The Jones Polynomial and Khovanov Homology of Weaving Knots $W(3,n)$

TL;DR

This paper computes the signature, Jones polynomial, and Khovanov homology ranks of weaving knots W(3,n), providing recursive formulas and evidence that these ranks are asymptotically normally distributed.

Contribution

It introduces recursive methods to compute invariants of weaving knots W(3,n) and explores the asymptotic distribution of their Khovanov homology ranks.

Findings

Signature calculation for weaving knots W(k,n)

Recursive formulas for Jones polynomial of W(3,n)

Evidence of normal distribution in Khovanov homology ranks

Abstract

In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to knots $W(3,n)$, we develop recursion relations that enable us to compute the Jones polynomial of $W(3,n)$. Using additional results of Lee, we compute the ranks of the Khovanov Homology of these knots. At the end we provide evidence for our conjecture that, asymptotically, the ranks of Khovanov Homology of $W(3,n)$ are {\it normally distributed}.

Tables

Table 1. Table 1: Data for W ​ ( 3 , n ) 𝑊 3 𝑛 W(3,n) with n ≡ 1 mod 3 𝑛 modulo 1 3 n\equiv 1\mod 3

$n$	Total dimension	$dim{\mathscr{H}}^{0,1}$	$\sigma$	$L^2$-comparison	$L^1$ comparison
10	7563	970	2.64088	0.040509	0.134828
13	135721	15418	2.95616	0.0411329	0.150599
16	2435423	250828	3.24564	0.040792	0.155995
19	43701901	4146351	3.51395	0.040145	0.161336
22	784198803	69337015	3.76322	0.039413	0.165763
25	14071876561	1169613435	3.99810	0.038678	0.167576
28	252509579303	19864129051	4.22032	.037971	0.167790
31	4531100550901	339205938364	4.43167	0.0373026	0.170736
34	81307300336923	5818326037345	4.63358	0.0366758	0.172391
37	1459000305513721	100173472277125	4.82718	0.036089	0.173119
40	26180698198910063	1730135731194046	5.01342	0.035541	0.173178
43	469793567274867421	29963026081609060	5.19305	0.035027	0.173811
46	8430103512748703523	520131503664409798	5.36671	0.034547	0.175059
49	$1.51272\cdot {10}^{20}$	$9.04765\cdot {10}^{18}$	5.53502	0.0340935	0.175779
52	$2.71447\cdot {10}^{21}$	$1.57670\cdot {10}^{20}$	5.69838	0.033667	0.176100
55	$4.87091\cdot {10}^{22}$	$2.75210\cdot {10}^{21}$	5.85721	0.033265	0.176098
58	$8.74050\cdot {10}^{23}$	$4.81071\cdot {10}^{22}$	6.01187	0.032885	0.175898
61	$1.56842\cdot {10}^{25}$	$8.42017\cdot {10}^{23}$	6.16267	0.032524	0.176778
64	$2.81441\cdot {10}^{26}$	$1.47552\cdot {10}^{25}$	6.30989	0.032182	0.177369
67	$5.05026\cdot {10}^{27}$	$2.58843\cdot {10}^{26}$	6.45376	0.031857	0.177716
70	$9.06233\cdot {10}^{28}$	$4.54520\cdot {10}^{27}$	6.59451	0.031547	0.177859
73	$1.62617\cdot {10}^{30}$	$7.98842\cdot {10}^{28}$	6.73233	0.031251	0.177831
76	$2.91804\cdot {10}^{31}$	$1.40517\cdot {10}^{30}$	6.86740	0.030968	0.177657
79	$5.23621\cdot {10}^{32}$	$2.47359\cdot {10}^{31}$	6.99986	0.030697	0.177995
82	$9.39600\cdot {10}^{33}$	$4.35747\cdot {10}^{32}$	7.12988	0.030437	0.178445
85	$1.68604\cdot {10}^{35}$	$7.68116\cdot {10}^{33}$	7.25757	0.030188	0.178746
88	$3.02548\cdot {10}^{36}$	$1.35483\cdot {10}^{35}$	7.38305	0.029948	0.178918
91	$5.42901\cdot {10}^{37}$	$2.39106\cdot {10}^{36}$	7.50645	0.029718	0.178976
94	$9.74196\cdot {10}^{38}$	$4.22211\cdot {10}^{37}$	7.62786	0.029496	0.178935
97	$1.74812\cdot {10}^{40}$	$7.45910\cdot {10}^{38}$	7.74736	0.029282	0.178807
100	$3.13688\cdot {10}^{41}$	$1.31840\cdot {10}^{40}$	7.86506	0.029075	0.178890
121	$1.87923\cdot {10}^{50}$	$7.18477\cdot {10}^{48}$	8.64424	0.027805	0.179577
142	$1.12580\cdot {10}^{59}$	$3.97500\cdot {10}^{57}$	9.35886	0.026769	0.180247
163	$6.74436\cdot {10}^{67}$	$2.22337\cdot {10}^{66}$	10.0227	0.025900	0.180596
184	$4.04037\cdot {10}^{76}$	$1.25398\cdot {10}^{75}$	10.6453	0.025156	0.180629
205	$2.42049\cdot {10}^{85}$	$7.11854\cdot {10}^{83}$	11.2334	0.024508	0.180907
247	$8.68689\cdot {10}^{102}$	$2.32816\cdot {10}^{101}$	12.3258	0.023423	0.181027
289	$3.11764\cdot {10}^{120}$	$7.72623\cdot {10}^{118}$	13.3289	0.022542	0.181268

Table 2. Table 2: Data for W ​ ( 3 , n ) 𝑊 3 𝑛 W(3,n) with n ≡ 2 mod 3 𝑛 modulo 2 3 n\equiv 2\mod 3

