| Min(phi) over symmetries of the knot is: [-3,-2,0,1,2,2,0,2,1,3,4,1,0,1,2,1,1,1,0,-1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.358'] |
| Arrow polynomial of the knot is: 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.212', '6.358', '6.588', '6.589', '6.603', '6.990'] |
| Outer characteristic polynomial of the knot is: t^7+69t^5+46t^3+7t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.358'] |
| 2-strand cable arrow polynomial of the knot is: 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 640*K1**2*K2**5 - 1984*K1**2*K2**4 + 4128*K1**2*K2**3 - 7888*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 576*K1**2*K2*K4 + 6144*K1**2*K2 - 96*K1**2*K3**2 - 128*K1**2*K4**2 - 4032*K1**2 - 512*K1*K2**4*K3 + 1376*K1*K2**3*K3 - 928*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 4936*K1*K2*K3 - 32*K1*K2*K4*K5 + 616*K1*K3*K4 + 136*K1*K4*K5 - 608*K2**6 + 448*K2**4*K4 - 2456*K2**4 - 304*K2**2*K3**2 - 104*K2**2*K4**2 + 1592*K2**2*K4 - 1412*K2**2 + 88*K2*K3*K5 + 48*K2*K4*K6 - 1088*K3**2 - 486*K4**2 - 48*K5**2 - 4*K6**2 + 2980 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.358'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11461', 'vk6.11765', 'vk6.12780', 'vk6.13117', 'vk6.20671', 'vk6.22109', 'vk6.28170', 'vk6.29593', 'vk6.31216', 'vk6.31565', 'vk6.32392', 'vk6.32795', 'vk6.39614', 'vk6.41853', 'vk6.46226', 'vk6.47831', 'vk6.52217', 'vk6.52495', 'vk6.53054', 'vk6.53371', 'vk6.57604', 'vk6.58764', 'vk6.62260', 'vk6.63204', 'vk6.63783', 'vk6.63899', 'vk6.64213', 'vk6.64396', 'vk6.67060', 'vk6.67926', 'vk6.69679', 'vk6.70360'] |
| 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 | O1O2O3O4O5U3U4O6U2U5U1U6 |
| R3 orbit | {'O1O2O3O4O5U3U4O6U2U5U1U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U6U5U1U4O6U2U3 |
| Gauss code of K* | O1O2O3O4U3U1U5U6U2O5O6U4 |
| Gauss code of -K* | O1O2O3O4U1O5O6U3U5U6U4U2 |
| 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 -1 -2 -2 0 2 3],[ 1 0 -1 -2 0 3 3],[ 2 1 0 -1 1 3 2],[ 2 2 1 0 1 2 1],[ 0 0 -1 -1 0 1 1],[-2 -3 -3 -2 -1 0 1],[-3 -3 -2 -1 -1 -1 0]] |
| Primitive based matrix | [[ 0 3 2 0 -1 -2 -2],[-3 0 -1 -1 -3 -1 -2],[-2 1 0 -1 -3 -2 -3],[ 0 1 1 0 0 -1 -1],[ 1 3 3 0 0 -2 -1],[ 2 1 2 1 2 0 1],[ 2 2 3 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,0,1,2,2,1,1,3,1,2,1,3,2,3,0,1,1,2,1,-1] |
| Phi over symmetry | [-3,-2,0,1,2,2,0,2,1,3,4,1,0,1,2,1,1,1,0,-1,-1] |
| Phi of -K | [-2,-2,-1,0,2,3,-1,-1,1,2,4,0,1,1,3,1,0,1,1,2,0] |
| Phi of K* | [-3,-2,0,1,2,2,0,2,1,3,4,1,0,1,2,1,1,1,0,-1,-1] |
| Phi of -K* | [-2,-2,-1,0,2,3,-1,1,1,3,2,2,1,2,1,0,3,3,1,1,1] |
| Symmetry type of based matrix | c |
| u-polynomial | -t^3+t^2+t |
| Normalized Jones-Krushkal polynomial | 3z^2+16z+21 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+5w^3z^2-6w^3z+22w^2z+21w |
| Inner characteristic polynomial | t^6+47t^4+17t^2 |
| Outer characteristic polynomial | t^7+69t^5+46t^3+7t |
| Flat arrow polynomial | 12*K1**3 - 6*K1**2 - 6*K1*K2 - 6*K1 + 3*K2 + 4 |
| 2-strand cable arrow polynomial | 256*K1**4*K2**3 - 640*K1**4*K2**2 + 704*K1**4*K2 - 832*K1**4 + 96*K1**3*K2*K3 + 32*K1**3*K3*K4 + 640*K1**2*K2**5 - 1984*K1**2*K2**4 + 4128*K1**2*K2**3 - 7888*K1**2*K2**2 + 64*K1**2*K2*K4**2 - 576*K1**2*K2*K4 + 6144*K1**2*K2 - 96*K1**2*K3**2 - 128*K1**2*K4**2 - 4032*K1**2 - 512*K1*K2**4*K3 + 1376*K1*K2**3*K3 - 928*K1*K2**2*K3 - 128*K1*K2*K3*K4 + 4936*K1*K2*K3 - 32*K1*K2*K4*K5 + 616*K1*K3*K4 + 136*K1*K4*K5 - 608*K2**6 + 448*K2**4*K4 - 2456*K2**4 - 304*K2**2*K3**2 - 104*K2**2*K4**2 + 1592*K2**2*K4 - 1412*K2**2 + 88*K2*K3*K5 + 48*K2*K4*K6 - 1088*K3**2 - 486*K4**2 - 48*K5**2 - 4*K6**2 + 2980 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{1, 6}, {5}, {4}, {3}, {2}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}]] |
| If K is slice | False |