| Min(phi) over symmetries of the knot is: [-4,-2,-1,2,2,3,1,0,2,3,5,0,1,2,4,0,0,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.345'] |
| Arrow polynomial of the knot is: -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.58', '6.76', '6.78', '6.90', '6.98', '6.154', '6.161', '6.162', '6.198', '6.280', '6.284', '6.345', '6.417', '6.421', '6.435', '6.511'] |
| Outer characteristic polynomial of the knot is: t^7+99t^5+132t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.345'] |
| 2-strand cable arrow polynomial of the knot is: -144*K1**4 + 384*K1**3*K2*K3 - 160*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3584*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 3232*K1**2*K2 - 272*K1**2*K3**2 - 2584*K1**2 + 1056*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 288*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4560*K1*K2*K3 + 544*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**6 + 448*K2**4*K4 - 2688*K2**4 + 256*K2**3*K3*K5 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 - 616*K2**2*K4**2 + 2488*K2**2*K4 - 240*K2**2*K5**2 - 8*K2**2*K6**2 - 1618*K2**2 + 1136*K2*K3*K5 + 168*K2*K4*K6 + 80*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 1508*K3**2 - 658*K4**2 - 288*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2618 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.345'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11424', 'vk6.11719', 'vk6.12736', 'vk6.13081', 'vk6.20610', 'vk6.22027', 'vk6.28081', 'vk6.29531', 'vk6.31169', 'vk6.31510', 'vk6.32333', 'vk6.32755', 'vk6.39490', 'vk6.41700', 'vk6.46084', 'vk6.47742', 'vk6.52180', 'vk6.52430', 'vk6.53006', 'vk6.53324', 'vk6.57486', 'vk6.58655', 'vk6.62164', 'vk6.63119', 'vk6.63749', 'vk6.63857', 'vk6.64173', 'vk6.64363', 'vk6.67010', 'vk6.67880', 'vk6.69631', 'vk6.70318'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3O4O5U3U2O6U1U6U4U5 |
| R3 orbit | {'O1O2O3O4O5U3U2O6U1U6U4U5'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U1U2U6U5O6U4U3 |
| Gauss code of K* | O1O2O3O4U1U5U6U3U4O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U1U2U6U5U4 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -3 -2 -2 2 4 1],[ 3 0 0 0 4 5 1],[ 2 0 0 0 2 3 0],[ 2 0 0 0 1 2 0],[-2 -4 -2 -1 0 1 0],[-4 -5 -3 -2 -1 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 4 2 1 -2 -2 -3],[-4 0 -1 0 -2 -3 -5],[-2 1 0 0 -1 -2 -4],[-1 0 0 0 0 0 -1],[ 2 2 1 0 0 0 0],[ 2 3 2 0 0 0 0],[ 3 5 4 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-4,-2,-1,2,2,3,1,0,2,3,5,0,1,2,4,0,0,1,0,0,0] |
| Phi over symmetry | [-4,-2,-1,2,2,3,1,0,2,3,5,0,1,2,4,0,0,1,0,0,0] |
| Phi of -K | [-3,-2,-2,1,2,4,1,1,3,1,2,0,3,2,3,3,3,4,1,3,1] |
| Phi of K* | [-4,-2,-1,2,2,3,1,3,3,4,2,1,2,3,1,3,3,3,0,1,1] |
| Phi of -K* | [-3,-2,-2,1,2,4,0,0,1,4,5,0,0,1,2,0,2,3,0,0,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^4+t^3+t^2-t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+9w^3z^2-8w^3z+26w^2z+17w |
| Inner characteristic polynomial | t^6+61t^4+33t^2+1 |
| Outer characteristic polynomial | t^7+99t^5+132t^3+16t |
| Flat arrow polynomial | -2*K1*K2 - 2*K1*K3 + K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -144*K1**4 + 384*K1**3*K2*K3 - 160*K1**3*K3 + 192*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 - 3584*K1**2*K2**2 + 128*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 384*K1**2*K2*K4 + 3232*K1**2*K2 - 272*K1**2*K3**2 - 2584*K1**2 + 1056*K1*K2**3*K3 + 384*K1*K2**2*K3*K4 - 672*K1*K2**2*K3 - 288*K1*K2**2*K5 - 256*K1*K2*K3*K4 - 32*K1*K2*K3*K6 + 4560*K1*K2*K3 + 544*K1*K3*K4 + 48*K1*K4*K5 + 8*K1*K5*K6 - 128*K2**6 + 448*K2**4*K4 - 2688*K2**4 + 256*K2**3*K3*K5 - 1344*K2**2*K3**2 - 32*K2**2*K3*K7 - 616*K2**2*K4**2 + 2488*K2**2*K4 - 240*K2**2*K5**2 - 8*K2**2*K6**2 - 1618*K2**2 + 1136*K2*K3*K5 + 168*K2*K4*K6 + 80*K2*K5*K7 + 8*K2*K6*K8 + 8*K3**2*K6 - 1508*K3**2 - 658*K4**2 - 288*K5**2 - 14*K6**2 - 4*K7**2 - 2*K8**2 + 2618 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}]] |
| If K is slice | False |