$n$	Total dimension	$dim{\mathscr{H}}^{0,1}$	$\sigma$	$L^2$-comparison	$L^1$ comparison
11	19801	2431	2.74903	0.040906	0.141925
14	355323	38983	3.05533	0.041079	0.153170
17	6376021	637993	3.33710	0.040595	0.156595
20	114413063	10591254	3.59850	0.039905	0.163190
23	2053059121	177671734	3.84305	0.039166	0.166596
26	36840651123	3004390818	4.07348	0.038438	0.167789
29	661078661101	51124396786	4.29190	0.037744	0.168941
32	11862575248703	874400336044	4.49997	0.037089	0.171411
35	212865275815561	15018149469823	4.69899	0.036476	0.172723
38	3819712389431403	258853011125599	4.89004	0.035903	0.173203
41	68541957733949701	4474997964407374	5.07400	0.035366	0.173083
44	1229935526821663223	77563025486587315	5.25158	0.034864	0.174290
47	22070297525055988321	1347390412214087833	5.42341	0.034392	0.175346
50	$3.96035\cdot {10}^{20}$	$2.34525\cdot {10}^{19}$	5.59000	0.033949	0.175926
53	$7.10657\cdot {10}^{21}$	$4.08927\cdot {10}^{20}$	5.75181	0.033531	0.176131
56	$1.27522\cdot {10}^{23}$	$7.14133\cdot {10}^{21}$	5.90921	0.033136	0.176037
59	$2.28829\cdot {10}^{24}$	$1.24888\cdot {10}^{23}$	6.06255	0.032763	0.176227
62	$4.10617\cdot {10}^{25}$	$2.18679\cdot {10}^{24}$	6.21213	0.032408	0.177005
65	$7.36823\cdot {10}^{26}$	$3.83347\cdot {10}^{25}$	6.35821	0.032072	0.177510
68	$1.32218\cdot {10}^{28}$	$6.72713\cdot {10}^{26}$	6.50102	0.031752	0.177785
71	$2.37255\cdot {10}^{29}$	$1.18163\cdot {10}^{28}$	6.64077	0.031446	0.177867
74	$4.25736\cdot {10}^{30}$	$2.07736\cdot {10}^{29}$	6.77765	0.031155	0.177787
77	$7.63953\cdot {10}^{31}$	$3.65504\cdot {10}^{30}$	6.91183	0.030876	0.177602
80	$1.37086\cdot {10}^{33}$	$6.43571\cdot {10}^{31}$	7.04347	0.030609	0.178163
83	$2.45990\cdot {10}^{34}$	$1.13397\cdot {10}^{33}$	7.17269	0.030353	0.178561
86	$4.41412\cdot {10}^{35}$	$1.99933\cdot {10}^{34}$	7.29963	0.030107	0.178817
89	$7.92082\cdot {10}^{36}$	$3.52717\cdot {10}^{35}$	7.42441	0.029871	0.178949
92	$1.42133\cdot {10}^{38}$	$6.22605\cdot {10}^{36}$	7.54714	0.029643	0.178972
95	$2.55048\cdot {10}^{39}$	$1.09958\cdot {10}^{38}$	7.66790	0.029424	0.178901
98	$4.57665\cdot {10}^{40}$	$1.94290\cdot {10}^{39}$	7.78679	0.029212	0.178747
119	$2.74175\cdot {10}^{49}$	$1.05696\cdot {10}^{48}$	8.57308	0.027914	0.179650
140	$1.64251\cdot {10}^{58}$	$5.84051\cdot {10}^{56}$	9.29316	0.026859	0.180257
161	$9.83989\cdot {10}^{66}$	$3.26385\cdot {10}^{65}$	9.96138	0.025977	0.180552
182	$5.89483\cdot {10}^{75}$	$1.83951\cdot {10}^{74}$	10.5875	0.025223	0.180539
203	$3.53144\cdot {10}^{84}$	$1.04367\cdot {10}^{83}$	11.1787	0.024566	0.180926
245	$1.26740\cdot {10}^{102}$	$3.41053\cdot {10}^{100}$	12.2759	0.023469	0.181064
287	$4.54858\cdot {10}^{119}$	$1.13115\cdot {10}^{118}$	13.2829	0.022580	0.181221
329	$1.63244\cdot {10}^{137}$	$3.79224\cdot {10}^{135}$	14.2187	0.021838	0.181399

Equations

$$v_{{\rm oct}}(p-2)\,q\,\biggl{(}1-\frac{(2\pi)^{2}}{q^{2}}\biggr{)}^{3/2}\leq{\rm vol}(W(p,q))<\bigl{(}v_{{\rm oct}}(p-3)+4\,v_{{\rm tet}})q.$$

$$\underset{p,q\to \infty }{\lim }\frac{\mathrm{vol}(W(p,q))}{c(W(p,q))}=v_{\mathrm{oct}}\approx 3.66$$

$$j=2i-\sigma (L)\pm 1.\text{qed}$$

$$\sigma\bigl{(}W(2k{+}1,n)\bigr{)}=o(D)-y(D)-1=(1+kn)-kn-1=0.$$

$$\sigma\bigl{(}W(2k,n)\bigr{)}=o(D)-y(D)-1=\bigl{(}(k{-}1)n+2\bigr{)}-kn-1=-n{+}1.\qed$$

$$j=2i\pm 1.$$

$$j=2i+n-1\pm 1$$

$$T_i{T}_j$$

$$T_i{T}_{i+1}{T}_i$$

$$T_i^2$$

$$\rho\bigl{(}(T_{1}T_{2}^{-1})^{n}\bigr{)}=q^{-n}\bigl{(}C_{n,0}+C_{n,1}\cdot T_{1}+C_{n,2}\cdot T_{2}+C_{n,12}\cdot T_{1}T_{2}+C_{n,21}\cdot T_{2}T_{1}+C_{n,121}\cdot T_{1}T_{2}T_{1}\bigr{)},$$

