JHEP12(2020)095 ) ) and 3.4 (4.1) (4.2) 3.13 1 3 in ( ) x ) or ( ( f 3.1 , and we have . m ) , so that one can 2 q ) 2 ; 9 x knot consists of one 1 x . − 1 − m 4 ) aq invariants in the case of (1 ( x  K r − m F  ), involve only a contribution (1 = 1 + m torus knot, which is thick. It is
x 3.13 ) knot. This example is particularly ! x 4 4) 2 , 7 in ( − consists of a single zig-zag of length 3, − ) (3 1 x (1 ( 3 m m x g ), its Alexander polynomial can be written + − . From the inverse binomial theorem we find 1 N x 3.14 1

are deformed into knot, colored by arbitrary symmetric represen- – 15 – knot, i.e. − ) — or 4 ) = x =0 ∞ m 7 X x 19 ) m ( = knot consists only of the shortest non-trivial zig-zag 8 x 1 3.12 f 1 − x 3 − N x and (1 1 +
= 2 − − ) 1 = 1 x . In particular, we start the presentation by discussing m m − ( p + 1 x , so that colored HOMFLY-PT polynomials are expected to take 1 N − 3.2 knot has the same Alexander polynomials as 1 rm N 3 1 4 q ) = 3 ∆ 7 x ) with = ( 1 3 m 3.3 x ∆ and , so as we already discussed in ( or in section ), the terms r r 1 ) reads q a 3.1 = 3.12 3.6 1 Let us reconstruct now colored HOMFLY-PT polynomials. Following our prescrip- First, we consider the “homological approach”. It is based on the middle expression Finally, to conclude, we analyze Note that among other examples, we also analyze We discuss different forms of expansion, following either the approach presented in − knots, not just because these are the simplest examples, but because they illustrate how N 1 tion ( x above, which is ( that ( In this simplest exampletomic”, we which discuss lead two tohave types different expressions form. of for The expansions, homological coloredsee “homological” diagram polynomials figure for and that “cyclo- areas of follows course equal, but reassuring to confirm explicitly that our reconstruction formalism works for thick knots too. 4.1 Trefoil knot, interesting, because clearly see differences that lead topolynomials their colored and superpolynomials. superpolynomials Moreover, HOMFLY-PT for tations, have not been determinedformalism before, and so contributes this new analysis explicit also results illustrates of the general power interest. of our that is involved in the inverse— binomial that expansion; in is consequence subsequently the deformedfrom function in one ( diamond.more Analysis diamonds, for essentially generalizes other these knots, two whose prototype cases. diagrams involve longer zig-zags or section 4 to deal with basic piecesIndeed, of Alexander a polynomial homological corresponding diagram to for (of a length zig-zag or 3) a and diamond. diamond no and diamond. a minimal On zig-zag the of other length hand, 1, a which diagram is for represented by “1” in ( torus knots. Ourwhenever they expressions are known. are Notethe consistent that Melvin-Morton-Rozansky this with asserts conjecture, earlier that which thesethus results results provides has are obtained a not also new in consistent been independent with literature, check verified of before, their and validity. which HOMFLY-PT polynomials and superpolynomials, as well as JHEP12(2020)095 . ]. γ 10 (4.4) (4.5) (4.6) (4.7) (4.3) , can and m ) q ; 1 α, β -degrees and − q . ) aq a , (

− , + 1 )(1 . 1 m 3 m ) − x ) q q . ; aq ; 2 1 t m that still need to be deter- − ) 1 ) − − q r − γ ; aq (1 1 ( 4 aq − q and compare with higher order − + 2) 2 γm ( and aq a a x ~ ( , and lead to the final result m 2 1) t βm 2 ) + +1) α, β + x = 1 -binomial expansion of r 1 − 2 ( 2 q ]. +1) − β x r m ( αm 14 q (( aq q m # q -dependence, leads to the following form of N polynomial is also known − r and t a m # rm 2 1 " q r ), after m – 16 – 1) # )(1 " ) is finite, it is sufficient to consider finite number = 2 =0 r = 0 q r m − 4.3 X r m " γ =0 r x knot. Each dot represents one homology generator or a r r X 3.12 m q a = =0 r 1 (1 + r r 1)( X 3 q m q a α 2 , and r r − a ) = q a 2 + N ) = ), one can expand it in ( a, q 2 ( ) = − − 1 4.3 with these first two colored polynomials suffices to fix q 3 r 2 a, q, t a, q P ( a ( ) = 1 ), if they would be known independently. In general, because the 1 = 2 3 r 3 r r P ) = P 1.1 x, N ( reduces to the result given in [ 1 a, q 3 1 ( in ( ) for 1 R 3 k 2 = 2 4.3 P R N follows from the table . Homological diagram for ) coefficients. For completeness we find that a, q k Furthermore, we note that the above analysis yields various information about the The above comparisons yield Alternatively, to fix ( ( 1 R 3 1 -degrees of generators. These results are in agreement with colored superpolynomials found by othercorresponding means quiver. in [ The expression ( Analogous computation, however including colored superpolynomials which for coefficients number of parameters to beof fixed in ( formula with known expressionsP for small colors. Uncolored HOMFLY-PT polynomial Comparing ( with additional potential deformationmined. specified These by parameters can be fixed in various ways. First, one can compare the above Figure 1 monomial in the superpolynomial.a Horizontal and vertical axes encode respectively form (note that the range of summation is in fact limited by JHEP12(2020)095 , , ). is to r − and α = 1 ) m 1.1 (4.9) (4.8) 3 j 3 (4.12) (4.10) (4.11) d , 1) 1 + − , we can . 2 − C ) we find d γ ( m m β 2 N = 4.8 ) , which thus ( m + = 2 = ) 2 x , x . and ]. Recall that ) 3 ! 2 1 3 . d m 3) in ( ) yields − d C 20 m ) . Note that the m − i + q q + j ) ). 2 d (1 ( α, β ; m, d q 2 d j 2 r ( 3.15 ; d m and ), which corresponds − q r α 4.6 , or by other means). j aq )( r 1) k ) ( 3 aq 4.1 a 3)+ ( R d m = 0 − = ) − − m 3 1 q 1 + ( ) , d . N ; d 1 q ( 2 ( 1 ; 3 d 2 C d − 1 m γm a − + quadratic in 1) )+ aq 1 3 ( aq − q d βm d ( ( + + 2 γm 2 2 / =0 a ) d = ( ∞ X m )( m αm 3 βm    + d + = 2 rm + + 2 m 2 rm m d q 2 +( αm + ) j 0 1 1 1 2 2 1 2 3 1 q ) x d m rm    ( q x − rm ; – 17 – q − ) we get q q = − 3 ( # q d j (1 # C ) in the Melvin-Morton-Rozansky expansion ( r m − 3 )+ 3.17 ! r 3 " d m m r 2 ]. Note that this quiver consists of 3 nodes, which are ) d ) 4.5 ) " m q + q q − 20 ; m 2 ; ; 1) d q -degrees of these generators. The fact that we immediately q q ( ( 1) t , so that finally m ( ( − γ m 2 ( − d m + + ( ) r − =0 r = 0 q a r X ; =0 r ) 3 N m γ q d X q m

