| Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,2,3,2,4,2,3,1,3,0,0,0,0,1,-1] |
| Flat knots (up to 7 crossings) with same phi are :['6.148'] |
| Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503'] |
| Outer characteristic polynomial of the knot is: t^7+120t^5+155t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.148'] |
| 2-strand cable arrow polynomial of the knot is: -128*K1**3*K3 + 128*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 2016*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 3304*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 3876*K1**2 + 320*K1*K2**3*K3 - 768*K1*K2**2*K3 - 416*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4744*K1*K2*K3 + 824*K1*K3*K4 + 328*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 2096*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 496*K2**2*K3**2 - 144*K2**2*K4**2 + 2792*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3386*K2**2 + 1072*K2*K3*K5 + 136*K2*K4*K6 + 56*K2*K5*K7 + 8*K2*K6*K8 + 16*K3**2*K6 - 2104*K3**2 - 1090*K4**2 - 460*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 3778 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.148'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73326', 'vk6.73466', 'vk6.74023', 'vk6.74575', 'vk6.75220', 'vk6.75474', 'vk6.76051', 'vk6.76773', 'vk6.78211', 'vk6.78438', 'vk6.79002', 'vk6.79567', 'vk6.80024', 'vk6.80174', 'vk6.80532', 'vk6.80992', 'vk6.81888', 'vk6.82354', 'vk6.82376', 'vk6.82602', 'vk6.83623', 'vk6.83667', 'vk6.84303', 'vk6.84373', 'vk6.84483', 'vk6.84584', 'vk6.84634', 'vk6.85225', 'vk6.85596', 'vk6.86757', 'vk6.88703', 'vk6.88984'] |
| 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 | O1O2O3O4O5U1O6U2U5U4U6U3 |
| R3 orbit | {'O1O2O3O4O5U1O6U2U5U4U6U3'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3O4O5U3U6U2U1U4O6U5 |
| Gauss code of K* | O1O2O3O4O5U6U1U5U3U2O6U4 |
| Gauss code of -K* | O1O2O3O4O5U2O6U4U3U1U5U6 |
| 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 -4 -3 2 1 1 3],[ 4 0 1 4 3 2 3],[ 3 -1 0 4 2 1 3],[-2 -4 -4 0 -1 -1 2],[-1 -3 -2 1 0 0 2],[-1 -2 -1 1 0 0 1],[-3 -3 -3 -2 -2 -1 0]] |
| Primitive based matrix | [[ 0 3 2 1 1 -3 -4],[-3 0 -2 -1 -2 -3 -3],[-2 2 0 -1 -1 -4 -4],[-1 1 1 0 0 -1 -2],[-1 2 1 0 0 -2 -3],[ 3 3 4 1 2 0 -1],[ 4 3 4 2 3 1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-3,-2,-1,-1,3,4,2,1,2,3,3,1,1,4,4,0,1,2,2,3,1] |
| Phi over symmetry | [-4,-3,1,1,2,3,0,2,3,2,4,2,3,1,3,0,0,0,0,1,-1] |
| Phi of -K | [-4,-3,1,1,2,3,0,2,3,2,4,2,3,1,3,0,0,0,0,1,-1] |
| Phi of K* | [-3,-2,-1,-1,3,4,-1,0,1,3,4,0,0,1,2,0,2,2,3,3,0] |
| Phi of -K* | [-4,-3,1,1,2,3,1,2,3,4,3,1,2,4,3,0,1,1,1,2,2] |
| Symmetry type of based matrix | c |
| u-polynomial | t^4-t^2-2t |
| Normalized Jones-Krushkal polynomial | 4z^2+17z+19 |
| Enhanced Jones-Krushkal polynomial | -2w^4z^2+6w^3z^2-8w^3z+25w^2z+19w |
| Inner characteristic polynomial | t^6+80t^4+44t^2+1 |
| Outer characteristic polynomial | t^7+120t^5+155t^3+16t |
| Flat arrow polynomial | 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1 |
| 2-strand cable arrow polynomial | -128*K1**3*K3 + 128*K1**2*K2**3 + 96*K1**2*K2**2*K4 - 2016*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 160*K1**2*K2*K4 + 3304*K1**2*K2 - 160*K1**2*K3**2 - 64*K1**2*K4**2 - 3876*K1**2 + 320*K1*K2**3*K3 - 768*K1*K2**2*K3 - 416*K1*K2**2*K5 - 224*K1*K2*K3*K4 + 4744*K1*K2*K3 + 824*K1*K3*K4 + 328*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 + 160*K2**4*K4 - 2096*K2**4 + 32*K2**3*K3*K5 - 64*K2**3*K6 - 496*K2**2*K3**2 - 144*K2**2*K4**2 + 2792*K2**2*K4 - 96*K2**2*K5**2 - 8*K2**2*K6**2 - 3386*K2**2 + 1072*K2*K3*K5 + 136*K2*K4*K6 + 56*K2*K5*K7 + 8*K2*K6*K8 + 16*K3**2*K6 - 2104*K3**2 - 1090*K4**2 - 460*K5**2 - 54*K6**2 - 8*K7**2 - 2*K8**2 + 3778 |
| 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}], [{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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {3, 4}, {1, 2}]] |
| If K is slice | False |