$$\rho(\sigma_{1}\sigma_{2}^{-1})=T_{1}T_{2}^{-1}=q^{-1}\cdot\bigl{(}T_{1}(-(q{-}1)+T_{2})\bigr{)}=q^{-1}\bigl{(}-(q{-}1)\cdot T_{1}+T_{1}T_{2}\bigr{)},$$

$$C_{1,0}(q)=0,C_{1,1}(q)=-(q-1),C_{1,2}(q)=0,C_{1,12}(q)=1,C_{1,21}(q)=0,\text{and}{C}_{1,121}(q)=0.$$

$$C_{n,0}(q)$$

$$C_{n,1}(q)$$

$$C_{n,2}(q)$$

$$C_{n,12}(q)$$

$$\begin{array}{rl}{C}_{n,21}(q)& =-(q-1)\cdot C_{(n-1),2}(q)+q\cdot C_{n-1,12}(q)\\ & -(q-1{)}^2\cdot C_{n-1,21}(q)+q(q-1)\cdot C_{n-1,121}(q)\end{array}$$

$$C_{n,121}(q)$$

$$\rho(T_{1}T_{2}^{-1})^{n}=\rho(T_{1}T_{2}^{-1})^{n-1}\cdot\rho(T_{1}T_{2}^{-1})\\ =q^{-n}\bigl{(}C_{n-1,0}+C_{n-1,1}\cdot T_{1}+C_{n-1,2}\cdot T_{2}+C_{n-1,12}\cdot T_{1}T_{2}+C_{n-1,21}\cdot T_{2}T_{1}+C_{n-1,121}\cdot T_{1}T_{2}T_{1}\bigr{)}\\ \cdot\bigl{(}-(q{-}1)\cdot T_{1}+T_{1}T_{2}\bigr{)}\\ =q^{-n}\biggl{(}-(q{-}1)C_{n-1,0}\cdot T_{1}-(q{-}1)C_{n-1,1}\cdot T_{1}^{2}-(q{-}1)C_{n-1,2}\cdot T_{2}T_{1}\\ -(q{-}1)C_{n-1,12}\cdot T_{1}T_{2}T_{1}-(q{-}1)C_{n-1,21}\cdot T_{2}T_{1}^{2}-(q{-}1)C_{n-1,121}\cdot T_{1}T_{2}T_{1}^{2}\\ +C_{n-1,0}\cdot T_{1}T_{2}+C_{n-1,1}\cdot T_{1}^{2}T_{2}+C_{n-1,2}\cdot T_{2}T_{1}T_{2}\\ +C_{n-1,12}\cdot T_{1}T_{2}T_{1}T_{2}+C_{n-1,21}\cdot T_{2}T_{1}^{2}T_{2}+C_{n-1,121}\cdot T_{1}T_{2}T_{1}^{2}T_{2}\biggr{)}\\ =q^{-n}\biggl{(}\Bigl{(}-(q{-}1)C_{n-1,0}\cdot T_{1}-(q{-}1)C_{n-1,2}\cdot T_{2}T_{1}-(q{-}1)C_{n-1,12}\cdot T_{1}T_{2}T_{1}+C_{n-1,0}\cdot T_{1}T_{2}\Bigr{)}\\ +\Bigl{\{}-(q{-}1)C_{n-1,1}\cdot T_{1}^{2}-(q{-}1)C_{n-1,21}\cdot T_{2}T_{1}^{2}-(q{-}1)C_{n-1,121}\cdot T_{1}T_{2}T_{1}^{2}+C_{n-1,1}\cdot T_{1}^{2}T_{2}\\ +C_{n-1,2}\cdot T_{2}T_{1}T_{2}+C_{n-1,12}\cdot T_{1}T_{2}T_{1}T_{2}+C_{n-1,21}\cdot T_{2}T_{1}^{2}T_{2}+C_{n-1,121}\cdot T_{1}T_{2}T_{1}^{2}T_{2}\Bigr{\}}\biggr{)}$$

$$\begin{array}{rl}-(q-1)C_{n-1,1}\cdot T_1^2& =-(q-1)C_{n-1,1}\cdot ((q-1)T_1+q)\\ & =-(q-1{)}^2{C}_{n-1,1}\cdot T_1-q(q-1)C_{n-1,1}\end{array}$$

$$\begin{array}{rl}-(q-1)C_{n-1,21}\cdot T_2{T}_1^2& =-(q-1)C_{n-1,21}\cdot T_2((q-1)T_1+q)\\ & =-(q-1{)}^2{C}_{n-1,21}\cdot T_2{T}_1-q(q-1)C_{n-1,21}\cdot T_2\end{array}$$

$$\begin{array}{rl}-(q-1)C_{n-1,121}\cdot T_1{T}_2{T}_1^2& =-(q-1)C_{n-1,121}\cdot T_1{T}_2((q-1)T_1+q)\\ & =-(q-1{)}^2{C}_{n-1,121}\cdot T_1{T}_2{T}_1-q(q-1)C_{n-1,121}\cdot T_1{T}_2\end{array}$$

$$\begin{array}{rl}{C}_{n-1,1}\cdot T_1^2{T}_2& =C_{n-1,1}\cdot ((q-1)T_1+q)T_2\\ & =(q-1)C_{n-1,1}\cdot T_1{T}_2+q{C}_{n-1,1}\cdot T_2\end{array}$$

