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The following is a list of comments of a purely mathematical nature. For errors and comments concerning references, exercises, or a page by page listing, select the desired option.

(p. = Page, e. = Exercise, l. = Line, t. = Theorem, c .= Corollary, f. = Figure, d. = Definition.)

• p.35, e.2.2: n is the number of half twists; for example, in Figure 3.6a, n is the number of crossings.
• p.52, l.5 & l.17: n must be odd to get a knot. To generalize the formula the formula to apply to links also, add (-1)^n and (-1)^(n-1) factors in front of the two t's.
• p.97, l.30: Knot 6_1 cannot be labeled with transposition, but can be labeled with 4-cycles. Knot 9_46 can be labeled in both ways, not just with transpositions. Finding the labeling with 4-cycles is challenging.(The point remains the same; these knots are distinguished using S_4 labelings only if the conjugacy classes of labels are considered.)
• p.113, l.3: The (2,1) entry of the matrix is 0, not -5. e.1.3: The 5-twisted double, not 3.
• p.117, l.3. The *'s in the matrix represent possibly distinct unknown entries.
• p.118, l.2: The matrix M can have determinant 1 or -1.
• p.121, l.24: The signature is 0, not 2. l.27: With respect to the usual basis we have used earlier, the matrix would have +1 on the off-diagonal, not -1.
• p.140, e.3.4: One can assume p is always positive. In this case, the knots are equivalent if and only if q - q' = 0 mod p, or qq' = 1 mod p.
• p.146, t.2: the value of brg(K) is greater than or equal to n-1, not n as stated.
• p.153, l.12: For a 3-stranded pretzel knot with all twist numbers odd, the knot is reversible only if two of the twist numbers are equal. If one of the twist numbers is even the knot is always reversible.
• p.154, d.: The last part of the sentence, "or reflection through the (y,x) plane," should be deleted.
• p.165, l.8 & l.10: For the first case, lambda is 2, and in the next 3.
• p.175, l.2: The (1,-3,5)-pretzel knot is the knot 8_1, not 7_2.
• p.183, l.18: The cross-sections of the (2,4)-torus link, not the trefoil, are illustrated.
• p.197, c.4: The statement and proof are correct for the signature. However, the omega-signatures may be nonzero for any values of omega that are roots of the Alexander polynomial. (In the given argument, for such omega the relevant matrix is not invertible.)

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