Dear Springer People,

My comments on the galley proofs of my paper "Balloons and Hoops...":

In the online abstract and in the actual abstract, change "background
field (BF)" back to BF. Except on Wikipedia I have not the term
"background field" used in this context and I think it is wrong.

In the footnote on page 1, remove the phrase "one of the handouts is at
the end of this paper", as it is no longer true.

Lines 25-29: I much prefer that the figure below will be returned to
its place as an inset on the right side of the paragraph. This saves a
significant amount of space and saves the reader the need to flip through
pages. This comment is repeated many times below, and always my goal is
to save space and make the paper more readable.

Line 30: I use "Page~\pageref{sol:piT}" to make sure that the page
reference remains true.

Line 76: "d-2D" should go back to "(d-2)-dimensional". Elsewhere in the
paper it doesn't matter which form to use, "2D" or "2-dimensional". But
here there is a difference in meaning and the current version is wrong.

Line 91: The figure below should be an inset on the right. Barring this,
replace "on the right" with "below".

The figure below line 128 should go back to being an inset to the right
of line 117. Barring this, in line 126 after " "unknotted balloon" ",
add "(see below)", in parenthesis.  Line 76: "d-2D" should go back to
"(d-2)-dimensional". Elsewhere in the paper it doesn't matter which
form to use, "2D" or "2-dimensional". But here there is a difference in
meaning and the current version is wrong.

Line 91: The figure below should be an inset on the right. Barring this,
replace "on the right" with "below".

The figure below line 128 should go back to being an inset to the right
of line 117. Barring this, in line 126 after " "unknotted balloon" ",
add "(see below)", in parenthesis, and increase the width of the figure
caption so that it will take up 2 lines rather than 4.

In the caption of figure 4, in the first line, replace the "\pm" with
"+" (or return that line to what it was in the original - "\pm" means
"plus or minus"). In line two do not italicize "lower part", and in line
3 add a period at the end of the sentence.

The figure above line 157 should go back to being an inset to the right
of line 147. Barring that remove the "on the right" on line 142 and
change the "on the right" in line 147 to "below".

The figure below line 177 is missing two arrows; please consult with
the original. Also, that figure should be moved back to being an inset
on the right of line 170. Barring that, the "on the right" in line 176
should be "below".

The figure below line 192 should go back to being an inset to the right of
line 186. Barring that, the "on the right" in line 187 should be "below".

The two parts of Fig 2 should be shrank; especially the second part
should be shrank so that the text in it is no bigger than the normal
text in the paper.

In line 269, too much space before the footnote mark "6".

The figure below line 282 should return to being an inset to the right
of line 271. Barring this it should be completely removed, along with
the phrase "or the even nicer ... on the right" in lines 275-6.

The italics in "lower part" in the caption of figure 4 should be removed.

Figure 5 should return to being an inset to the right of line 287. Barring
this the "on the right" in lines 287-8 should become "in Fig 5". The
original form of figure 5 was much more elegant.

A whole phrase is missing in line 288, please compare with the original.
Should be "... its $\zeta$ formulae of Section~\ref{sec:zeta}). Whenever
we have disjoint label sets $T_1\cap T_2=\emptyset=H_1\cap H_2$, ...".

Please remove the parenthetical "(Fig 5)" at the end of line 291.

The figure below line 337 should go back to being an inset to the right
of line 329. Barring this, on lines 332-3, "on the right" -> "below".

Throughout the paper, the operations "tm", "hm" and "tha" should appear
in italics, as they had been in the original - for unclear reasons it
was changes to roman but only in about half the places. You should be
able to search for all the places where that change was made. A partial
list includes footnote 8, equations 7, lines 351-352, 375-385., 375-385,
407, 411-420, 448-451, 469, 480-486, 513-519, 556-557, 627-635, 647-650,
705, 716-717, 782, 804-805, 962-970, 1006, 1141-1157, 1150-1152, 1410,
1432, 1434.

The "Solution of Riddle" mini-paragraph should return to being an inset
to the right of line 342. Barring this it should become an ordinary text
paragraph placed right above line 342.

A long phrase in line 347-9 is repeated twice. Remove everything in
these 3 lines except for the word "by" which should appear once.

Change the "on the right" in line 355 to "below" (given the change in
the page width it will be impractical to move the figure there again).

Add a "\qed" at the end of line 356. At the end of propositions that
are given without a proof it is normal to put a \qed symbol to indicate
to the reader that no further proof will appear later. The same comment
also applies to lines 388, 423, 1274, 1325.

In lines 363-4, change "$\rho^-_{ux}$ and $\rho^{\pm}_{ux}$" either back
to the original form or to "$\rho^+_{ux}$ and $\rho^-_{ux}$".