$$C_{n-1,2}\cdot T_2{T}_1{T}_2$$

$$\begin{array}{rl}{C}_{n-1,12}\cdot T_1{T}_2{T}_1{T}_2& =C_{n-1,12}\cdot T_1^2{T}_2{T}_1=C_{n-1,12}((q-1)T_1+q)T_2{T}_1\\ & =(q-1)C_{n-1,12}\cdot T_1{T}_2{T}_1+q{C}_{n-1,12}\cdot T_2{T}_1\end{array}$$

$$\begin{array}{rl}{C}_{n-1,21}\cdot T_2{T}_1^2{T}_2& =C_{n-1,21}\cdot T_2((q-1)T_1+q)T_2=(q-1)C_{n-1,21}\cdot T_2{T}_1{T}_2+q{C}_{n-1,21}\cdot T_2^2\\ & =(q-1)C_{n-1,21}\cdot T_1{T}_2{T}_1+q{C}_{n-1,21}\cdot ((q-1)T_2+q)\\ & =(q-1)C_{n-1,21}\cdot T_1{T}_2{T}_1+q(q-1)C_{n-1,21}\cdot T_2+q^2{C}_{n-1,21})\end{array}$$

$$\begin{array}{rl}{C}_{n-1,121}\cdot T_1{T}_2{T}_1^2{T}_2& =C_{n-1,121}\cdot T_1{T}_2((q-1)T_1+q)T_2\\ & =(q-1)C_{n-1,121}\cdot T_1{T}_2{T}_1{T}_2+q{C}_{n-1,121}\cdot T_1{T}_2^2\\ & =(q-1)C_{n-1,121}\cdot T_1^2{T}_2{T}_1+q{C}_{n-1,121}\cdot T_1((q-1)T_2+q)\\ & =(q-1)C_{n-1,121}\cdot ((q-1)T_1+q)T_2{T}_1+q{C}_{n-1,121}\cdot T_1((q-1)T_2+q)\\ & =(q-1{)}^2{C}_{n-1,121}\cdot T_1{T}_2{T}_1+q(q-1)C_{n-1,121}\cdot T_2{T}_1\\ & +q(q-1)C_{n-1,121}\cdot T_1{T}_2+q^2{C}_{n-1,121}\cdot T_1\end{array}$$

$$C_{n,0}$$

$$C_{n,1}$$

$$C_{n,2}$$

$$C_{n,12}$$

$$C_{n,21}$$

$$C_{n,121}$$

$$+(q-1)C_{n-1,12}+(q-1)C_{n-1,21}+(q-1{)}^2{C}_{n-1,121}$$

$$=C_{n-1,2}+(q-1)C_{n-1,21}$$

$$C_{2,0}(q)$$

$$C_{2,1}(q)$$

$$C_{2,2}(q)$$

$$C_{2,12}(q)$$

$$C_{2,21}(q)$$

$$C_{2,121}(q)$$

$$C_{n+1,121}(q)=C_{n,2}(q)+(q{-}1)\cdot C_{n,21}(q)\\ =C_{n,2}(q)\\ +(q{-}1)\Bigl{(}-(q{-}1)\cdot C_{n-1,2}(q)+q\cdot C_{n-1,12}(q)-(q{-}1)^{2}\cdot C_{n-1,21}(q)+q(q{-}1)\cdot C_{n-1,121}\Bigr{)}\\ =C_{n,2}(q)+(q{-}1)\bigl{(}-(q{-}1)\cdot C_{n-1,2}(q)+q\cdot C_{n-1,12}(q)-(q{-}1)^{2}\cdot C_{n-1,21}(q)\bigr{)}.$$

Full text

1 Introduction

Weaving knots originally attracted interest, because it was conjectured that their complements would have the largest hyperbolic volume for a fixed crossing number. Here is the weaving knot $W(3,4)$.

Enumerating strands $1,\ldots,p$ from the outside inward, our example is the closure of the braid $(\sigma_{1}\sigma_{2}^{-1})^{4}$ on three strands. Thus, $\sigma_{1}$ is a righthand twist involving strands 1 and 2, and $\sigma_{2}$ is a righthand twist involving strands 2 and 3, and so on.

In words, the weaving knot $W(p,q)$ is obtained from the torus knot $T(p,q)$ by making the standard diagram of the torus knot alternating. Symbolically, $T(p,q)$ is the closure of the braid $(\sigma_{1}\sigma_{2}\cdots\sigma_{p-1})^{q}$, and $W(p,q)$ is the closure of the braid $(\sigma_{1}\sigma_{2}^{-1}\cdots\sigma_{p-1}^{\pm 1})^{q}$. Obviously, the parity of $p$ is important. If the greatest common divisor $\gcd(p,q)>1$, then $T(p,q)$ and $W(p,q)$ are both links with $\gcd(p,q)$ components. In general we are interested only in the cases when $W(p,q)$ is an actual knot.

In [1] the main result is the following theorem.

Theorem 1.1** (Theorem 1.1, [1]).**

If $p\geq 3$ and $q\geq 7$, then

[TABLE]

Champanerkar, Kofman, and Purcell call these bounds asymptotically sharp because their ratio approaches 1, as $p$ and $q$ tend to infinity. Since the crossing number of $W(p,q)$ is known to be $(p{-}1)q$, the volume bounds in the theorem imply

[TABLE]

Their study raised the question of examining the asymptotic behavior of other invariants of weaving knots. In this paper we start a study of the asymptotic behavior of Khovanov homology of weaving knots.