( ; 1 1) ). In this case the inverse binomial expansion ( q m d ) = ( ) ]. This shows that even without fixing parameters − and ) = 1) q ( need to be determined by further corrections m r ; 2 19 3.13 − a, q ) ≤ q γ arises only from the coefficient as above, by comparison to the first and the second colored polynomial, ( / ( j q ( ∞ a, q 1 X ; ( 1 ≤ γ 3 =0 r 1 q in ( 0 d ∞ 3 X ( = 1 r P m and r r r P , as explained above, from quadratic terms in the power of q a X β = β = and 3 − , and taking into account ( ) = d = 0 x + m ) = α ( 2 ) X α α, β ) = 1 d , so that the second line takes form of a quiver generating series [ q 1 j ; − + a, q X 1 1 1 Let us now discuss the second, “cyclotomic” form of expansion. In this case we consider ( ( N 3 d 1 − = g 3 r ∆ = 3 aq P Fixing we find To determine colored HOMFLY-PT polynomial,( we deform the Pochhammer obtain this particular quiver is the feature of theAlexander particular polynomial form written of as in ( to the expression on the right in ( in agreement with the resultsin one-to-one in correspondence [ with generatorsements of of the HOMFLY-PT homology, above and matrix are diagonal el- parameters Fixing that the whole quiver matrix takes form determines the first row (and column)in of agreement the with quiver [ matrix as deduce at least athe partial only information parameter about thatalready the determines remaining corresponding from quadratic quiver. the terms,Thus first Furthermore, and this it correction first can ( correction be is determined sufficient to specify the whole quiver (even though additional where summations in the secondd line are rewritten inthe terms of underlying quiver isdependence determined on only by powers of be written as JHEP12(2020)095 . , ). X 2 4.9 (4.14) (4.15) (4.16) (4.13) ) = ). This X ( . g 3.19 , and taking m ) -independent arises already . m q x ) . Therefore its ; x m q 3 2 ; m t ) m 1 r ), with x 1) x − ! ), but of course both aq − − aq ( arises only in positive m − 3.13 ( . Removing the overall ] ( 4.7 x . (1 x ) as above. We simply m to 16 knot, shown in figure m m ) , ) q ), and also leave an infinite 4.2 2 m 1 q 1) ; r ) ; t 4 q 1) x 1 1 − , leads to colored superpolyno- − − x = − t − ) straightforward manipulations m N ) aq aq x ( N q ( (1 ( ; − =0 m ( 4.12 q ∞ ) X − 2 ( m 1 / ), is not suitable in this case and does ) ) only by a sign − = , as already advertised in ( m q ) = 1 ; + 4.10 m 2 x 4.10 x 2 ( m ) − x m m +( x, a, q q x ( − m – 18 – rm ), which is different than ( K + 3 x − (1 F 1 q . However, for ( − ! =0 # 2.14 ∞ ) 2 X ]), which has two extra generators compared to ( ). x m r m in this way is possible, because − . It is not hard to deduce its structure from Alexander " − 20 ) ) both yield m 2 4.10 ) = a, q, t m m ( (i.e. without substituting 1) ) = 1 + x 3 − x r 3.15 ( 1 ( x, a, q 1 P 4 N ( we obtain then the same result as in [ 4 . We again stress that the entire dependence on x, a, q 1 =0 r (

3 1 ∆ 1 X m m 3 invariant, because of the presence of both positive and negative − F ), while the subsequent deformation introduces only r =0 F ∞ N ) ) and ( − X m x ) 4.2 t 3.6 -deformed invariant = − a 1 x, a, q − ( 1 N 3 ), so that we obtain a well defined series in powers of ) = ( ) F in the summand of ( 1 x ( 4.2 x 1 4 a, q, t ( ∆ 1 3 r The formulae ( Finally, we stress that for trefoil knot, the “homological” reconstruction scheme enables P Note that it differsdeformation from is analogous; the we result promote for the Pochhammer trefoil ( polynomial, which we write accordingly which is automatically in the form relevant for cyclotomic expansion ( Figure-eight knot is a prototype example ofcorresponding how to to deal with diamonds. pieces ofconsists Alexander polynomial of The one homologicalthe diamond diagram “homological” for and and asuperpolynomial “cyclotomic” zig-zag is expansions given of essentially in minimal table overlap. length 1, The corresponding and for the latter reason Note that the second approach,not which lead leads to to ( powers of 4.2 Figure-eight knot, powers in ( normalization factor invariant arises from similarexpress the deformations result of in terms therange of of expression summation ( over from the expression ( corrections. Identifying these results yield the same lead to a quiver ofTherefore size we 5 obtain (see the alsobetween [ cyclotomic quiver form nodes at and the homology expense generators. of loosing ato nice determine correspondence the mial This expression is of cyclotomic form ( Analogous computation, however including dependence of JHEP12(2020)095 , so (4.18) (4.19) (4.20) (4.21) (4.17) = 1 γ . and m ) . q ) , ; = 0 3 3 . m . t ) aq r q m 1) , β ; ) 1 − aq r q − − ; − 2 r aq ( )(1 ( x )+ = ( m a aq 2 m )
( ) α q knot. − aq q m ; ; ) 1 t + 1 1 4 q 1 − ]. Note that, as expected, we get )(1 ; − x − 2 1 20 − 3) )(1 aq aq aq ( 1 ). For example, the first subleading 2 − aq − − ( − ( x rm x 1.1 2 m aq m − ( 2 q + )(1 − − 1 t in ( 1) rm γm − m ) − a + − )(1 1 + 2 q aq q -1 – 19 – m N rm − βm x, N − − q ( a q 2)( (1 k (1 + # 2 m 1 αm R − r − m − a − q " q a # 2 N 1 # . Homological diagram for r ( − m − =0 r r a m " a − X m " ). + =0 r X =0 r m ) = ) = X m 2.14 Figure 2 a, q ) = ) = 1 + x, N ( ) = ( 1 1 4 r 4 a, q 1 a, q ( ( P R r ) we get 1 encode potential further deformations yet to be specified. These param- 4 a, q, t 2 P ( γ P 1 4 3.17 r ) with the polynomials P , and the polynomial in the second symmetric representation and -dependence in the above computation, we analogously find the formula for t 2 4.17 α, β We can also relate the above analysis to the knots-quivers correspondence. Because in These expressions are in agreementa with cyclotomic the expression results ( in [ this case “cyclotomic” and “homological” expansions overlap, we now find a quiver whose Including colored superpolynomials In particular, from thethat first two colored polynomials we fix Alternatively, one can comparesion subleading of terms in ( thecorrection Melvin-Morton-Rozansky in expan- this expansion takes form by where eters can be fixedfrom by table comparison with the uncolored HOMFLY-PT polynomial following into account ( JHEP12(2020)095 1 β (in (4.23) (4.24) (4.25) (4.22) ) using = 1 r 4.17 × , knot consists = 2 m 1 ) k 1 p . q ) 5 1 ; q k q ; ) ( = 1 x gives rise to the full , which implies that m − ) m − brings this expression − ) 4 ) with γ j aq 2 d ) . ( x m q (1 2 3.1 ; 2 − αk k k q 2 and = + 2 : this first row arises from ( β j l , d γ 5 +2 + )
(1 + + knot. 1 d r 1 − ) q k ) m k m 2 ; 1 α, β 1 ) q )( x + 5 q x β l 5 ; and x ( ! l a q a and d ) reads 1 2 ) ( − l − 2 k k rk j + , which together with parameters ,k 3.6 − 2 − α, β = 1 (1 )(1 + k k k rj ! ) ... x 4 ( m 2 q α x ; + − 1)+ − q ! )+ and 1 − ( 2 m, d 2 1 j (1 d 1 k k 1 ( k x k − j 2 a k − − 3 2 r +2 j + + ) 1 − – 20 – m k q jl , and then ( m ) = ( ( = 1 ; ) q N r k + q 2 3 q

2 ( x . Fixing parameters γk 1 # − 1 x 1 N 1 2 m + k l, d j k k 2 +

( − ≤ l ]. − 2 r ∞ βk + (1 + k . Homological diagram for #" X ) = j = =0 j q 20 1 ∞ x ≤ x r X k )) 0 m ( − 1) 14 1 " m k 5 1) 1) − 1 C + ) = ( k − − l ∆ = x ≤ = N N 3( ( =0 j r 2 2 − k f Figure 3 2( 2( X X m 2 12 ≤ x x , so its Alexander polynomial takes form ( j 0 of the form m k, d 5 − =0 ,C + = = r l r X k q 2 − 2 2 l knot is analogous to the trefoil. However, in this case extra summations k 1 ( q a r ≤ 1 2 1 − = 0 ) j ∞ 5 q = is a quadratic polynomial in X N ≤ ) 15 0 1 ) = ) × 1 x 2.16 2 d C ( 1 ) takes form , k a, q 5 = ) = ( 1 1 knot ∆ k 3.4 5 13 ( r still needs to be fixed. By comparing with HOMFLY-PT polynomials for 1 a, q C α P ( 5 2 r β For colored HOMFLY-PT polynomials, our quantization rules now yield = P 11 where and so that ( The analysis for in expressions for colored polynomialsfocusing arise, so only we on also discuss the thisof “homological” example a in expansion. zig-zag detail, of albeit Homological length diagram for quadratic powers of C quiver then, in agreement with [ 4.3 Then setting to the form ofmatrix a is quiver determined generating even series. before As fixing before, parameters the first row (column) of the quiver size agrees with thethe number identity of ( HOMFLY-PT generators. We simply rewrite ( JHEP12(2020)095 ) , , 1 1 3.6 k k torus i ) (4.31) (4.29) (4.30) (4.26) (4.27) (4.28) β x i − P . + 1) a 1 ) (1 k -deformation p p p . ) q 2 . k q 1 , 1 . ; k k ,...,k t ) +2  ) (2 m 1 1 ) q q ... ) k ; − ( we then reproduce j ; 12 + 1 2 α 1 2 q − q aq 1) x k 1 − k ) − 1 + − ) p aq +2 aq ( − =0 k N q ( 9 1 ( ) p j . The whole dependence ; 1 k q 2 2 2( ) k ) q +2 k k x ) x P ( q 1 + ... 1 )+ ! k 1 x 7 1 + − k − 2 q p q 2 − k 2( k p ; k x, a, q t = ( ( + k x + 2 +2 1 ( 1 k 3 1 5