In line 373, I think "Conjugation Relation" should be in caps, but I
won't argue.

In lines 411-412, the symbol "/" appears 3 times. In all cases there
should be more space around it as it was in the original.

There is an awkward line break in line 477.

The figure below line 512 should move back to being an inset to the right
of line 501. Barring that the "on the right" in lines 501-2 should be
"below".

The diagram below line 539 should be made wider. As it is it is hard to
tell which label corresponds to which of the diagonal arrows and where
exactly does the middle space begins and ends. In the original I had
"\xymatrix@R=0.5in@C=-0.3in{" and I recommend the same.

Also, that same diagram should move back to being an inset to the right of
line 531. Barring this the "on the right" in line 531 should be "below"
and the "above diagram" in line 539 should be "diagram below".

In Equation 15 and line 549 "t^{uv}_w$ should be "tm^{uv}_w". This
mistake was in the original; sorry.

The diagram below line 550 should be made wider (I used
"\xymatrix@R=0.5in@C=-0.3in{"). Also it  should be back as an inset to the
right of line 545. Barring that, "on the right" -> "below" in line 545.

The diagram below line 563 should move back to being an inset on the right
of line 560. Barring that, the "on the right" in lines 562-3 should be
"below" and the "\qed" in line 563 should be moved to after the diagram.

In line 562 the should be a space between the "ad" and the "bch".

In line 570 there should be a space between the "ad" and the "v".

The figure below line 603 should move back to being an inset on the
right of line 596. Barring that "on the right" -> "below" on line 600.

The "bch" in Equation 19 should be in roman font.The "bch" in Equation
19 should be in roman font.

There's an odd line break in line 631.

All computer code should be flushed to the left and with very little
space above, below, and between code snippets (3-4mm, perhaps). This
applies to the code following lines 671, 673, 684, 696, 700, 703, 705,
706, 709, 718, 723, 724 (several), 737, 742, 745, 754, 757, 945.

The "top brackets" in line 676 are too big. I used
"$\overbracket[0.5pt][1pt]{u\overbracket[0.5pt][1pt]{uv}}$".

There's an odd line break in line 744. If necessary, the long formula
there can be displayed.

The figure below line 754 should be brought to below line 749 or should
return to being an inset on the right of line 750.

The two figures below line 776 should be made smaller and if possible
they should fit side by side as in the original.

The figure above line 890 should go back to being an inset on the right
of line 890. Barring that, "on the right" -> "above" in line 891.

Line 919, "Lemma definition" -> "Lemma-Definition". Also in lines 958-959.

Line 927: Funny spacing in "FL". I used "\FL" everywhere, with
"\def\FL{\text{\it FL}}" in defs.tex. Similarly for the FL and FA in
line 1197.

In line 944, parenthesis missing around the matrix towards the end of
the line. See the original. Similarly for the vectors in lines 969, 972,
and below 973. Especially in the last case, the entries of that vector
should be centred.

The matrix below line 970 should return to being an inset on the right
of line 968. Barring this, "on the right -> "below" in line 968.

The figure below line 1036 should move back to being an inset on the
right of line 1034. Barring that, "on the right -> "below" in line 1036.

It is better if the two figures below line 1043 could fit side by side.
Barring that, the figures should be properly centred.

The operators "bch" and "div" should be in Roman font, everywhere. See
lines 1214-1215, equation 37.

The usual comment about figures and "on the right" -> "below" in lines
1241, 1249, 1255 (multiple "on the right").

The first line in each of the equations below lines 1316,
1320, and 1334 should be flushed further to the left. I use
"\begin{multline*}"..."\end{multline*}".

In line 1343, I used "In \LaTeX,
$\sslash=\verb$\sslash$\in\verb$stmaryrd.sty$$.".

In line 1453, I used "$\overbrace[L1R]{uv}$" which requires
"\usepackage[overload]{abraces}" in the preamble.

In the references, I very very much dislike numerical citations and
much prefer that you go back to letter-based citations, as it was in the
original. I care about my readers and as a reader myself I know that it is
much easier to remember which is which if it is an acronym rather than a
numeral. For the same reason we do not name all variables in a paper a1,
a2, a3, etc.

References 3, 4, and 9 (in the current numbering) indeed can be dropped.
Note that this will make reference 5 "BN2" instead of "BN4".

Reference #32 used to be labelled "Web" and should return to being "Web".
The short Web is used throughout the paper to refer to URLs of a certain
form, and that form is defined within that reference line. It is also
defined in a footnote at the beginning of the paper but as it is easy not
to note that footnote, I felt it would be to the benefit of the reader
to also include it within the list of references. Please keep it this way.

I believe above I have answered your questions Q1-Q7.