Briefly, since weaving knots are alternating knots by definition, we may specialize certain properties of the Khovanov homology of alternating knots to get started. This is accomplished in section 2, where we explain how to calculate the signature of weaving knots. The second main ingredient in our analysis is the fact that for alternating knots knowing the Jones polynomial is equivalent to knowing the Khovanov homology. How this works explicitly in our examples is explained in section 5. In section 3 we prepare to follow the development of the Jones polynomial in [2], starting from representations of braid groups into Hecke algebras. For weaving knots $W(3,n)$, which are naturally represented as the closures of braids on three strands, we develop recursive formulas for their representations in the Hecke algebras. These formulas are used in computer calculations of the Jones polynomials we need. Section 4 builds on the recursion formulas to develop information about the Jones polynomials $V_{W(3,n)}(t)$. After we explain how to obtain the two-variable Poincaré polynomial for Khovanov homology in section 5, we present the results of calculations in a few relatively small examples. Our observation is that the distributions of dimensions in Khovanov homology resemble normal distributions. We explore this further in section 7, where we present tables displaying summaries of calculations for weaving knots $W(3,n)$ for selected values of $n$ satisfying $\gcd(3,n)=1$ and ranging up to $n=326$. The standard deviation $\sigma$ of the normal distribution we attach to the Khovanov homology of a weaving knot is a significant parameter. The geometric significance of this number is an open question.

2 Generalities on Weaving knots

We have already mentioned that weaving knots are alternating by definition. Various facts about alternating knots facilitate our calculations of the Khovanov homology of weaving knots $W(3,n)$. For example, we appeal first to the following theorem of Lee.

Theorem 2.1** (Theorem 1.2, [4]).**

For any alternating knot $L$ the Khovanov invariants ${\mathcal{H}}^{i,j}(L)$ are supported in two lines

[TABLE]

We will see that this result also has several practical implications. For example, to obtain a vanishing result for a particular alternating knot, it suffices to compute the signature. Indeed, it turns out that there is a combinatorial formula for the signature of oriented non-split alternating links. To state the formula requires the following terminology.

Definition 2.2**.**

For a link diagram $D$ let $c(D)$ be the number of crossings of $D$, let $x(D)$ be number of negative crossings, and let $y(D)$ be the number of positive crossings. For an oriented link diagram, let $o(D)$ be the number of components of $D(\emptyset)$, the diagram obtained by $A$-smoothing every crossing.

In words, $A$-regions in a neighborhood of a crossing are the regions swept out as the upper strand sweeps counter-clockwise toward the lower strand. An $A$-smoothing removes the crossing to connect these regions. With these definitions, we may cite the following proposition.

Proposition 2.3** (Proposition 3.11, [4]).**

For an oriented non-split alternating link $L$ and a reduced alternating diagram $D$ of $L$, $\sigma(L)=o(D)-y(D)-1$. ∎

We now use this result to compute the signatures of weaving knots. For a knot or link $W(m,n)$ drawn in the usual way, the number of crossings $c(D)=(m{-}1)n$. In particular, for $W(2k{+}1,n)$, c\bigl{(}W(2k{+}1,n)\bigr{)}=2kn; for $W(2k,n)$, c\bigl{(}W(2k,n)\bigr{)}=(2k{-}1)n. Evaluating the other quantities in definition 2.2, we calculate the signatures of weaving knots.

Proposition 2.4**.**

For a weaving knot $W(2k{+}1,n)$, \sigma\bigl{(}W(2k{+}1,n)\bigr{)}=0, and for $W(2k,n)$, \sigma\bigl{(}W(2k,n)\bigr{)}=-n{+}1.

Proof.

Consider first the example $W(3,n)$, illustrated by figures 3 and 4 for $W(3,4)$ drawn below. After $A$-smoothing the diagram, the outer ring of crossings produces a circle bounding the rest of the smoothed diagram. On the inner ring of crossings the $A$-smoothings produce $n$ circles in a cyclic arrangement. Therefore o\bigl{(}W(3,n)\bigr{)}=1+n. The outer ring of crossings consists of positive crossings and the inner ring of crossings consists of negative crossings, so $x(D)=y(D)=n$. Applying the formula of theorem 2.3, we obtain the result \sigma\bigl{(}W(3,n)\bigr{)}=0.

For the general case $W(2k{+}1,n)$, we have the following considerations. The crossings are organized into $2k$ rings. Reading from the outside toward the center, we have first a ring of positive crossings, then a ring of negative crossings, and so on, alternating positive and negative. Thus $y(D)=kn$. Considering the $A$-smoothing of the diagram of $W(2k{+}1,n)$, as in the special case, a bounding circle appears from the smoothing of the outer ring. A chain of $n$ disjoint smaller circles appears inside the second ring. No circles appear in the third ring, nor in any odd-numbered ring thereafter. On the other hand, chains of $n$ disjoint smaller circles appear in each even-numbered ring. Since there are $k$ even-numbered rings, we have $o(D)=1+kn$. Applying the formula of proposition 2.3

[TABLE]

These figures illustrate the main points of the $W(3,n)$-cases, and, as explained above, the main points of the $W(2k{+}1,n)$-cases.

For the case $W(2k,n)$, we show $W(4,5)$ below in figures 5 and 6 as an example. Our standard diagram may be organized into $2k{-}1$ rings of crossings. In each ring there are $n$ crossings, so the total number of crossings is $c(D)=(2k{-}1)n$. In our standard representation, there is an outer ring of $n$ positive crossings, next a ring of $n$ negative crossings, alternating until we end with an innermost ring of $n$ positive crossings. There are thus $k$ rings of $n$ positive crossings and $k{-}1$ rings of $n$ negative crossings. Therefore, $y(D)=kn$ and $x(D)=(k{-}1)n$. Considering the $A$-smoothing of the diagram, a bounding circle appears from the smoothing of the outer ring. As before, a chain of $n$ disjoint smaller circles appears inside the second ring and in each successive even-numbered ring. As previously noted, there are $k{-}1$ of these rings. No circles appear in odd-numbered rings, until we reach the last ring, where an inner bounding circle appears. Thus, $o(D)=1+(k{-}1)n+1=(k{-}1)n+2$. Consequently,

[TABLE]

Theorem 2.5**.**

For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology {\mathcal{H}}^{i,j}\bigl{(}W(2k{+}1,n)\bigr{)} lies on the lines

[TABLE]

For a weaving knot $W(2k,n)$ the non-vanishing Khovanov homology {\mathcal{H}}^{i,j}\bigl{(}W(2k,n)\bigr{)} lies on the lines

[TABLE]

Proof.

