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Flat knot 6.1243

Min(phi) over symmetries of the knot is: [-2,-2,1,1,1,1,0,0,1,2,2,1,2,2,2,-1,-1,0,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.1243']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.565', '6.1229', '6.1243', '6.1920']
Outer characteristic polynomial of the knot is: t^7+36t^5+54t^3+7t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1243']
2-strand cable arrow polynomial of the knot is: -1152*K1**4*K2**2 + 6144*K1**4*K2 - 5984*K1**4 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 1984*K1**3*K3 - 128*K1**2*K2**4 + 3680*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 9840*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 1408*K1**2*K2*K4 + 7568*K1**2*K2 - 928*K1**2*K3**2 - 320*K1**2*K3*K5 - 128*K1**2*K4**2 - 2288*K1**2 + 704*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7128*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1816*K1*K3*K4 + 632*K1*K4*K5 + 144*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1808*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 816*K2**2*K3**2 - 64*K2**2*K3*K7 - 368*K2**2*K4**2 - 32*K2**2*K4*K8 + 1920*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 2374*K2**2 - 32*K2*K3**2*K4 + 1008*K2*K3*K5 + 432*K2*K4*K6 + 120*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1516*K3**2 - 788*K4**2 - 380*K5**2 - 122*K6**2 - 24*K7**2 - 2*K8**2 + 3044
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1243']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3250', 'vk6.3274', 'vk6.3299', 'vk6.3374', 'vk6.3409', 'vk6.3434', 'vk6.3478', 'vk6.3513', 'vk6.4625', 'vk6.5912', 'vk6.6031', 'vk6.7961', 'vk6.8080', 'vk6.9391', 'vk6.17841', 'vk6.17856', 'vk6.19064', 'vk6.19871', 'vk6.24358', 'vk6.25682', 'vk6.25694', 'vk6.26310', 'vk6.26755', 'vk6.37785', 'vk6.43783', 'vk6.43798', 'vk6.45047', 'vk6.48106', 'vk6.48122', 'vk6.48145', 'vk6.48195', 'vk6.50669']
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 O1O2O3O4U3U5U4O5O6U2U1U6
R3 orbit {'O1O2O3O4U3U5U4O5O6U2U1U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U4U3O5O6U1U6U2
Gauss code of K* O1O2O3U2U4O5O6O4U6U5U1U3
Gauss code of -K* O1O2O3U1U4O5O4O6U5U6U3U2
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 -1 -1 2 -1 2],[ 1 0 0 -1 2 0 2],[ 1 0 0 -1 2 0 1],[ 1 1 1 0 1 0 0],[-2 -2 -2 -1 0 -2 0],[ 1 0 0 0 2 0 2],[-2 -2 -1 0 0 -2 0]]
Primitive based matrix [[ 0 2 2 -1 -1 -1 -1],[-2 0 0 0 -1 -2 -2],[-2 0 0 -1 -2 -2 -2],[ 1 0 1 0 1 1 0],[ 1 1 2 -1 0 0 0],[ 1 2 2 -1 0 0 0],[ 1 2 2 0 0 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,1,1,1,1,0,0,1,2,2,1,2,2,2,-1,-1,0,0,0,0]
Phi over symmetry [-2,-2,1,1,1,1,0,0,1,2,2,1,2,2,2,-1,-1,0,0,0,0]
Phi of -K [-1,-1,-1,-1,2,2,-1,-1,0,2,3,0,0,1,1,0,1,2,1,1,0]
Phi of K* [-2,-2,1,1,1,1,0,1,1,1,2,1,1,2,3,0,0,-1,0,0,-1]
Phi of -K* [-1,-1,-1,-1,2,2,-1,0,0,1,2,0,1,0,1,0,2,2,2,2,0]
Symmetry type of based matrix c
u-polynomial -2t^2+4t
Normalized Jones-Krushkal polynomial 9z^2+30z+25
Enhanced Jones-Krushkal polynomial 9w^3z^2+30w^2z+25w
Inner characteristic polynomial t^6+24t^4+34t^2+4
Outer characteristic polynomial t^7+36t^5+54t^3+7t
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 4*K1*K2 - 4*K1*K3 - K1 + K3 + K4
2-strand cable arrow polynomial -1152*K1**4*K2**2 + 6144*K1**4*K2 - 5984*K1**4 - 384*K1**3*K2**2*K3 + 1280*K1**3*K2*K3 - 1984*K1**3*K3 - 128*K1**2*K2**4 + 3680*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 9840*K1**2*K2**2 + 384*K1**2*K2*K3**2 - 1408*K1**2*K2*K4 + 7568*K1**2*K2 - 928*K1**2*K3**2 - 320*K1**2*K3*K5 - 128*K1**2*K4**2 - 2288*K1**2 + 704*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 2208*K1*K2**2*K3 + 96*K1*K2**2*K4*K5 - 416*K1*K2**2*K5 - 576*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 7128*K1*K2*K3 - 288*K1*K2*K4*K5 - 32*K1*K2*K4*K7 - 64*K1*K2*K5*K6 + 1816*K1*K3*K4 + 632*K1*K4*K5 + 144*K1*K5*K6 + 24*K1*K6*K7 - 32*K2**6 - 32*K2**4*K4**2 + 192*K2**4*K4 - 1808*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 816*K2**2*K3**2 - 64*K2**2*K3*K7 - 368*K2**2*K4**2 - 32*K2**2*K4*K8 + 1920*K2**2*K4 - 128*K2**2*K5**2 - 16*K2**2*K6**2 - 2374*K2**2 - 32*K2*K3**2*K4 + 1008*K2*K3*K5 + 432*K2*K4*K6 + 120*K2*K5*K7 + 16*K2*K6*K8 + 8*K3**2*K6 - 1516*K3**2 - 788*K4**2 - 380*K5**2 - 122*K6**2 - 24*K7**2 - 2*K8**2 + 3044
Genus of based matrix 1
Fillings of based matrix [[{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{4, 6}, {2, 5}, {1, 3}]]
If K is slice False
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