Homology for virtual knot diagram

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I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology group $H_1(Gamma,mathbbZ_2)$. I don't know how, is it defined? Thought first that this is simply the homology of the graph understood as a 1-dimensional CW-complex. With this definition, I don't know, however, how to interpret this sentence:




"The homology group $H_1(Gamma,mathbbZ_2)$ is generated by
"halves" corresponding to vertices: for every vertex $v$ we have two
halves of the graph $Gamma_v,1$ and $Gamma_v,2$, obtained by
smoothing at this vertex".





homology-cohomology knot-theory knot-invariants




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    I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology group $H_1(Gamma,mathbbZ_2)$. I don't know how, is it defined? Thought first that this is simply the homology of the graph understood as a 1-dimensional CW-complex. With this definition, I don't know, however, how to interpret this sentence:




    "The homology group $H_1(Gamma,mathbbZ_2)$ is generated by
    "halves" corresponding to vertices: for every vertex $v$ we have two
    halves of the graph $Gamma_v,1$ and $Gamma_v,2$, obtained by
    smoothing at this vertex".





    homology-cohomology knot-theory knot-invariants




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      I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology group $H_1(Gamma,mathbbZ_2)$. I don't know how, is it defined? Thought first that this is simply the homology of the graph understood as a 1-dimensional CW-complex. With this definition, I don't know, however, how to interpret this sentence:




      "The homology group $H_1(Gamma,mathbbZ_2)$ is generated by
      "halves" corresponding to vertices: for every vertex $v$ we have two
      halves of the graph $Gamma_v,1$ and $Gamma_v,2$, obtained by
      smoothing at this vertex".





      homology-cohomology knot-theory knot-invariants




      share|cite|improve this question











      I am studying now the article of V. O. Manturov about free knots ("Free Knots and Parity", https://arxiv.org/pdf/0912.5348.pdf). I got stuck on the section "Parity as homology" (p.6), at the homology group $H_1(Gamma,mathbbZ_2)$. I don't know how, is it defined? Thought first that this is simply the homology of the graph understood as a 1-dimensional CW-complex. With this definition, I don't know, however, how to interpret this sentence:




      "The homology group $H_1(Gamma,mathbbZ_2)$ is generated by
      "halves" corresponding to vertices: for every vertex $v$ we have two
      halves of the graph $Gamma_v,1$ and $Gamma_v,2$, obtained by
      smoothing at this vertex".






      homology-cohomology knot-theory knot-invariants





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