Substitute the calculations made in lemma 2.4 into the formula of theorem 2.1. ∎

3 Recursion in the Hecke algebra

We review briefly the definition of the Hecke algebra $H_{N+1}$ on generators $T_{1}$ through $T_{N}$, and we define the representation of the braid group $B_{3}$ on three strands in $H_{3}$. Theorem 3.2 sets up recursion relations for the coefficients in the expansion of the image in $H_{3}$ of the braid $(\sigma_{1}\sigma_{2}^{-1})^{n}$, whose closure is the weaving knot $W(3,n)$. The recursion relations are essential for automating the calculation of the Jones polynomial for the knots $W(3,n)$. Proposition 3.4 uses these relations developed in theorem 3.2 to prove a vanishing result for one of the coefficients. Being able to ignore one of the coefficients speeds up the computations slightly.

Definition 3.1**.**

Working over the ground field $K$ containing an element $q\neq 0$, the Hecke algebra $H_{N+1}$ is the associative algebra with $1$ on generators $T_{1}$, …, $T_{N}$ satisfying these relations.

[TABLE]

It is well-known [2] that $(N{+}1)!$ is the dimension of $H_{N+1}$ over $K$.

Recasting the relation $T_{i}^{2}=(q{-}1)T_{i}+q$ in the form q^{-1}\bigl{(}T_{i}-(q-1)\bigr{)}\cdot T_{i}=1 shows that $T_{i}$ is invertible in $H_{N+1}$ with T_{i}^{-1}=q^{-1}\bigl{(}T_{i}-(q-1)\bigr{)}. Consequently, the specification $\rho(\sigma_{i})=T_{i}$, combined with relations (3.1) and (3.2), defines a homomorphism $\rho\colon B_{N+1}\rightarrow H_{N+1}$ from $B_{N+1}$, the group of braids on $N{+}1$ strands, into the multiplicative monoid of $H_{N+1}$.

For work in $H_{3}$, choose the ordered basis $\{1,T_{1},T_{2},T_{1}T_{2},T_{2}T_{1},T_{1}T_{2}T_{1}\}$. The word in the Hecke algebra corresponding to the knot $W(3,n)$ is formally

[TABLE]

where the coefficients $C_{n,*}=C_{n,*}(q)$ of the monomials in $T_{1}$ and $T_{2}$ are polynomials in $q$. For $n=1$,

[TABLE]

so we have initial values

[TABLE]

Theorem 3.2**.**

These polynomials satisfy the following recursion relations.

[TABLE]

Proof.

We have

[TABLE]

after collecting powers of $q$ and expanding. In the last grouping, the first four terms inside the parentheses $(\ )$ involve only elements of the preferred basis; the second eight terms in the pair of braces $\{\ \}$ all require further expansion, as follows.

[TABLE]

Collecting the constant terms from (3.13) and (3.19), we get

[TABLE]

Up to simple rearrangements and expansion of notation, these are formulas (3.6) through  (3.11). ∎

Example 3.3**.**

Applying the recursion formulas just proved to the table of initial polynomials, or by computing \rho\bigl{(}(\sigma_{1}\sigma_{2}^{-1})^{2}\bigr{)} directly from the definitions, we find

[TABLE]

As a first application, we have the following vanishing result.

Proposition 3.4**.**

For all $n$, $C_{n,121}(q)=0$.

Proof.

For $n\geq 1$, we claim $C_{n+1,121}(q)=0$. Make the inductive assumption that $C_{k,121}(q)=0$ for $1\leq k\leq n$. Apply (3.11), (3.10), and the inductive hypothesis to write

[TABLE]

Using (3.8) to replace the first term $C_{n,2}(q)$ and (3.9) to replace the third term factor $C_{n-1,12}(q)$ on the right,

[TABLE]

and using (3.8) in reverse to rewrite the term $qC_{n-2,1}(q)$. Making the obvious cancellation,

[TABLE]

since

[TABLE]

by (3.7) and the inductive hypothesis. Therefore,

[TABLE]

using (3.11) and the inductive hypothesis. ∎

4 Obtaining the Jones Polynomial

Following the construction given in [2, p.288] we work over the function field $K={\mathbf{C}}(q,z)$, and we put $w=1{-}q{+}z$. Let $H_{N+1}$ be the Hecke algebra over $K$ corresponding to $q$ with $N$ generators as in definition 3.1. The starting point is the following theorem.

Theorem 4.1**.**

For $N\geq 1$ there is a family of trace functions $\operatorname{Tr}\colon H_{N+1}\rightarrow K$ compatible with the inclusions $H_{N}\rightarrow H_{N+1}$ satisfying

$\operatorname{Tr}(1)=1$, 2. 2.

$\operatorname{Tr}$* is $K$-linear and $\operatorname{Tr}(ab)=\operatorname{Tr}(ba)$,* 3. 3.

If $a,b\in H_{N}$, then $\operatorname{Tr}(aT_{N}b)=z\operatorname{Tr}(ab)$.