m + (1 + k 1 2 k q ) k ( F k 6 x r )+ 1 )( q − ··· ( 2 k q 2 1 k f )( k − 1 − ! , so that 2 k a + 1 2 +2 k ) 1 k k 1 aq q k + − k ; 1 ), and the inverse binomial theorem ( = 0 ( − k 1 )+ r 2 ! 2 q q − )(1 k 3.1 2 2 β # 1 k )(1 1 2 aq − +2 − q = ( k k – 21 – 1 +2 # 1 k 1 1 1 aq ( 1 k #" k β r k p 1 − x q r − p k k + # # k " 2 1 2 1 " 1 k k and N k k k " 1 + (1 + ), and similarly as in the case of trefoil, the analogous + (1

2 ≤ 1 #"  ... k 1 r 2 k 1 2 k 1 k X r # ≤ k ). Removing normalization factor − k ≤ 1 ≤ 3.19 r 2 r " q 0 k k X ... 2 1 − in the computation, we analogously reconstruct colored super- " r r ≤ ∞ ≤ k a X 2 2 0 2 1 p 4.24 t ≤ q k a k k r 2 ≤ k ≤ case, we can reconstruct colored polynomials for X ]) 0 ) = + ) = ≤ ... 1 0 1 ∞ ) = ≤ 1) 16 5 X p k r r a, q − k 2 2 ( = 2 q N a ≤ a, q 1 torus knots ( 0 r ( 5 x, a, q 2 p ) = . For brevity, we also focus only on the “homological” expansion. For a ( 1 2 P x 5 1 p r pr pr 5 ) = q a P , k = F + 1) 1 k 1 p ( 2 − ) = a, q, t , α N , Alexander polynomial takes form ( ( ) and for ) 1 1 p (2 -deformation of this expression yields the formula for colored HOMFLY-PT polyno- 5 a, q x r follows simply from ( 2 a ( Furthermore, following ( P r x ∆( P and mials where multiple binomials arise from the expansion of 4.4 Analogously to the knots for any given yields (in agreement with [ expansion), however it can be5 immediately nodes, rewritten which in are the in quiver one-to-one form, correspondence for with a HOMFLY-PTtransformations quiver generators. enable with us to reconstructon the invariant polynomials This expression is not cyclotomic (to find such form one should follow the “cyclotomic” Including dependence on we find table JHEP12(2020)095 - ) 1) = p x . . − 1 , so 1 2 p k l k N 1 − ) ( ) (4.34) (4.35) (4.32) (4.33) − p p q )) = q p 2 − ; p x ; k , . . . , k l t , and in 1 ) p 1 1 q q − k − − ; ( q p aq = 2 α ( aq ( k , . . . , d p p l 1 − l k ) ( × +1 1 p q i + ( l p 1 − k ; p − 2 k p p , k q i l k 1 2 ( ( β case. The whole , which is the only =1 − p i 1 − i k i r C ... 1 − d ... 2 p P 5 1) i P l l − − = 2 2 ) a 2 we can also reconstruct t k 1 ) q P − k p p k − 1 ; 2 ) p , +1 k q and i p 1 = +1 ( l k − = p i 1 ,...,k 1 1 + 2 k 1 r − , 3 k 3 rk 1 ... p + (2 2 l ( − − + =1 α T − p i ) , the denominators in the above , d 2 l p q p p 1 ... ,...,C ( k l l k ) P 1 = p − + k − k . 1 − q 1 = ...
= 3 ) − ) )) + p i K p + +2 rk that needs to be specified together with l 2 2 k 1 k 3 l ... − i ) 15 k +1 ) − k q i + , d p C 2 i = − 2 k 2 ; 1 l ( k q d k 3 +1)( ( = =1 r k =1 p i 1 +1 +2 p i + − p 1 p (2 l − 2 14 k p P q – 22 – ) r ( P 2 2 # q r 1 2 d , d 1 ; k q ,C p q = , and the whole expression takes form of a quiver p q +1)( − l i p ( r p ), and analogously to k p l 1)( d =0 1 + ( k k ) p (2 + X = 1 = l d " q 2 q − ...... l ; p # ( 3.19 1 ) in the form p 13 + q 2 l 1 ... r 1 ··· ( ) l p C − 2 ) q , d ). Such a comparison yields the final result # p k k ), and after appropriate deformation, and removing ; 1 +1 a 1 k 4.31 =0 + (2 − = r p q =1 1 k − X 1 " 2 i − ( l ]. 1.1 p k " ( 1 ]. l )) + 3 l 12 4.30 1 Q 1 ... 1 20 ··· k l − k 20 -dependence in the above computation leads to the form of colored − ) = 2 in ( ≤ t ≤ + ,C # p k p − 1 k ...... k 3 r − k k r 2 1 d ∞ ≤ ≤ R X X " k k is a quadratic polynomial in p p = 0 − ) k k 1 + 2 ) q k − ≤ 1 ≤ + 3 p 11 ; 0 0 ≤ 1 − 3 q ... C p k ... ( d pr r pr pr , either by comparison with the first few colored polynomials, or with the pr k ≤ i + X q a q a p × + β + ( . There is one such term k 2 invariants, following ( ) we can rewrite ( = , . . . , k ) 2 1 k ≤ 1 1 ) d l d 0 1 k ( ) = ) = ( − r r 2.16 p pr + 2 pr . Once these parameters are fixed, we can then read off the whole quiver matrix, α 2 i q a, q, t a 1 a, q a, q = = ( β ( ( k Finally, for this whole family of torus knots In this general case let us also identify the information about the corresponding quiver. x, a, q , d r r r ( ( = 1 P r P P − K p and which is consistent with [ F dependence follows from ( It follows that that we can fix the first row (and column) of the quiver even without fixing generating function. Theparticular form the of first the rowsource quiver of arises arises from from terms quadratic proportional powers to of Changing the variables asl follows expression are turned to Using ( matching the results in [ Furthermore, including superpolynomials parameters first few coefficients where JHEP12(2020)095 . 1 k ) . For q is the (4.39) (4.36) (4.37) (4.38) ; p 1 , i − 1 k k = 1 i ) aq β p , Alexander q ( ; 1 p . =1 k q p i ) 1 ( k 1 P 2 − ) a 1 q ) x k ; p x x − ( i ,...,k . The case k 1 (1 1 k ( − ! i . For a given α ,... 1 k q 3 p − 4 1 , p k =2 knot. 2 i p k rk , 1 −

P 6 q − 1 i = 1 k ... . k ) 2 p q generalizes to other values of ! ) =1 ; 2 3 i p x r k k x P 4.2 − aq , for q knot in figure ( 1 i ! k (1 1 1 1 2 diamonds displaced vertically and a zig-zag k 6 p k k ] ) =2 a p i p q + 2) 1 2 − ; 16 -1 p ! P 1 – 23 – 2 − (2 +2 − 1 aq ) = 1 k ( 1 1 x x k # k # 1 1 + p ∆( − . Homological diagram for p − p k p k k N k "

