Amendments to the book
"Introduction to Vassiliev knot invariants"
by S.Chmutov, S.Duzhin and J.Mostovoy
(Cambridge University Press, 2012)
1. p. xvi: include M.Karev, B.Pittel and L.Traldi in the list "It is our pleasure
to thank..." (in alphabetical order).
2. p. 32, end of Sec. 2.3:
Change "does not follow any known rules" to "is more intricate".
After the picture of 4 links add the following sentence:
"The exact rule for this transformation of Conway's polynomial
can be stated in terms of the multivariate Alexander polynomial, see
\cite{Tra}."
3. p. 49, line 6
Change "old Perko's notation" to "Perko's old notation".
4. p.87, theorem 4.5:
(a) add \index{Fundamental theorem!on Vassiliev invariants}
(b) after the statement of the theorem, add:
Remark. In fact, this statement is valid with $\C$ replaced by any field of
characteristic zero; this follow from the rationality of the Kontsevich
integral (see Section \ref{kifieqki} below).
5. p. 93, line 4 (picture formula):
invert the signs of the last two terms of the sum.
6. p.108, line 7 (not counting the pictures). After the sentence ending in
"double occurence words" add two more sentences:
"A complete proof was first discovered in \cite{Ghi}. Below, we reproduce
the argument of \cite{CL}."
7. p. 246, Exercises 8.10 and 8.11: change "Compute" to "Compute to order 2".
8. p. 289:
Replace the two last lines before Sec. 10.2.7 by the following:
"It was recently proved (see \cite{Br}) that every MZV is equal
to a sum of rational multiples of MZV's of the same weight whose exponent
strings involve only 2's and 3's. Useful information about generators and
relations in small degrees is available on the web pages of M.Petitot and
J.Vermaseren (see References)."
9. p. 291, line 2: replace "-\zeta(k+2)" by "-\zeta(k+1)".
10. p.295: in the first formula (line 2), change the sign \pm to \mp
(in front of 1/24).
11. p. 307: add \index{Kontsevich integral!rationality}
after the title of the section.
12. p.307: add \label{thnLM2} to Theorem 10.37.
13. p. 308: after Remark 10.38, add one more
Remark. The existence of a rational associator and Theorem \ref{thmLM2}
imply that the Fundamental theorem on Vassiliev invariants \ref{fund_thm}
is valid for invariants with values in any field of characteristic zero.
No analogue of this theorem for integer coefficients is yet known.
14. p. 431, Theorem 14.25.
There are two errors here (noticed by B.Pittel). The correct statement is:
"For fixed $n$ and $k$, congruent modulo 2, the diagrams $PN_{a_1,...,a_k,b}$
with $0\le a_1\le...\le a_k\le b$, $a_1+...+a_k+2b=n-k$
are linearly independent."
15. p. 441: change \index{Vassiliev invariant}
to \index{Vassiliev!invariant}
16. p.465: in the third displayed formula from above insert the symbol of
tensor product into the last two summands (with Q).
17. p. 482, line 6:
Replace "(up to signs)" by "(up to a linear change by an upper triangular
matrix with $\pm1$ on the diagonal)".
18. p. 485.
Replace the reference
\bibitem{Br}
D.~Broadhurst, {\it Conjectured enumeration of Vassiliev
invariants}, Online at {\tt arXiv:q-alg/9709031}.
by the reference
\bibitem{Br} Francis Brown. {\it Mixed Tate motives over $\Z$}.
\texttt{arXiv:1102.1312v1 [math.AG]}, Annals of Math. 175 (2012),
949--976.
19. p. 487: add the reference
\bibitem{Ghi}
L. Ghier, Double occurrence words with the same alternance graph, Ars
Combin. 36 (1993) 57--64.
20. p. 492: change "Poincar\'{e} A." to "Poincar\'{e} H."
21. p. 493: add the reference
\bibitem{Tra}
L.Traldi. Conway's potential function and its Taylor series.
\textit{Kobe J. Math.} \textbf{5} (1988), 233--264.
22. p. 82, Exercise 3.16.
(1) Change "a given chord diagram D" to "a degree n chord diagram D".
(2) At the end, add "In particular, for any diagram of an odd degree the
obtained curve is disconnected".