Property 3 enables the calculation of $\operatorname{Tr}$ on basis elements of $H_{N+1}$ through use of the defining relations and induction. For $H_{3}$, note that

[TABLE]

The next step toward the Jones polynomial of the knot that is the closure of the braid $\alpha\in B_{N+1}$ is given by the formula

[TABLE]

where $e(\alpha)$ is the exponent sum of the word $\alpha$. The expression defines an element in the quadratic extension $K(\sqrt{q/zw})$. For the weaving knot $W(3,n)$, viewed as the closure of $(\sigma_{1}\sigma_{2}^{-1})^{n}$, we have the exponent sum $e=0$, and $N=2$, and

[TABLE]

thanks to proposition 3.4, which says the expression for $(T_{1}T_{2}^{-1})^{n}$ requires only the use of the basis elements $1$, $T_{1}$, $T_{2}$, $T_{1}T_{2}$ and $T_{2}T_{1}$. Then we have

[TABLE]

using the facts that $\operatorname{Tr}T_{1}=\operatorname{Tr}T_{2}=z$ and $\operatorname{Tr}T_{1}T_{2}=\operatorname{Tr}T_{2}T_{1}=z^{2}$. Consequently, the sums $C_{n,1}+C_{n,2}$ and $C_{n,12}+C_{n,21}$ are essential for understanding the two-variable Jones polynomial of $W(3,n)$, the closure of $\alpha=(\sigma_{1}\sigma_{2}^{-1})^{n}$. Making the substitutions

[TABLE]

leads to the one-variable Jones polynomial

[TABLE]

Example 4.2**.**

For $W(3,1)$, which is the unknot, we have

[TABLE]

Example 4.3**.**

For $W(3,2)$, which is the figure-8 knot, we have

[TABLE]

Now we take a closer look at the formal expression

[TABLE]

for the Jones polynomial of the weaving knot $W(3,n)$.

Proposition 4.4**.**

We have a uniform bound on the degrees of the polynomials $C_{n,*}$. Namely,

[TABLE]

and the sharper bounds

[TABLE]

Proof.

These are easy arguments by induction, using either the formulas (3.5) or the formulas in example 3.3 to start the inductions. Use the recursion formulas (3.6) through (3.10) along with the fact that $C_{n,121}=0$, proved in proposition 3.4, for the inductive step. We have

[TABLE]

∎

Accordingly, set

[TABLE]

Lemma 4.5**.**

In the polynomial $C_{n,0}(q)$, the constant term $c_{n,0,0}=0$ for all $n\geq 1$, and the degree one coefficient $c_{n,0,1}=(-1)^{n-2}$ for $n\geq 2$.

Proof.

The first polynomial $C_{1,0}(q)=0$, and setting $q=0$ in the recurrence relation (3.6) immediately yields

[TABLE]

Differentiate the recursion relation (3.6) with respect to $q$, obtaining

[TABLE]

Substituting $q=0$ yields immediately $c_{n,0,1}=C_{n,0}^{\prime}(0)=C_{n-1,1}(0)=c_{n-1,1,0}$. We have

[TABLE]

simplifying relation (3.7) using proposition 3.4 to set $C_{n-1,121}(q)=0$. Now we prove $c_{n,1,0}=(-1)^{n-1}$ for all $n\geq 1$. We have $C_{1,1}(q)=-(q-1)$, so $c_{1,1,0}=1$ as claimed. Substituting $q=0$ and using $c_{n,0,0}=0$ we get

[TABLE]

Thus $c_{n,0,1}=c_{n-1,1,0}=(-1)^{n-2}$, as claimed. ∎

Thus, we improve the expression for $C_{n,0}(q)$ slightly, obtaining $C_{n,0}(q)=\sum_{i=1}^{2n-1}c_{n,0,i}q^{i}$. Now we examine

[TABLE]

In the expression for $V_{W(3,n)}(t)$ the highest degree term is apparently

[TABLE]

but we claim this term is actually zero.

Proof.

Using $\deg(C_{n-1,1})\leq 2n-3$, $\deg(C_{n-1,0})\leq 2n-3$, the fact that $C_{n,121}=0$, and simplifying the recursion formulas (3.7) and (3.9), respectively, to

[TABLE]

we compute

[TABLE]

So the degree of the highest term in $V_{W(3,n)}$ is no more than the degree of $t^{-n-1}\cdot t^{2n+1}$ which is $n$. Lemma 4.5 shows that the term of lowest degree is $\pm t^{-n-1}\cdot t=\pm t^{-n}$, so the span of the Jones polynomial $V_{W(3,n)}$ is no more than $2n$. Of course, Kauffman [3, Theorem 2.10] has proved that the span of the polynomial is, in fact, precisely $2n$.

5 From the Jones Polynomial to Khovanov homology

In this section we amplify Theorem 2.5, at least the first part of it.

Theorem 2.5**.**

For a weaving knot $W(2k{+}1,n)$ the non-vanishing Khovanov homology {\mathcal{H}}^{i,j}\bigl{(}W(2k{+}1,n)\bigr{)} lies on the lines

[TABLE]

For a weaving knot $W(2k,n)$ the non-vanishing Khovanov homology {\mathcal{H}}^{i,j}\bigl{(}W(2k,n)\bigr{)} lies on the lines

[TABLE]

We have the following definition of the bi-graded Euler characteristic associated to Khovanov homology.

[TABLE]

Theorem 5.1** (Theorem 1.1, [4]).**

For an oriented link $L$, the graded Euler characteristic

[TABLE]

of the Khovanov invariant ${\mathcal{H}}(L)$ is equal to $(Q^{-1}{+}Q)$ times the Jones polynomial $V_{L}(Q^{2})$ of $L$.

In terms of the associated polynomial $Kh(L)$,

[TABLE]

Theorem 5.2** (Compare Theorem 1.4 and subsequent remarks, [4]).**

For an alternating knot $L$, its Khovanov invariants ${\mathcal{H}}^{i,j}(L)$ of degree difference $(1,4)$ are paired except in the [math]th cohomology group.