" ,... 1 1 k ··· 8 ··· ≤ # , # ... 1 1 Figure 4 r 1 2 k ∞ 6 ≤ X k k p , " " k 1 1 1 4 ≤ k k , and the analysis from section 0 ≤ 1 ≤ 4 ... = ... ∞ ≤ X ≤ 1 p X p k − k ≤ N ≤ 0 ) 0 1 x = , a homological diagram consists of ) = p ∆( +1 p a, q 2 Following the quantization and deformation prescription we find that colored HOMFLY- , ( 2 r T P F PT polynomials take form Because the zig-zag is oftomic” length expansion. 1, In this this form case is the relevant for inverse both binomial “homological” expansion and takes “cyclo- form of minimal length 1, aspolynomial seen takes in the form example of 4.5 Twist knots We consider now a familyfigure-eight of knot twist knots a fixed prefactor, we obtain the same result as in [ JHEP12(2020)095 i × − = β 1) × ) ] 1 i 2 p l )+ − n k − 1 3 p (4.42) (4.43) (4.40) (4.41) 11 − p + − and , 2 − ( , p 2 i ... 1 ) k p × 10 + m . p m C × n 2 p 3 ) l ) p k − p p − k + q = + 1 2 i k ; 1 + m n k − q k ... + ( ...... − + ... 2 i , . . . , k p = + − a 1 2 1 m + m 2 p 1 k ( k = l ) − k 4 ( 2 p + p q , d k + =1 n 1 α ; p i 14 n 1 m 1 k − − q k ) C p − ( rk P p l + + l a − 1 2 − − 2 1 = + q p ]. q k ··· r n 1 k ...... p 1 + ) k 12 n + 20 1 ) n = q + , which we label respectively . + 2 × ) + C q l ; ) are parameters. Fixing them rk 1 ( 1 p 1 q ; p 1 q 1 d − ; k 3 k i ( n n m k r t ,... ( q q p ( β + − ) q + r ( 2 1 3 1 r ) q m l ··· k 1 i ) 7 ... − l ; ) − aq , q + p m + q ), and we can also relate it to the q p − 3 2 l ; ) 1 l ( 2 − i ; k 5 q ) ) + l q n ( r ( ( , p + , and + q ; 1 + i i 1 2 ; q k 1 2.13 =1 k k aq k m 3 ( ) q p i i ( m ( ... q β 1 ) ; + P k +2( a t 1 ) )+ 1 =1 1 2 1 ··· F p i p k q − ) + q T 2 ... − 1 ; ( n 1 p q P n 1 l − p − p t + a aq – 24 – l − − k + p . ) ) 1 =0 ) yields ) − 3 l p p − ... p p p m . In this way we find X k aq − ( − n + k k ( m r 2 + 1 # l + + k # ,...,k 1 2.16 ... ) ... + =0 1 1 ... − p p r − l + p k 1 p 1 − + X p ( k 3 , and so on, leads to the form of the quiver generating ... k − p k 3 m k n α k aq 1 ) k k ( 2 q q + " n + − 2 + " 1 2 n ; ··· l ( 2 1 5 k 2 q l ( k rk = ( ··· k ( ,... p − m =0 ··· − 1 − k 2 2 ) − p # 5 =0 X ) + q l ) k 7 # 2 − p m 1 1 p + X 2 p r 1 k 1 , − n k ) , d − r k k 2 3 1 k + 1 p 1 m " ··· l k + 5 l ≤ " n ) n ... 1 2 , proportional to ... 1 ... k + q k 1 + k =0 + = ; + 1 ∞ q 2 ≤ + =0 2 ≤ is a quadratic polynomial in 3 − l 1 X 2 2 ≤ 1 p q k X 1 2 X l 4 ... ( k ( k ... ) k m ... ( n q ∞ p ≤ ... ≤ q X ∞ ≤ , d . Their homological diagrams consist of a zig-zag of length 3 and p 0 + X × × × p 1 k + × + k 2 × n ≤ 1 ≤ 1 0 ,... m n 0 ) = n − ( ( 3 , . . . , k ( , and all other entries are zero. Furthermore, after fixing 1 4 , ), we reconstruct the full quiver in agreement with [ 1 1 1 l l 1 l 2 ) = a, q k k ) = , − ( ( + + 1 = α 4.40 = ( = a, q 3 = 1 ( a, q, t Note that this result is of cyclotomic form ( +2) 1 p r ( p p 2 , d F , r P 1 (2 1 r T P as in ( 4.6 Twist knots The second family of twistby knots consists of knots Changing the summation variables inm the above formula as function. As before, we can nowquadratic identify powers the first of row of theC quiver matrix, which arises from where corresponding quiver. The identity ( P The generalization to the colored superpolynomials yields by comparing with several first colored polynomials yields the same result as in [ where JHEP12(2020)095 , ). i k i β i 4.37 (4.48) (4.44) (4.45) (4.46) (4.47) P a 1 ) k p . take form ) 1 q 2 k ; 9 2 ,...,k r × ) 1 1 1 k x k k , which leads to ( aq i ) ( ), yields . By comparison x α i q − β 1 q . ; k k 1 4 r ) k (1 q rk 3.17 + ; aq ! and − 3 1 knot and ( 1 k q 1 ) − p 1 − 2 k + p k p k 2 ) 5 ) aq k k q q ( a ; )

; ) 1 2 knot. r 4 k − k 2 , . . . , k ··· − aq − 7 1 2 2 aq ( 3 k ( k ! 1 k ( 2 3 k . # − ) 4 3 k k 2 α 2 2+( q ) k k k / ; ) x − 1 1 ! 2 4 #" x k − 1 2 k 3 2 − generators, as shown in the example in + k k + k k 2 1 aq 2 3 (1 1 k ( k #" p # ! + 2 1 +( − 1 2 a 2 2 r k k 3 2 1 4 k p 1 − p k p k − – 25 – 4 #" k − 1 2+( 1 1 " q r / k k ) k 2 1 " k ) = 1 + k + 1 ··· a x k + # # 2 1 1 2 N 1 2 1) k ∆( k k . Homological diagram for k k

− 1 +( ( r k #" #" 1 1 ]. We have already explicit formula for colored polynomials 1 1 k r k r 1) k k ≤ − 11 " 2 " − q 1 k 1 ( k 1 k ≤ × k Figure 5 3 ∞ 1) 1) k X ≤ ≤ − ... − 4 ( ( ∞ ≤ is a quadratic polynomial in summation variables k 1 1 X p ). As another example, explicit formulae for k k ≤ ) k 0 ≤ ≤ p ≤ r 2 0 ... k X 4.12 r ≤ ≤ X ) = p 0 = k ) ≤ , . . . , k x a, q 0 1 ) = ( ( k 1 2 ( 9 1 r . Trefoil, the first knot in this family, is a little special, and its diagram consists of − α ) = a, q knot in ( P N ( 5 1 2 As usual, appropriate deformation of this expression, including ( ∆ 5 3 r a, q ( P r P and with first colored polynomialsthe one can results fix consistent with the formfor [ of where Note that it differs only by aFrom sign the from inverse the result binomial for theorem the previous we family get of twist knots ( diamonds displaced vertically, andfigure have a zig-zag only. We writefor Alexander polynomial “cyclotomic” for expansion knots in this family in the form relevant JHEP12(2020)095 ], so (4.52) (4.51) (4.49) (4.50) 12 ), with , and for × 3.15 1 k ) = 1 = q ; × ) p } . 3 1 2 1 t k − k r ) x q . X aq ; r 1 r + k − − i c ( {z x k aq 1 ( k =0 1 ) =2 ∞ p i 1 k X q 1 + . k ) ; i q P t k − zig-zag of length 3 ; = 1 2 | 1 t − =2 i − p i m + ( k ) ), one can also find corresponding ) } aq P aq 2 =2 X ( p i − a − ) ( # i x 2.16 1 − # P k 1 p a − − − ) p p k − (1 2 i i ), we would obtain a quiver whose size is p k k k 1 k m ( k " − " 2 − i X =2 p i k 4.49 {z 1 ( ··· ! ··· − 2 P # =2 3 = x p i 1 2 # diamond 2 – 26 – 1 2 − 2+ k k / − P + . k k ) 4 1 1 m 1 #" m ) 2+ k 1 #" − − / r x 1 k + + ) x 2 x r 2 1 1 k " | x k k − , yields 1 " N , which results in a bit more involved analysis than in k 1 2 + + ( + 3 +( ) k 2 1

