| Min(phi) over symmetries of the knot is: [-3,-3,-1,2,2,3,0,0,1,2,4,0,2,3,5,0,0,1,0,0,0] |
| Flat knots (up to 7 crossings) with same phi are :['6.346'] |
| Arrow polynomial of the knot is: -2*K1*K2 + K1 + K3 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.1', '4.3', '6.59', '6.66', '6.112', '6.215', '6.297', '6.306', '6.346', '6.351', '6.352', '6.353', '6.368', '6.393', '6.398', '6.402', '6.420', '6.422', '6.524', '6.529', '6.630', '6.632', '6.633', '6.642', '6.684', '6.707', '6.708', '6.717', '6.719', '6.721', '6.722', '6.737', '6.793', '6.835', '6.837', '6.847', '6.849', '6.857', '6.858', '6.883', '6.902', '6.913', '6.1084', '6.1092', '6.1097', '6.1136', '6.1146', '6.1155', '6.1159', '6.1374', '7.349', '7.365', '7.690', '7.2260', '7.2269', '7.2612', '7.2624', '7.2972', '7.2975', '7.4214', '7.4542', '7.4546', '7.9686', '7.9695', '7.9947', '7.10639', '7.10643', '7.10829', '7.10833', '7.13433', '7.15124', '7.15128', '7.15638', '7.15647', '7.15703', '7.15845', '7.16115', '7.16120', '7.16150', '7.19418', '7.19470', '7.19474', '7.19871', '7.20310', '7.20362', '7.20421', '7.20424', '7.23942', '7.24011', '7.24100', '7.24114', '7.24116', '7.24445', '7.26258', '7.26318', '7.26811', '7.26827', '7.27967', '7.28040', '7.28124', '7.28138', '7.29092', '7.29107', '7.29452', '7.29853', '7.30091', '7.30098', '7.30140', '7.30193', '7.30339', '7.30350', '7.30354'] |
| Outer characteristic polynomial of the knot is: t^7+96t^5+144t^3+12t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.346'] |
| 2-strand cable arrow polynomial of the knot is: 512*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2912*K1**2*K2**2 - 608*K1**2*K2*K4 + 3400*K1**2*K2 - 128*K1**2*K4**2 - 3232*K1**2 + 640*K1*K2**3*K3 - 832*K1*K2**2*K3 - 544*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4072*K1*K2*K3 + 864*K1*K3*K4 + 248*K1*K4*K5 - 128*K2**6 + 320*K2**4*K4 - 1664*K2**4 - 96*K2**3*K6 - 608*K2**2*K3**2 - 328*K2**2*K4**2 + 2264*K2**2*K4 - 2566*K2**2 - 32*K2*K3**2*K4 + 776*K2*K3*K5 + 336*K2*K4*K6 + 40*K3**2*K6 - 1432*K3**2 - 900*K4**2 - 256*K5**2 - 106*K6**2 + 2866 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.346'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11516', 'vk6.11847', 'vk6.12864', 'vk6.13173', 'vk6.20616', 'vk6.22038', 'vk6.28094', 'vk6.29539', 'vk6.31286', 'vk6.31678', 'vk6.32441', 'vk6.32858', 'vk6.39506', 'vk6.41723', 'vk6.46109', 'vk6.47758', 'vk6.52287', 'vk6.52551', 'vk6.53126', 'vk6.53438', 'vk6.57498', 'vk6.58678', 'vk6.62183', 'vk6.63135', 'vk6.63798', 'vk6.63928', 'vk6.64240', 'vk6.64440', 'vk6.67016', 'vk6.67888', 'vk6.69641', 'vk6.70324'] |
| 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 | O1O2O3O4O5U3U2O6U1U6U5U4 |
| R3 orbit | {'O1O2O3O4O5U3U2O6U1U6U5U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U2U1U6U5O6U4U3 |
| Gauss code of K* | O1O2O3O4U1U5U6U4U3O6O5U2 |
| Gauss code of -K* | O1O2O3O4U3O5O6U2U1U6U5U4 |
| 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 3 3 1],[ 3 0 0 0 5 4 1],[ 2 0 0 0 3 2 0],[ 2 0 0 0 2 1 0],[-3 -5 -3 -2 0 0 0],[-3 -4 -2 -1 0 0 0],[-1 -1 0 0 0 0 0]] |
| Primitive based matrix | [[ 0 3 3 1 -2 -2 -3],[-3 0 0 0 -1 -2 -4],[-3 0 0 0 -2 -3 -5],[-1 0 0 0 0 0 -1],[ 2 1 2 0 0 0 0],[ 2 2 3 0 0 0 0],[ 3 4 5 1 0 0 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-3,-1,2,2,3,0,0,1,2,4,0,2,3,5,0,0,1,0,0,0] |
| Phi over symmetry | [-3,-3,-1,2,2,3,0,0,1,2,4,0,2,3,5,0,0,1,0,0,0] |
| Phi of -K | [-3,-2,-2,1,3,3,1,1,3,1,2,0,3,2,3,3,3,4,2,2,0] |
| Phi of K* | [-3,-3,-1,2,2,3,0,2,2,3,1,2,3,4,2,3,3,3,0,1,1] |
| Phi of -K* | [-3,-2,-2,1,3,3,0,0,1,4,5,0,0,1,2,0,2,3,0,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+2t^2-t |
| Normalized Jones-Krushkal polynomial | 6z^2+19z+15 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+10w^3z^2-4w^3z+23w^2z+15w |
| Inner characteristic polynomial | t^6+60t^4+32t^2+1 |
| Outer characteristic polynomial | t^7+96t^5+144t^3+12t |
| Flat arrow polynomial | -2*K1*K2 + K1 + K3 + 1 |
| 2-strand cable arrow polynomial | 512*K1**2*K2**3 + 192*K1**2*K2**2*K4 - 2912*K1**2*K2**2 - 608*K1**2*K2*K4 + 3400*K1**2*K2 - 128*K1**2*K4**2 - 3232*K1**2 + 640*K1*K2**3*K3 - 832*K1*K2**2*K3 - 544*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4072*K1*K2*K3 + 864*K1*K3*K4 + 248*K1*K4*K5 - 128*K2**6 + 320*K2**4*K4 - 1664*K2**4 - 96*K2**3*K6 - 608*K2**2*K3**2 - 328*K2**2*K4**2 + 2264*K2**2*K4 - 2566*K2**2 - 32*K2*K3**2*K4 + 776*K2*K3*K5 + 336*K2*K4*K6 + 40*K3**2*K6 - 1432*K3**2 - 900*K4**2 - 256*K5**2 - 106*K6**2 + 2866 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {3, 5}, {1, 4}], [{3, 6}, {2, 5}, {1, 4}], [{5, 6}, {1, 4}, {2, 3}]] |
| If K is slice | False |