| Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,1,1,1,2,3,1,0,0,2,-1,0,0,-1,0,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1061'] |
| Arrow polynomial of the knot is: 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['4.2', '6.303', '6.338', '6.381', '6.432', '6.468', '6.558', '6.583', '6.597', '6.607', '6.634', '6.637', '6.643', '6.654', '6.667', '6.701', '6.709', '6.712', '6.718', '6.728', '6.767', '6.801', '6.825', '6.827', '6.974', '6.994', '6.1042', '6.1061', '6.1069', '6.1181', '6.1271', '6.1286', '6.1287', '6.1289', '6.1297', '6.1337', '6.1355'] |
| Outer characteristic polynomial of the knot is: t^7+40t^5+130t^3+24t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1061'] |
| 2-strand cable arrow polynomial of the knot is: -192*K1**4 + 1440*K1**2*K2**3 - 4992*K1**2*K2**2 - 128*K1**2*K2*K4 + 5056*K1**2*K2 - 3612*K1**2 + 640*K1*K2**3*K3 - 928*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3960*K1*K2*K3 + 112*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 992*K2**4*K4 - 3096*K2**4 - 256*K2**3*K6 - 464*K2**2*K3**2 - 88*K2**2*K4**2 + 2152*K2**2*K4 - 1088*K2**2 + 200*K2*K3*K5 + 32*K2*K4*K6 - 840*K3**2 - 246*K4**2 - 4*K5**2 + 2460 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1061'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71622', 'vk6.71784', 'vk6.72203', 'vk6.72349', 'vk6.73376', 'vk6.73539', 'vk6.75285', 'vk6.75552', 'vk6.77244', 'vk6.77328', 'vk6.77575', 'vk6.77683', 'vk6.78268', 'vk6.78518', 'vk6.80080', 'vk6.80230', 'vk6.81113', 'vk6.81178', 'vk6.81197', 'vk6.81234', 'vk6.81326', 'vk6.81515', 'vk6.82011', 'vk6.82432', 'vk6.82745', 'vk6.85445', 'vk6.86341', 'vk6.86927', 'vk6.87145', 'vk6.88085', 'vk6.88666', 'vk6.88771'] |
| 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 | O1O2O3O4U5U2O6U1O5U6U3U4 |
| R3 orbit | {'O1O2O3O4U5U2O6U1O5U6U3U4'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4U1U2U5O6U4O5U3U6 |
| Gauss code of K* | O1O2O3U4U5U2U3O6O5U1O4U6 |
| Gauss code of -K* | O1O2O3U4O5U3O6O4U1U2U6U5 |
| 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 -2 -1 1 3 -1 0],[ 2 0 0 2 3 1 0],[ 1 0 0 0 1 1 -1],[-1 -2 0 0 1 0 -1],[-3 -3 -1 -1 0 -2 -1],[ 1 -1 -1 0 2 0 0],[ 0 0 1 1 1 0 0]] |
| Primitive based matrix | [[ 0 3 1 0 -1 -1 -2],[-3 0 -1 -1 -1 -2 -3],[-1 1 0 -1 0 0 -2],[ 0 1 1 0 1 0 0],[ 1 1 0 -1 0 1 0],[ 1 2 0 0 -1 0 -1],[ 2 3 2 0 0 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-1,0,1,1,2,1,1,1,2,3,1,0,0,2,-1,0,0,-1,0,1] |
| Phi over symmetry | [-3,-1,0,1,1,2,1,1,1,2,3,1,0,0,2,-1,0,0,-1,0,1] |
| Phi of -K | [-2,-1,-1,0,1,3,0,1,2,1,2,1,1,2,2,2,2,3,0,2,1] |
| Phi of K* | [-3,-1,0,1,1,2,1,2,2,3,2,0,2,2,1,1,2,2,-1,0,1] |
| Phi of -K* | [-2,-1,-1,0,1,3,0,1,0,2,3,1,-1,0,1,0,0,2,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+8w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+24t^4+53t^2+9 |
| Outer characteristic polynomial | t^7+40t^5+130t^3+24t |
| Flat arrow polynomial | 4*K1**3 - 2*K1**2 - 2*K1*K2 - 2*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -192*K1**4 + 1440*K1**2*K2**3 - 4992*K1**2*K2**2 - 128*K1**2*K2*K4 + 5056*K1**2*K2 - 3612*K1**2 + 640*K1*K2**3*K3 - 928*K1*K2**2*K3 - 224*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3960*K1*K2*K3 + 112*K1*K3*K4 + 8*K1*K4*K5 - 864*K2**6 + 992*K2**4*K4 - 3096*K2**4 - 256*K2**3*K6 - 464*K2**2*K3**2 - 88*K2**2*K4**2 + 2152*K2**2*K4 - 1088*K2**2 + 200*K2*K3*K5 + 32*K2*K4*K6 - 840*K3**2 - 246*K4**2 - 4*K5**2 + 2460 |
| Genus of based matrix | 1 |
| Fillings of based matrix | [[{2, 6}, {4, 5}, {1, 3}]] |
| If K is slice | False |