(1 1 ) } 6 x 1) k -dependent colored superpolynomials for this class of knots + x x 2 t − r rk − =0 x − +( ∞ ( (1 − 1 X 1) m 1 2 q + 3 − k ) rk − = 2 = ≤ 1 ( x − × − 1 q ... , we find x 1 k {z x X r − − 1 ≤ − ≤ X × p x − k ... N (1 diamond 1 knot. Interestingly, diamonds in its homological diagram are displaced r ) 1 2 ≤ ≤ + X and p > 0 2 x p x − k 6 x ) − | ≤ ∆( 0 X ) = = ( = 1 ) = ) = a, q x ( (1 + ( knot for r 2 knot 6 X 2 P 2 ∆ a, q, t 6 ( For this family of twist knots, using the identity ( r ) = + 1) P X p ( polynomial in the cyclotomicg form, so that the inverse binomial expansion ( In the middle linezag we and indicated two explicitly diamonds how in various monomials the are homology associated diagram. to In a the zig- third line we rewrite Alexander we will illustrate howrewrite Alexander those polynomial results in can the form be reproduced from our perspective. First, we 4.7 We consider now horizontally as shown inother figure examples. Colored superpolynomials for this knot have been determined in [ quivers, and deduce their partialeters. structure even However, before starting determining from all thelarger deformation expression than param- ( the number ofshould homology find generators. an To expression obtainwhich for a we quiver colored skip of polynomials for appropriate following brevity. size the we “homological” expansion, These results are of cyclotomic form. Furthermore, we find that the read (2 For arbitrary twist knot in this family, i.e. for trefoil corresponding to JHEP12(2020)095 , 2 (4.56) (4.54) (4.55) (4.53) , can be i β

= , 1 × 1 1 i k k k ) 2 rk ! , i ) q ) − 2 1 β 1 ; x 3 k q r rk 1 x − =1 + k 3 i q − 3 ) aqt N q k P (1 q ; a ( 2 r ) + ! k 2 3 2 2 k aq ,k )(1 + k ( − − 2 t − 2 1 1 1 ,k and take form k , knot. k 2 − 3 1 k − ) 1 k − k k 2 1 k q ( 1 − 6 X ) k ; k aq α 2 1 + q ) k k + 1) ; q ) + 2 1 r − ; q k − 3 1 ; rk N k aq 1 q − − − )(1 + ( + ( 3 k q q 2 k 1 N k # ! k ( 2 3 2 3 + − aq q 1 k k = k k ( ( 1 + k 2 ) 3 − k ! #" k t q l ) a N ! 1 2 ) ; 2 1 0 2 q 2 k k q ) as follows 1 + k ; − l – 27 – ; ) 1 − ! 2 − 1 q #" q 3 2 3 − ( − k 1 k ; 2 k k 3 r 2 t 4.52 k k k 1 1) aq k aq " ( − − − ( ! 3 + 2 ! # 2 q k 2 N k 2 3 2 k ( ) + ) . Homological diagram for k k N q 2 − q specialization of the uncolored superpolynomial in table q − ( + ( k ; l

; l 1 #" 2 q 1 3 1 2 1) 1 2 ( k k − − − − k − k k t 1 + − 1 2 k 1 ( + = r aq − k k 1 #" k − ) ( k t 1 r r Figure 6 N 1) + ) ≤ k × 2 "

) = − q, q k 3 3 ( N ( r k k q, q ≤ 2 X ( 3

arise from assembling fixed powers of k + + is a quadratic polynomial. Its form, as well as parameters a, q k c 3 2 2 ≤ ( 1 1 2 k k k ≤ ) 3 k k /  0 k 3 X + c =0 1 1) 1) 2 l X P ≤ k k , k 0 b − − 2 in this form) ) = 1) = = , k 6 − = ( = ( 1 1 ( a, q k k 1 ( c ( k r

α P X 1 We deform now the summand in ( k c fixed by comparison with which can also be rewrittenfrom as figure follows (it is instructive to identify a zig-zag and diamonds where which then yields where we used a non-trivial identity to get the expression in the second line. The coefficients JHEP12(2020)095 - ). q × 4.53 (4.58) (4.59) (4.57) 1 ) than k ) q ; 4.55 3 t r × 1 aq k )+ ) − 3 q ( ; 2 aq r k − − . aq )+ 1 ( r k 3 2 ) − k . 1 2 )(1 aq ) k − q 2 3 1 , ; − − k t 1 2 aq ) 1 aq k )+ 2 knot. rk 2 − + q − − )(1 3 − 3 2 ; 2 k 2 k 6 )+ aq 1 k t 3 1 1 − )(1 aq aq )(1 k − 3 a , we find the final result 2 aq rk k − i − ( − β 1)+ − aq − 2 2 aq )(1 k k − ( ) 1 2 2 )(1 − k k q k )(1 1 )(1 ], and proves that this expression is con- ( a/q ) ; and 1 2 2 − t q k − )(1 ) 1)+ 12 1 ; − 1 2 3 aq aq a 1 − − aq − 2 a , whose analysis is very similar to the previous − , k k 1 0 − − − 3 )(1 aq ( 2 -Pochhammers in the second line of ( − 6 -1 – 28 – 1 2 aq +1) q − k ( 3 − , k ( ) and motivate the identity that we used in ( 1 2 )(1 )(1 )(1 k # q 1 # ( )(1 1 1 a 2 3 − k 3 2 3 2 − − k k ( k − k k q − 1 2 4.55 α +1) aq aq #" q 3 #" 1 2 3 k 1 2 . Homological diagram for + 1 ) + )(1 k ( − − k k k k 3 1 q 2 + k 2 − #" − 1 2 #" k )(1 )(1 1 q 1 q r aq q q k r 1 1) k )( (1 + " k " q − − 1 − 1 ( − k r Figure 7 k (1 (1 + (1 + ≤ 1 ≤ × 1) 2 2 1 4 − 2 k − − − k r q − r ≤ q q q ( X ≤ 3 X 3 k k + + ( − + × ≤ ≤ 0 0 ) = 1 + (1 + ) = ) = a, q ( knot a, q 2 ( P 3 -colored polynomial r a, q, t 6 2 ( P r S P example. Its homological diagram consists of three diamonds and a zig-zag made of one 2 -dependent refinement. An analogous computation as above yields As expected, these final expressions have the cyclotomic4.8 form. For completeness we also discuss the6 knot which is in agreementsistent with with the the expression in Melvin-Morton-Rozanskyt [ theorem. It is also not difficult to include the in other exampleshomological is diagram. a Having consequence fixed of horizontal displacement of two diamonds in the In particular, the termsPochhammers in in the the second last line twoNote of lines that ( above, more complicated independently structure fix of the structure of and JHEP12(2020)095 - 1 ), . q − 1 k 2 4.54 ) = (4.65) (4.62) (4.63) (4.64) (4.60) (4.61) ), just 1 x k t x − 4.53 ) 3 = (1 ) } × ! , × aq x 1 i ) 2 k k 3 − i ) − − β q ; 3 aqt r =1 )(1 k 3 i . 2 + 2 {z 3 ) 2 aq in this expression) + 1 k P 3 ( k aq 1 2 a − 2 2 7 diamond 3 . ) )(1 + k aq k + x ) − 3 − t 2 3 − 1 1 ,k k − 1 − − ) 2 | by comparing with − k k ). Therefore we can conduct − aq 2 1 )(1 ) 1 ,k 3 . ) + ( 1 k q k a aq 6 1 3 )(1 − aq k ; 3 2 } k ( 2 ) 1 ) + − + α , which can also be written as (it , k 3 − q − 1 2 aq + )(1 3 ; 2 + 1) N 1 2 k aq r rk )(1 )(1 + , k ), also using the identity ( − x )(1 − rk 1 1 1 aq aq − 2 2 1 aq {z − − ! k − ( ( k − ( 2 3 q 2 1 2 )(1 − − 4.52 qt k k k k k aq # α 1 )(1 diamond 2 ) aq 2 3 2 − − 2 − q 2 + 3 1 x k k − )(1 ; k | k − ! 2 aq 1 aq a ) 1 2 2 #" − q − + ( 2 1 + )(1 t ; − − − )(1 } 2 1 − 2 k k 1 aq 1 k – 29 – + − 2 ), which is a consequence of replacing a zig-zag of ), whose more complicated structure is a conse- ( − 1 2 2 − q #" k 3 )(1 = 3 # )(1 )(1 k q + 1) q 1 k 3 2 1 1 a 1 r − 2 k 1 + + t k k k . − − − 4.51 4.62 x aq " 1 − 4 − − 4 ( 3 x ) − N #" aq aq − a 2 k + {z 1 2 q 2 k 3 x q

