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nLab > Latest Changes: weight systems are the associated graded objects of Vassiliev invariants

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 25th 2019

    starting something, in order to record Theorem 1 in

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2019

    added pointer to

    • Joan S. Birman, Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent Math (1993) 111: 225 (doi:10.1007/BF01231287)

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 23rd 2021

    I’m finding it hard to parse this page title.

    Should the ’graded’ be ’grades’?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021
    • (edited Apr 23rd 2021)

    By the infinite linguistic wisdom of the maths community, “the associated graded” has become the standard term for what started life as “the associated graded vector space”.

    What the entry title adds to this abuse is only to abbreviate “vector space of weight systems” to “weight systems”.

    The statement in full English is:

    The vector space of weight systems is the associated graded vector space to the filtered vector space of Vassiliev invariants.

    (as it does say a little further down in the entry).

    Whatever entry title seems to both capture as well as abbreviate this I am fine with.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 23rd 2021

    Ok, ugly, but at least it’s not a misprint.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021

    In principle we could add the word “elements” after “associated graded” to make the entry title read

    weight systems are the associated graded elements of Vassiliev invariants

    While closer to meaningful English, I can easily imagine experts on “the associated graded” wondering what an “associated graded element” might be.

    But I am agnostic about it. The entry title here is more like a file name to me, not like something you need to read out.

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 23rd 2021
    • (edited Apr 23rd 2021)

    I agree that “associated graded of”, while being the standard mathematical slang by now, does sound clumsy.

    One can also say

    weight systems are the associated graded group of Vassiliev invariants

    or

    weight systems form the associated graded group of Vassiliev invariants

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 23rd 2021

    Right, we could add “elements” as suggested in #6 or “group” as in your #7 or else “space” or “vector space”.

    But of all these I find “elements” would fit best, since both the “weight systems” and the “Vassiliev invariant” are the elements of their respective groups/spaces

    • CommentRowNumber9.
    • CommentAuthorTim_Porter
    • CommentTimeApr 24th 2021
    • (edited Apr 24th 2021)

    As the use of ’associated graded of’ in lots of contexts can be thought of as being the spoken (perhaps lazy) form of ’associated graded object’. In written material it might be preferable to include the extra word, ’object’. The suggestion that Urs makes would work as well, but how do you ‘grade’ an element, so that may be a bit confusing. (The advantage of ’object’ is that it covers a multitude of cases, so is very versatile.)

    I have heard the slang version many times and it is so clumsy (and to my ears ugly) that it stops my appreciation of the mathematics. I stop to think ’what is being graded’ and then have to work harder to catch up the thread of the seminar.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2021

    okay, have changed the page name to “… the associated graded objects …”

    also, I am exporting the list of “Related theorem” to a new entry

    which then to !include back into this and related entries, for ease of synchronizing and updating their cross-links

    diff, v10, current

1 to 10 of 10

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