KTp’s, l=0
K-theoretic Thom polynomials of contact singularities of relative dimension 1, in Grothendieck expansions
(Hierarchy of these singularities)
codimension 0
| Local algebra | C |
| Thom-Boardman class | \Sigma^0 |
| Codimension | 0 |
| K-Thom polynomial in Grothendieck expansion | g_{}=1 |
| Remarks |
This contact singularity is open in its Thom-Boardman class. Source: Implicit function theorem. |
codimension 1
| Local algebra | C[x]/(x^2) |
| Thom-Boardman class | \Sigma^{1,0} |
| Codimension | 1 |
| K-Thom polynomial in Grothendieck expansion | g_{1} |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
codimension 2
| Local algebra | C[x]/(x^3) |
| Thom-Boardman class | \Sigma^{1,1,0} |
| Codimension | 2 |
| K-Thom polynomial in Grothendieck expansion | g_{1,1}+ 2g_{2} – ( 2g_{2,1}+g_{3} ) + g_{3,1} |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
codimension 3
| Local algebra | C[x]/(x^4) | |
| Thom-Boardman class | \Sigma^{1,1,1,0} | |
| Codimension | 3 | |
| K-Thom polynomial in Grothendieck expansion |
g_{1,1,1}+ 5g_{2,1}+ 6g_{3} – ( 5g_{2, 1, 1} + 16g_{3, 1}+ 9g_{4} ) + ( 10g_{3, 1, 1} + 19g_{4, 1} + 5g_{5} ) – ( 10g_{4, 1, 1} +10g_{5, 1} + g_{6} ) + g_{6, 1, 1} |
– 4g_{2, 2} + ( 4g_{2, 2, 1} + 10g_{3, 2} ) – ( 10g_{3, 2, 1} + 10g_{4, 2} ) + ( 10g_{4, 2, 1} + 5g_{5, 2} ) – ( 5g_{5, 2, 1} + g_{6, 2} ) + g_{6, 2, 1} |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
codimension 4
| Local algebra | C[x]/(x^5) |
| Thom-Boardman class | \Sigma^{1,1,1,1,0} |
| Codimension | 4 |
| K-Thom polynomial in Grothendieck expansion | ? |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
| Local algebra | C[x,y]/(x^2,y^2) |
| Thom-Boardman class | \Sigma^{2,0} |
| Codimension | 4 |
| K-Thom polynomial in Grothendieck expansion | g_{2,2} |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
codimension 5
| Local algebra | C[x]/(x^6) |
| Thom-Boardman class | \Sigma^{1,1,1,1,1,0} |
| Codimension | 5 |
| K-Thom polynomial in Grothendieck expansion | ? |
| Remarks | This contact singularity is open in its Thom-Boardman class. |
| Local algebra | C[x,y]/(xy,x^2+y^3) |
| Thom-Boardman class | \Sigma^{2,0} |
| Codimension | 5 |
| K-Thom polynomial in Grothendieck expansion |
2g_{2,2,1} + 4g_{3,2} – ( g_{2, 2, 2} + 5g_{3, 2, 1} + 4g_{4,2} ) + ( g_{3,2,2} + 4g_{4,2,1} + g_{5, 2} ) – g_{5,2,1} |