+ 2 ( + 1 ) k ) k k )(1 x − − 1 3 1 q x 1 q − + k 2 − 3 k diamond 1 ; − 1 − #" 2 − + 1 k a 1) 1 1 − )(1 )(1 q (1 − r − k k aq q q 1) − x )( example ( " (1 + ( | + 1) aq q − 1 − 1 2 ( + 5 ( k k 2 − − + ( 6 ) 1 ≤ ( ≤ (1 (1 + (1 + 5 × × 2 1 2 x − 2 5 4 k k − ) = 1 + k (1 + r x x − − − q ∞ ≤ ≤ 1 ≤ − X knot, the terms in the last two lines above fix the structure of and a quadratic polynomial 3 X q ( q q 3 2 3 − 2 2 2 2 k k k i 2 a (1 − ∞ − − − − ≤ ≤ ≤ β |{z} X 6 a, q, t 0 3 a a a a 0 2 ( . Its Alexander polynomial can be rewritten as follows − k − ) by a diamond and a zig-zag of length 1 (for  ≤ + + + − 2 7 trivial zig-zag x 0 P 6 ) = ) = = 1 + = 1 = -colored polynomial ) = 1 2 ) = a, q ) a, q ( x ( S x ( r r ( a, q 3 ( P 1 P 6 2 1 − 3 ∆ P N 6 ∆ comparison we get the final result Similarly as for Pochhammers in the secondquence line of of a ( horizontal displacement of diamonds in the homological diagram. From this and the As usual, we fix specialization of the uncolored superpolynomialis in again table instructive to identify a zig-zag and diamonds from figure which results in Furthermore, we take advantage of the same deformation of this expression as in ( In the middle line we identifydiffers elements only of in the signs homological from diagram.length Note 3 that (for the third line the inverse binomial expansion analogouslytaking as care in of ( the signs dot, see figure JHEP12(2020)095 × (4.67) (4.68) (4.69) (4.70) (4.66) . This 1 k ]. This ], which 8 ) q 34 12 ; . 3 × t 1 ) r k , 4 2 ) ) 5 aq 1 t x k ,...,k 3 − 1 x ( a − k , 2 ( these diamonds are ) k + α . 3 (1 − . 2 3 q t 3 1 1 k 1 2 2 9 t k k k / 1 2 2 a ) ) ) a 1 q + 1) q . k 2 ; + ; t k − 2 + + 3 1 ( ) 2 1 t − at k 1 r − x ! at k 3 3 4 x k 3 2 +( aq − k k r + 1 − aq 1 k (1 ( knot is derived e.g. in [ ! 1 − − rk )(1 + 2 )(1 + ( k 2 3 q 4 3 3 − ) 2 k k 7 # 2 k q 3 4 k ) ; 1 aqt aqt k k t q k ! 1 ; . As usual we fix them by comparing to 1 2 − t , which can also be written as follows = 1 + 4 i #" − 2 3 k k 1 γ 2 3 k 2 x − k k 1 2 aq )(1 + )(1 + ! t t + − aq #" 2 1 1 1 2 ( and 1 2 – 30 – i k for both knots. Ultimately we get the following − − − 7 + 4 k k k 2 1 ( i − i i # q γ β γ aq aq 1 − 2 3 1 #" 1 , k k 1 k k k 1 =1 ) r 4 i k − 4 − + " a #" and x P 1 1 1 2 t k . Homological diagrams for these two knots consist of a k i two of these diamonds overlap, as shown in figure N k k i 2 + (1 + + (1 + k β − 4

1) i 9 2 1 1 #" , β 7 1 k , . . . , k ) = 4 − 1 ) − − 2 k r 1 ( , which is of course the reason why Alexander polynomials of 4 t t x k =1 − ≤ 1 k 4 i " 3 2 k ( − − ), our deformation procedure leads to the following form of su- k k 1 ∆( ≤ α t P = 1 k ≤ 2 a 3 k ≤ ∞ a × k X 2 ) = ) = 3.17 ≤ , . . . , k k × ≤ 3 1 ∞ ∞ 4 k ≤ X knot. Its colored HOMFLY-PT polynomials or superpolynomials have k X k 3 ( ≤ 4 k ≤ 4 α 7 a, q, t a, q, t 0 k ≤ ( ( 0 ≤ 4 2 0 = -colored superpolynomial, which for 7 9   2 ) P P ) = S x ) = ( 1 knot 1 − a, q 4 a, q, t N ( knot is the same as for ( 7 r r ∆ 4 The inverse binomial expansion then yields P 7 P and to the comparison fixes Interestingly, the difference infrom colored deformation superpolynomials terms forthe uncolored these superpolynomials two given knots in arises table only Taking advantage of ( perpolynomials displaced uniformly, while for difference vanishes for these two knots are the same, and in the “cyclotomic” form they read Now we consider not been written explicitly before,to provide so such we new may results. take Thisfor advantage example of is our also instructive, reconstruction because scheme Alexanderzig-zag polynomial of length 3 and three vertically displaced diamonds. For These results are of cyclotomicin form, addition and proves are that consistent this with expression the has expression correct in4.9 Melvin-Morton-Rozansky [ limit. Furthermore, an analogous computation reveals the form of colored superpolynomials JHEP12(2020)095 ) . 4.72 × 1 4.6 × (4.71) (4.72) 4 k 4 k ) k 3 q k − ; 3 − 3 k 4 t k ) and ( − r 2 2 k k aq + − 3 4.71 2 4 − k k ( 2 1 + k k 2 3 ) − k 4 q + k . ; 2 2 t 1 − k 1 k 2 ) k − 2+ q a − / (right) knots. 4 3 2 1 ; 2 4 ) aq 3 1 k 2 t k 9 − + r . Note that ( ( 2 + 3 r 2 1 k , so all these terms are irrelevant aq k − + ], by checking conditions imposed t 2 2 N − ) +( k ( in the summand. In the Melvin- q 4 const r 1 k 36 1 k , 3 2+ k = + ) = / k 3 − ) q 2 (left) and 35 a k 1 q ; x − k 4 t + # t 7 1 2 + 3 4 3 k 2 = 1 k − k k k 2( − r t aq #" a q +( 4 2 3 4 r k − 1 k k k – 31 – ( + 3 k r 3 k − k #" and − − q t 1 2 + 4 and identify ) # 1 2 k k k 4 3 4 k 2 1 k knot found in [ k k k a #" + − 1 → 4 + 1 2 k r 3 #" 7 k k q k 2 3 + = , in agreement with results for twist knots in section " 2 2( r k k 2 k t t 1 for 9 4 k ), and the consistency with the exponential growth. 1) − k #" 3 ≤ 5 1 2 − k + 2 , ( knot we obtain the following colored superpolynomials 2 2.8 q k k k 4 k a ≤ 4 , 1 3 2 4 × a 3 #" 7 ∞ . Homological diagram for 1 k X 1 k r ≤ k = 3 + 4 " r k r 1 ≤ k 1) 0 ≤ − 2 ( k Figure 8 ≤ ) = ) × 3 ∞ k X ≤ 4 k a, q, t a, q, t ( ≤ ( 2 0 4 9 r 7 r In this example it is also worth illustrating the difference between the Melvin-Morton- = P P Rozansky limit and theconjecture. limit that Both yields limitsdiffer A-polynomial, involve which only is by relevantMorton-Roznasky the for limit the we terms set volume This result is cyclotomic,perpolynomials and for we verifiedby its canceling correctness differentials by ( comparison with colored su- On the other hand, for colored superpolynomials for JHEP12(2020)095 ) i x,z ( (4.74) (4.73) e W i z ∂ i , z knots have 3 e z 2 i 9 k q log 1 = that reduces the t = and i 0 z 4 d and 7 2 log ) i torus knot. This knot is − x,z 3 ( z 4) , e . W is determined by eliminating x )) (3 log ~ x∂ ( . In the homological diagram a e 1 O − )+ ) = 0 = log i y = − x,z x, y ( 4 t ( z e ) implies that superpotentials associated W A ( 1 ~ log e and 4.72 i 3 z a dz 4 3 5 – 32 – ) is not the same for all generators. It is therefore i log = 1 Y ) and ( − a 2.5 Z 4 knot. The pairs of generators shown in red, connected by knot, or in other words z ∼ 4.71 19 19 ) 8 8 log 2 , the action of this differential is represented by vertical a, q z ( 9 -grading ( r , which can be determined by approximating the summations δ , which is why we obtain the same Alexander polynomial for P 0 f W + log 3 → . z 1 q − log ) by integrals and introducing continuous variables = 2 z t = log 4.72 ~ log knots differ by and 4 knot − 7 . Homological diagram for = = 1 19 knot shown in figure ) and ( 8 a f and W 19 For thick knots, in consequence of the action of the differential 2 8 ∆ 4.71 9 from this set of equations, so this explains why A-polynomials for i for purple segments, and generators thathomological cancel diagram are consists shown in of red. athe zig-zag total Therefore, even made number though of of this 7 HOMFLY-PTAlexander generators generators polynomial is and has 11), one only after 5 diamond canceling monomial (so terms. 3 that pairs Nonetheless, of it generators, can be rewritten in the As the final example wehomologically consider thick, i.e. the instructive to demonstrate that our formalism works for suchHOMFLY-PT knots homology too. tosuperpolynomial the cancel knot upon Floer setting homology, some pairs of monomials in the z different forms. 4.10 to and these terms affectfor the these form two knots. of Nahm Furthermore, A-polynomial equations The difference in the summands in ( when we set these knots. On thepoints of other the hand, prepotential toin determine ( A-polynomial, we need to consider saddle Figure 9 vertical purple segments, correspond tosetting monomials in the superpolynomial that are cancelled upon JHEP12(2020)095 i = k . i 1 2 γ k k
(4.80) (4.77) (4.78) (4.79) (4.75) (4.76) − ) =1 1 4 i 4 k 4 x 2 t , P ) ) 3 , 4 + x + a 1 t j ) 2 6 k = 0 1 − = − i q x k 4 − 2 δ = ) ij k (1 + ) ) β ) 6 10 2 atq 4 4 = . 34 t t t k (1 + x =1 6  β 3 3 1 = + 15
q q δ ) 3 4 i,j ) − q + } 2 8 k = ) 2 ) t + 2 = )(1 + P x x + x 3 4 x q 12 14 2 +2( t 8 (1 + q 2 β δ )) − . t 4 − {z 4 4 k 8 q aqt t k t + 12 − = = 1 (1 (1 + (1 4 diamond 1 + q (1 + 4 1 2 + , − 1 3 k q 2 1 t x | 2 13 2 x − k q 6 2 k 6 + k 6 t x a x q − − 2 6 − ) a + 1 t ! +2( 1 q ) } 3
k 2 9 3 4 k , β t ) x 3 ) + (1 + 2 k ) t ) q 3 4 k k ) 1 + 4 3 t − x (1 + x x , q 1 4 1 + aq 2 = = 0 r , δ k (1 + k t q aq ! − + 4 − 4 2 3 2 1 2 2 3 (2 t ) 23 2 aq + x r − 6 k k qt ( x β − (1 q x q 2 q 6 1 2 t 6 + # k )(1 + a k 2 − 3 4 = = a ) ! ) )(1 + + 3 2 q − ) t k k 1 2 3 3 q 1 + (1 4 x 2 1 3 t t 2 ; (1 k k 12 t k t 2 2 ) 1 )(1 + 2 #" 3 2 – 33 – − k aq x {z 2 3 q x x 2 (1 + aq aq ! a/q k k aq -colored polynomial, whose form follows from the 2 x ! , γ , β ( − 2 − t 2 i 1 2 1 2 + 1 ! 1 #" k S k k i − 2 1 2 (1 2 − (1 + (1 δ = 1 )(1 + k k 3 q = 1 x zig-zag of length 7 − x )(1 + )(1 + 1 t − 3 k )(1 + ! 1 1 =1 x at 3 k 4 24 4 i still to be fixed. We fix them by comparing with the a #" 1 − 1 2 − 1 − − ) − 1 β + k − − r i 1 k 1 q 1 P k ) and − δ 7  3 − a " = + atq atq N 1 a 3 3 2 − atq 1 1 1 1 )(1 + ) k

