Trefoil Knot Group

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I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling curves Bela Bollobas is labeling the closed region and then come up with the presentation
$$<a,b,c,d mid ad^-1b, cd^-1a, cbd^-1>$$



enter image description here



I didn't get how this presentation we got with the given labeling, or I say what is this method called and how we write presentation of knot group via this method.






group-theory knot-theory group-presentation







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  • There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
    – Arthur
    Aug 30 at 5:52







  • 1




    I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
    – Touseef
    Aug 30 at 5:58






  • 4




    Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
    – Arthur
    Aug 30 at 6:08






  • 1




    The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
    – Adam Lowrance
    Aug 31 at 3:03















up vote
2
down vote

favorite
1












I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling curves Bela Bollobas is labeling the closed region and then come up with the presentation
$$<a,b,c,d mid ad^-1b, cd^-1a, cbd^-1>$$



enter image description here



I didn't get how this presentation we got with the given labeling, or I say what is this method called and how we write presentation of knot group via this method.






group-theory knot-theory group-presentation







share|cite|improve this question























  • There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
    – Arthur
    Aug 30 at 5:52







  • 1




    I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
    – Touseef
    Aug 30 at 5:58






  • 4




    Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
    – Arthur
    Aug 30 at 6:08






  • 1




    The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
    – Adam Lowrance
    Aug 31 at 3:03













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling curves Bela Bollobas is labeling the closed region and then come up with the presentation
$$<a,b,c,d mid ad^-1b, cd^-1a, cbd^-1>$$



enter image description here



I didn't get how this presentation we got with the given labeling, or I say what is this method called and how we write presentation of knot group via this method.






group-theory knot-theory group-presentation







share|cite|improve this question















I am studying Knot theory and have gone through the Wirtinger Presentation for the Knot Group. However, I come across the different(at least for me) way of finding the Knot Group. Instead of labeling curves Bela Bollobas is labeling the closed region and then come up with the presentation
$$<a,b,c,d mid ad^-1b, cd^-1a, cbd^-1>$$



enter image description here



I didn't get how this presentation we got with the given labeling, or I say what is this method called and how we write presentation of knot group via this method.






group-theory knot-theory group-presentation




group-theory knot-theory group-presentation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 30 at 9:07









Paul Frost

4,838424




4,838424










asked Aug 30 at 5:48









Touseef

464




464











  • There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
    – Arthur
    Aug 30 at 5:52







  • 1




    I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
    – Touseef
    Aug 30 at 5:58






  • 4




    Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
    – Arthur
    Aug 30 at 6:08






  • 1




    The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
    – Adam Lowrance
    Aug 31 at 3:03

















  • There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
    – Arthur
    Aug 30 at 5:52







  • 1




    I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
    – Touseef
    Aug 30 at 5:58






  • 4




    Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
    – Arthur
    Aug 30 at 6:08






  • 1




    The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
    – Adam Lowrance
    Aug 31 at 3:03
















There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
– Arthur
Aug 30 at 5:52





There are two parts to this, one easy and one difficult. The easy part is to see that the given relations are true. The difficult part is to see that the given relations suffice to give the correct group. Can you at least see how $ad^-1b=e$? (And is the last one supposed to be $cd^-1b$?)
– Arthur
Aug 30 at 5:52





1




1




I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
– Touseef
Aug 30 at 5:58




I didn't get how the given relation work? Is $ad^-1b$ operation of region or curves?
– Touseef
Aug 30 at 5:58




4




4




Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
– Arthur
Aug 30 at 6:08




Any letter represents a loop going through that region once (in your favourite positive direction; mine is down in this case, as is the author's) and no other region. So $ad^-1b$ is a loop going down through $a$, up through $d$ and down through $b$ (and then to the side, to join up with itself outside the trefoil). This would be a lot easier if I could paint a picture, sorry.
– Arthur
Aug 30 at 6:08




1




1




The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
– Adam Lowrance
Aug 31 at 3:03





The name of this presentation is the Dehn presentation of the knot group. See these links: nLab, a paper by Jim Hoste, and a paper by Scott Carter, Dan Silver, and Susan Williams.
– Adam Lowrance
Aug 31 at 3:03
















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