This fact may be expressed in terms of the polynomial $Kh(L)$, as follows. There is another polynomial $Kh^{\prime}(L)$ in one variable and an equality

[TABLE]

When we combine theorems 5.1 and 5.2, we find that the bi-graded Euler characteristic and the Jones polynomial of an alternating link determine one another. Obviously, the equality (5.1) shows that one knows $V_{L}$ if one knows $Kh(t,Q)$.

To obtain $Kh(t,Q)$ from $V_{L}(Q^{2})$ requires a certain amount of manipulation. Implementing these manipulations in Maple and Mathematica is an important step in our experiments. Setting $t{=}-1$ in (5.2) and combining with equation (5.1), one has

[TABLE]

Replacing $Q^{2}$ in the last equation by $-tQ^{2}$ is the last step to obtain $Kh^{\prime}(L)$ from the Jones polynomial. Within a computer algebra system, one must first replace $Q^{2}$ by $-X$ and then replace $X$ by $tQ^{2}$. Once one has $Kh^{\prime}(L)(tQ^{2})$, one obtains $Kh(t,Q)$ directly from equation (5.2).

Example 5.3**.**

We have computed $V_{W(3,2)}(t)=t^{-2}-t^{-1}+1-t+t^{2}$ in example 4.3, so

[TABLE]

It follows that Kh^{\prime}\bigl{(}W(3,2)\bigr{)}(tQ^{2})=t^{-2}Q^{-4}+tQ^{2}, and

[TABLE]

6 Khovanov homology examples

Once one has the Khovanov polynomial one can make a plot of the Khovanov homology in an $(i,j)$-plane as in this example. The Betti number \dim KH^{i,j}\bigl{(}W(3,11)\bigr{)} is plotted at the point with coordinates $(i,j)$.

Clearly, as $n$ gets larger, it is going to be harder to make sense of such plots. Notice that the $(1,4)$-periodicity of the Khovanov homology for these knots makes the information on one of the lines $j-2i=\pm 1$ redundant.

Taking advantage of this feature, we simplify by recording the Betti numbers along the line $j-2i=1$. In order to study the asymptotic behavior of Khovanov homology we have to normalize the data. This is done by computing the total rank of the Khovanov homology along the line and dividing each Betti number by the total rank. We obtain normalized Betti numbers that sum to one.

This raises the possibility of approximating the distribution of normalized Betti numbers by a probability distribution. For our baseline experiments we choose to use the normal $N(\mu,\sigma^{2})$ probability density function

[TABLE]

Fit a quadratic function $q_{n}(x)=-(\alpha\,x^{2}-\beta\,x+\delta)$ to the logarithms of the normalized Khovanov dimensions along the line $j=2i+1$ and exponentiate the quadratic function. Since the total of the normalized dimensions is 1, we normalize the exponential, obtaining

[TABLE]

To obtain a formula for $A_{n}$, complete the square

[TABLE]

Then consider

[TABLE]

Thus, the expression for $A_{n}$ is

[TABLE]

Equating the expressions

[TABLE]

$\mu_{n}=\beta/2\alpha$ and the efficient way to the parameter $\sigma_{n}$ is by solving the equation

[TABLE]

obtaining $\sigma_{n}=1/\sqrt{2\alpha}$.

Working this out for $W(3,10)$, and carrying only 3 decimal places, the raw dimensions are

[TABLE]

and, to three significant digits, the logarithms of the normalized dimensions are

[TABLE]

Fitting a quadratic to this information, we get

[TABLE]

To three significant digits $\mu_{10}=0.500$ and $\sigma_{10}=2.64$.

By the symmetry of Khovanov homology, the mean $\mu_{n}$ approaches $1/2$ rapidly, so this parameter is of little interest. On the other hand, relating the parameter $\sigma_{n}$ to some geometric quantity, say, some hyperbolic invariant of the complement of the link, is a very interesting problem.

For $W(3,10)$, the density function is

[TABLE]

When placed into standard form, $\mu_{10}=0.5000054030$ and $\sigma_{10}=2.640882970$. Here is the comparison plot.

For the knot $W(3,11)$ the expression for the density function is

[TABLE]

When placed into standard form, $\mu_{11}=0.5000017550$ and $\sigma_{11}=2.749031276$. Here is the comparison plot.

For $W(3,22)$, the density function is

[TABLE]

When placed into standard form, $\mu_{22}=0.500000000$ and $\sigma_{22}=3.763224354$. Here is the comparison plot.

For $W(3,23)$, the density function is

[TABLE]

When placed into standard form, $\mu_{23}=0.5000000000$ and $\sigma_{23}=3.843052143$. Here is the comparison plot.

Maple worksheets and, later, Mathematica notebooks will be available at URL prepared by the second-named author.

7 Data Tables

This section contains tables of data generated using Maple to implement some of the results of earlier sections. The first table collects data for weaving knots $W(3,n)$ for $n\equiv 1\mod 3$. The first column lists the dimension; the second column lists the total dimension of the Khovanov homology lying along the line $j=2i{+}1$; and the third column lists the dimension of the vector space {\mathcal{H}}^{0,1}\bigl{(}W(3,n)\bigr{)}. In section 6 we approximate the distribution of normalized Khovanov dimenstions by a standard normal distribution, and we have displayed graphics comparing the actual distribution with the approximation.

To quantify those visual impressions, we compute two total deviations. Let

[TABLE]

For the $L^{2}$-comparison, we compute

[TABLE]

For the $L^{1}$-comparison, we compute

[TABLE]

The $L^{2}$ comparisons appear to tend to 0, whereas the $L^{1}$ comparisons appear to be growing slowly.