x x × k | x k 44 1 1 , γ N atq 1 ≤ β − − and + k 2

= = = k ≤ i )(1 + )(1 + ) = = 0 2 ≤ N =0 γ q atq q atq x k r 3 ∞ ( 2 1 + (1 + X

k X ≤ 33 k , + (1 + 19 3 1 6 ≤ ∞ 8 ij k k 4 X 3 1) − k β ≤ ≤ q − ∆ − , β 4 2 ≤ -deformation of this expression leads to the following form of colored 6 q ∞ k k N 0 X 1 2 + (1 + + (1 + + (1 + + (1 + 3 a a ≤ ≤ 3( a r r 0 0 3 3 x = q a ) = 1) 1) = 22 ) = − − , γ ) that makes the whole homological structure manifest β ) N N - and ) = x q 3( 3( = 1 = ( 3.3 a, q, t x x ( 1 a, q, t 1 2 a, q 1 ( − 11 γ ( 19 = = P  β r -colored superpolynomial N 8 2 P P ∆ From this comparison we find (which is also givenS in table with parameters uncolored HOMFLY-PT polynomial Then, HOMFLY-PT polynomial Now, following our “homological” approach, the inverse binomial expansion yields form ( JHEP12(2020)095 ) × 2 + k . 2.8 . 2 44 − 2 k . 1 q × (4.85) (4.84) (4.81) (4.82) (4.83) k ) k − 2 − )) ) 1 − k 1 4 1 k . k k ) − − ) = 42 1 + 70 q k 3 q , q q ; ( , q k )) 3 1 r 4 × − t + + ) k − 2 r 2 +2( k aq − k 2 39 qt ( 3 69 − aq k 1 1 q + k 1 q ( 3 k − k − − ) k 1 2 × 1 ) ( − a − q 2 k 1 + q + k ; k 4 ; − 1 × (2 37 1 ) k 65 ( − k r q )) q 1 q 1 − q 2( +2 4 k ; k ) +( r k ) − ) 1 + 4 4 + q aq t q k q + k − ( + ( 1 3 3 ; , ; ) 1 + 35 64 k k 3 q 4 3 k 27 rk q q 43 t ( k ) k − q atq r − 1 q 2 4 q − ) − +2( k k − 3 ; 3 2 − ) 2 aq ( +2( k − − k q k k q ( ) 2 , 34 ( − 4 ; 1 2 − k − 26 + q # 2 60 41 k ( 1 23 4 q + . 3 4 k + q q k k # q − ) ( 1 +2 1 − k k 1 3 3 4 3 4 r k k (2 − k k t k k + + q r k k + q ( ]. We have also confirmed their validity 32 ( #" # − q − 4 1 2 25 3 )+ q 2 3 3 4 2 # k 59 40 # #" 22 2 q k k 3 13 + 3 4 k k k k 3 4 q q 2 3 ( q 1 k k 1 + k k 4 k k k k k k . For completeness, let us also write down + + − k #" #" − + 3 − q − 3 4 1 2 30 2 3 #" 2 k 4 4 #" #" k 23 )+ q 2 3 k k k k k − k 2 3 2 57 39 1 2 + 20 2 q 3 k k k 4 q q k k + k k k q – 34 – k − k 2 3 #" #" + = + ( − k − 4 1 1 2 #" + − 3 1 #" #" + r 4 k 1 2 29 k k k s k k 1 2 1 k 1 )+ 22 r q k k " 2 k k 37 55 + k k − 2 2 19 q ) k 4 #" q q 2 k " q − 2 2 k − 1 k #" + 3 #" + 1 ( k r − 4 1 k k + + 2 1 1 1 + k − r + k r 1 1 27 k k k " k ≤ 2 1 1 k ( 21 k q " 1 2 − 35 54 k " k 1 2 16 q − ( 2 k 1 k 1 q q 1 − q 1) 2 4 1 2 k + k k ≤ ≤ 3 k )+ + 2 q r 3 + k − ≤ − − 4 ≤ − + k 1 k 1 3 ( X 2 25 k 2 2 3 k k 1 k 19 k ≤ q ≤ k 33 52 + rk k 13 r 3 4 q − − ≤ 3 ≤ q q k q − k 2 4 4 specialization of the above formulas ≤ X r 3 k r 3 − )+ k 2 k -dependence, and analogously determine colored superpolynomials q k X k ≤ − ≤ 2 2 X t k + + 1 1 4 − 0 − + k ≤ ≤ 22 k k 1 2 3 ≤ +2( 4 4 r , 17 − + q k r 3 2 k k 31 49 − 6 k ≤ 8 12 2 1 q k k 7 2 3 X t 0 q q k ≤ q q ≤ k = + )+ 2 ( ≤ − 0 0 − r r + r 2 2 1 2 4 1 − t 2 − − 3 2 3 − + 3 k k k r 4 r k 17 q r q a q 3 3 k + 5 9 13 5 2 ≤ q (2 r 29 47 2 q 1 a 0 q q q r − q × 3 +3 q q k q q r r 2 ( + and a 3 6 k + + + 1 2 ) = + + 2 q q × × − q 3 6 9 12 ) = q 1 q q q q k ) = = × a = ) = a, q a, q, t ( q a ( ) = ) = ) = ) = × ( r r Several first of those colored Jones polynomials take form The above expressions match the results in [ q q q q r P a, q, t ( ( ( ( P J ( 3 1 2 4 r J J J J is satisfied, with Rasmussenexplicitly invariant colored Jones polynomialsfrom (in particular in the cyclotomic form), which follow from other perspectives discussed earlier; in particular we verified that the relation ( We can also provide the cyclotomic form of coloredP polynomials We can also introduce so that colored HOMFLY-PT polynomials for this knot take form JHEP12(2020)095 . , ] , , (1995) 501 Invent. (2016) 137 , 169 680 hep-th/9401061 knot. S.B. is sup- Proc. Symp. Pure [ , ]. 19 8 . ]. SPIRE IN (1996) 275 SPIRE ][ (1998) 1 IN Contemp. Math. ][ 175 , Comm. Math. Phys. 134 , Super-A-polynomials for Twist Knots analogs of Argyres-Douglas theories and . ]. ]. d 3 Volume Conjecture: Refined and Categorified The Superpolynomial for knot homologies SPIRE SPIRE Adv. Math. IN arXiv:1203.2182 IN , – 35 – [ ][ (1997) 291 ][ arXiv:1209.1416 [ 183 On the Melvin-Morton-Rozansky conjecture Commun. Math. Phys. , An analytic version of the melvin-morton-rozansky Lectures on knot homology ]. ), which permits any use, distribution and reproduction in (2012) 1669 (2013) 175 The coloured Jones function . 16 Homological Algebra of Knots and BPS States 01 ]. SPIRE arXiv:1209.1409 arXiv:1112.0030 IN . [ [ ][ manifolds. 1. CC-BY 4.0 SPIRE JHEP D , IN [ This article is distributed under the terms of the Creative Commons − A Contribution to the trivial connection to Jones polynomial and Witten’s Higher order terms in the Melvin-Morton expansion of the coloredThe Jones universal R-matrix, Burau representation, and the Melvin-Morton 3 Commun. Math. Phys. math/0503641 (1996) 103 , (2012) 157 (2012) 125 , ]. 