Table of flat knot invariants
Invariant Table Check a Knot Higher Crossing Crossref Virtual Knots Please cite FlatKnotInfo
Glossary Reference List

Flat knot 6.1151

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,0,0,1,1,-1,0,1]
Flat knots (up to 7 crossings) with same phi are :['6.1151', '7.24431']
Arrow polynomial of the knot is: -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.406', '6.410', '6.412', '6.1151', '6.1175', '6.1176']
Outer characteristic polynomial of the knot is: t^7+36t^5+66t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1151']
2-strand cable arrow polynomial of the knot is: -1408*K1**4*K2**2 + 2752*K1**4*K2 - 4160*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 - 1056*K1**3*K3 + 384*K1**2*K2**5 - 2944*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5408*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 1088*K1**2*K2**2*K4 - 13328*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 1472*K1**2*K2*K4 + 9288*K1**2*K2 - 928*K1**2*K3**2 - 32*K1**2*K3*K5 - 2324*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4704*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2400*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 1088*K1*K2**2*K5 + 64*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7984*K1*K2*K3 - 32*K1*K2*K4*K5 + 680*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 576*K2**4*K3**2 - 192*K2**4*K4**2 + 1664*K2**4*K4 - 4192*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 256*K2**3*K6 + 64*K2**2*K3**2*K4 - 1552*K2**2*K3**2 - 32*K2**2*K3*K7 - 376*K2**2*K4**2 + 2600*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 378*K2**2 + 624*K2*K3*K5 + 104*K2*K4*K6 + 8*K3**2*K6 - 1000*K3**2 - 238*K4**2 - 36*K5**2 - 14*K6**2 + 2564
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1151']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.44', 'vk6.89', 'vk6.186', 'vk6.245', 'vk6.280', 'vk6.659', 'vk6.666', 'vk6.1241', 'vk6.1332', 'vk6.1389', 'vk6.1426', 'vk6.1910', 'vk6.2364', 'vk6.2420', 'vk6.2623', 'vk6.2968', 'vk6.10090', 'vk6.10101', 'vk6.14589', 'vk6.15811', 'vk6.16214', 'vk6.17764', 'vk6.24268', 'vk6.29835', 'vk6.33399', 'vk6.33468', 'vk6.33539', 'vk6.36574', 'vk6.43687', 'vk6.53712', 'vk6.53771', 'vk6.63291']
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 O1O2O3O4U1U3U4O5O6U5U6U2
R3 orbit {'O1O2O3O4U1U3U4O5O6U5U6U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U3U5U6O5O6U1U2U4
Gauss code of K* O1O2O3U4U5O4O5O6U1U6U2U3
Gauss code of -K* O1O2O3U2U3O4O5O6U4U5U1U6
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 1 0 2 -1 1],[ 3 0 3 1 2 0 0],[-1 -3 0 -1 1 -1 1],[ 0 -1 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 1 0 1 0 0 0 1],[-1 0 -1 0 0 -1 0]]
Primitive based matrix [[ 0 2 1 1 0 -1 -3],[-2 0 0 -1 -1 0 -2],[-1 0 0 -1 0 -1 0],[-1 1 1 0 -1 -1 -3],[ 0 1 0 1 0 0 -1],[ 1 0 1 1 0 0 0],[ 3 2 0 3 1 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-1,-1,0,1,3,0,1,1,0,2,1,0,1,0,1,1,3,0,1,0]
Phi over symmetry [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,0,0,1,1,-1,0,1]
Phi of -K [-3,-1,0,1,1,2,2,2,1,4,3,1,1,1,3,0,1,1,-1,0,1]
Phi of K* [-2,-1,-1,0,1,3,0,1,1,3,3,1,0,1,1,1,1,4,1,2,2]
Phi of -K* [-3,-1,0,1,1,2,0,1,0,3,2,0,1,1,0,0,1,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^3-t^2-t
Normalized Jones-Krushkal polynomial 6z^2+25z+27
Enhanced Jones-Krushkal polynomial -2w^4z^2+8w^3z^2+25w^2z+27w
Inner characteristic polynomial t^6+20t^4+29t^2
Outer characteristic polynomial t^7+36t^5+66t^3+9t
Flat arrow polynomial -8*K1**4 + 8*K1**3 + 8*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 3*K2 + K3 + 4
2-strand cable arrow polynomial -1408*K1**4*K2**2 + 2752*K1**4*K2 - 4160*K1**4 - 384*K1**3*K2**2*K3 + 2112*K1**3*K2*K3 - 1056*K1**3*K3 + 384*K1**2*K2**5 - 2944*K1**2*K2**4 - 384*K1**2*K2**3*K4 + 5408*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 1088*K1**2*K2**2*K4 - 13328*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 128*K1**2*K2*K3*K5 - 1472*K1**2*K2*K4 + 9288*K1**2*K2 - 928*K1**2*K3**2 - 32*K1**2*K3*K5 - 2324*K1**2 + 256*K1*K2**5*K3 - 256*K1*K2**3*K3*K4 + 4704*K1*K2**3*K3 + 416*K1*K2**2*K3*K4 - 2400*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 1088*K1*K2**2*K5 + 64*K1*K2*K3**3 - 512*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7984*K1*K2*K3 - 32*K1*K2*K4*K5 + 680*K1*K3*K4 + 56*K1*K4*K5 - 128*K2**8 + 256*K2**6*K4 - 1472*K2**6 - 576*K2**4*K3**2 - 192*K2**4*K4**2 + 1664*K2**4*K4 - 4192*K2**4 + 416*K2**3*K3*K5 + 64*K2**3*K4*K6 - 256*K2**3*K6 + 64*K2**2*K3**2*K4 - 1552*K2**2*K3**2 - 32*K2**2*K3*K7 - 376*K2**2*K4**2 + 2600*K2**2*K4 - 48*K2**2*K5**2 - 8*K2**2*K6**2 - 378*K2**2 + 624*K2*K3*K5 + 104*K2*K4*K6 + 8*K3**2*K6 - 1000*K3**2 - 238*K4**2 - 36*K5**2 - 14*K6**2 + 2564
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {5}, {4}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {4}, {2, 3}, {1}], [{6}, {1, 5}, {4}, {2, 3}]]
If K is slice False
Contact