11 125 85 SPIRE arXiv:1510.01795 IN knot homologies JHEP [ conjecture Math. Adv. Theor. Math. Phys. Math. math/0505662 polynomial expansion of the colored Jones polynomial invariant of [ S. Nawata, P. Ramadevi, Zodinmawia and X. Sun, S. Nawata and A. Oblomkov, H. Fuji, S. Gukov, M. Stosic and P. Sulkowski, S. Gukov and M. Stošić, H. Awata, S. Gukov, P. Sulkowski and H. Fuji, S. Garoufalidis and T.T.Q. Le, N.M. Dunfield, S. Gukov and J. Rasmussen, L. Rozansky, L. Rozansky, D. Bar-Natan and S. Garoufalidis, P. Melvin and H. Morton, L. Rozansky, [8] [9] [6] [7] [3] [4] [5] [1] [2] [11] [12] [10] Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References of Warsaw where thisScience project Centre was (NCN) initiated. grantthe 2016/23/D/ST2/03125. TEAM J.J. programme was of The supportedUnion the work under Foundation by of for the the Polish P.S. European Polish Science is Regional National co-financed Development supported Fund by (POIR.04.04.00-00-5C55/17-00). by the European We thank forStavros discussions Garoufalidis, and Angus acknowledgeSunghyuk Park, Gruen, insightful Ramadevi comments Sergei Pichai, VivekOsburn from Gukov, Singh for Tobias and Piotr drawing Marko Ekholm, our Kucharski, Stošić.ported attention by We Satoshi to Humboldt also the postdoctoral Nawata, thank cyclotomic Robert grant. expansion He of also acknowledges his stay at the University Acknowledgments JHEP12(2020)095 , , : from Phys. ]. JHEP Adv. ]. , , , N dualities Nucl. ]. , , , SPIRE = 2 SPIRE IN (2019) 124 IN SPIRE N (2011) 231 at large ][ (2015) 277 ][ ]. IN d [ 5 01 b Z arXiv:1904.06057 3 , 90 arXiv:1711.03333 , SPIRE JHEP IN ]. ]. , ][ Sequencing BPS Spectra SPIRE SPIRE IN IN [ arXiv:1909.13002 arXiv:1811.03110 ]. ][ arXiv:1401.7107 BPS states, knots and quivers Knots-quivers correspondence [ [ , ]. ]. ]. SPIRE . IN SPIRE Proc. Symp. Pure Math. SPIRE arXiv:1802.04573 , ][ IN SPIRE (2020) 044 ]. [ (2020) 361 IN Commun. Num. Theor. Phys. [ IN ][ Cyclotomic Expansion of Generalized Jones , 16 ][ Physics and geometry of knots-quivers Multi-cover skeins, quivers, and – 36 – 379 Donaldson-Thomas invariants, torus knots, and SPIRE arXiv:2005.13349 arXiv:1707.04017 Super-A-polynomial for knots and BPS states IN Quadruply-graded colored homology of knots , Duality, Mirror Symmetry, and a Q-deformed [ ][ Cohomological Hall algebra, exponential Hodge structures N SIGMA . , (2018) 026022 arXiv:1102.3978 ]. K A two-variable series for knot complements , ]. ]. ]. Topological strings, strips and quivers F ]. . 98 Tangle addition and the knots-quivers correspondence Rational links and DT invariants of quivers Large arXiv:1707.02991 Super-A-polynomial arXiv:1205.1515 [ arXiv:1204.4709 [ and SPIRE (2019) 1849 , SPIRE SPIRE SPIRE arXiv:1910.06193 ˆ IN Z SPIRE [ IN IN IN 23 [ ][ IN (2007) 1 [ ][ ][ Commun. Math. Phys. arXiv:1512.07883 An introduction to knot Floer homology , 171 [ arXiv:1908.04415 (2013) 506 Phys. Rev. D Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas , , A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres (2017) 121902 (2020) 018 ]. ]. Higher rank 867 96 02 (2016) 004 SPIRE SPIRE arXiv:1811.03556 IN arXiv:1006.2706 arXiv:1303.3709 IN correspondence JHEP [ [ arXiv:2004.10837 [ invariants for m-loop quivers lattice paths arXiv:1304.3481 and motivic Donaldson-Thomas invariants Phys. B [ A-polynomial for Knots Rev. D Theor. Math. Phys. curve counts to quantum modularity Invent. Math. Polynomials 03 [ T. Ekholm, P. Kucharski and P. Longhi, M. Panfil and P. Sułkowski, M. Stosic and P. Wedrich, M. Stosic and P. Wedrich, T. Ekholm, P. Kucharski and P. Longhi, M. Reineke, M. Panfil, M. Stošić and P. Sułkowski, C. Manolescu, E. Gorsky, S. Gukov and M. Stosic, M. Kontsevich and Y. Soibelman, H. Fuji and P. Sulkowski, M. Aganagic and C. Vafa, P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, P. Kucharski, M. Reineke, M. Stosic and P. Sułkowski, H. Fuji, S. Gukov and P. Sulkowski, K. Habiro, Y. Berest, J. Gallagher and P. Samuelson, S. Gukov and C. Manolescu, S. Park, T. Ekholm, A. Gruen, S. Gukov, P. Kucharski, S. Park and P. Sułkowski, S. Gukov, S. Nawata, I. Saberi, M. Stošić and P. Sułkowski, [32] [33] [29] [30] [31] [27] [28] [24] [25] [26] [22] [23] [19] [20] [21] [17] [18] [14] [15] [16] [13] JHEP12(2020)095 Nucl. ]. , SPIRE IN ][ arXiv:1302.5144 [ ]. SPIRE IN ][ Colored HOMFLY polynomials from (2013) 1350078 – 37 – quantum Racah coefficients \& non-torus links 22 ) N SU( arXiv:1107.3918 [ J. Knot Theor. , (2013) 205 870 Phys. B Chern-Simons theory S. Nawata, P. Ramadevi and Zodinmawia, P. Ramadevi and V. Singh, private communication (2020). Zodinmawia and P. Ramadevi, [35] [36] [34]