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Flat knot 6.1150

Min(phi) over symmetries of the knot is: [-3,-1,0,0,2,2,0,1,3,1,2,0,0,1,0,1,0,1,2,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.1150']
Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + K4 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.1150', '6.1153']
Outer characteristic polynomial of the knot is: t^7+41t^5+112t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1150']
2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 320*K1**3*K2*K3 - 384*K1**2*K2**4 + 288*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 2208*K1**2*K2**2 + 2200*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K4**2 - 1664*K1**2 + 960*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2944*K1*K2*K3 + 368*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 816*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1008*K2**2*K3**2 - 136*K2**2*K4**2 + 416*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 1116*K2**2 + 440*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 48*K3**2*K6 - 1100*K3**2 - 296*K4**2 - 100*K5**2 - 68*K6**2 - 2*K8**2 + 1832
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1150']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.3611', 'vk6.3680', 'vk6.3873', 'vk6.3996', 'vk6.7037', 'vk6.7076', 'vk6.7253', 'vk6.7370', 'vk6.17701', 'vk6.17748', 'vk6.24252', 'vk6.24311', 'vk6.36551', 'vk6.36626', 'vk6.43661', 'vk6.43766', 'vk6.48239', 'vk6.48304', 'vk6.48389', 'vk6.48420', 'vk6.49999', 'vk6.50030', 'vk6.50115', 'vk6.50144', 'vk6.55741', 'vk6.55796', 'vk6.60317', 'vk6.60398', 'vk6.65441', 'vk6.65468', 'vk6.68573', 'vk6.68600']
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 O1O2O3O4U1U3U4O5O6U5U2U6
R3 orbit {'O1O2O3O4U1U3U4O5O6U5U2U6'}
R3 orbit length 1
Gauss code of -K O1O2O3O4U5U3U6O5O6U1U2U4
Gauss code of K* O1O2O3U4U5O4O6O5U1U6U2U3
Gauss code of -K* O1O2O3U1U3O4O5O6U4U5U2U6
Diagrammatic symmetry type c
Flat genus of the diagram 2
If K is checkerboard colorable False
If K is almost classical False
Based matrix from Gauss code [[ 0 -3 0 0 2 -1 2],[ 3 0 3 1 2 0 1],[ 0 -3 0 -1 1 0 2],[ 0 -1 1 0 1 0 0],[-2 -2 -1 -1 0 0 0],[ 1 0 0 0 0 0 1],[-2 -1 -2 0 0 -1 0]]
Primitive based matrix [[ 0 2 2 0 0 -1 -3],[-2 0 0 0 -2 -1 -1],[-2 0 0 -1 -1 0 -2],[ 0 0 1 0 1 0 -1],[ 0 2 1 -1 0 0 -3],[ 1 1 0 0 0 0 0],[ 3 1 2 1 3 0 0]]
If based matrix primitive True
Phi of primitive based matrix [-2,-2,0,0,1,3,0,0,2,1,1,1,1,0,2,-1,0,1,0,3,0]
Phi over symmetry [-3,-1,0,0,2,2,0,1,3,1,2,0,0,1,0,1,0,1,2,1,0]
Phi of -K [-3,-1,0,0,2,2,2,0,2,3,4,1,1,3,2,1,1,0,1,2,0]
Phi of K* [-2,-2,0,0,1,3,0,0,2,2,4,1,1,3,3,-1,1,0,1,2,2]
Phi of -K* [-3,-1,0,0,2,2,0,1,3,1,2,0,0,1,0,1,0,1,2,1,0]
Symmetry type of based matrix c
u-polynomial t^3-2t^2+t
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial -4w^3z+15w^2z+23w
Inner characteristic polynomial t^6+23t^4+40t^2
Outer characteristic polynomial t^7+41t^5+112t^3
Flat arrow polynomial 4*K1**3 + 4*K1**2*K2 - 8*K1**2 - 2*K1*K2 - 4*K1*K3 - 2*K1 + 4*K2 + K4 + 4
2-strand cable arrow polynomial -192*K1**4*K2**2 + 224*K1**4*K2 - 864*K1**4 + 128*K1**3*K2**3*K3 + 320*K1**3*K2*K3 - 384*K1**2*K2**4 + 288*K1**2*K2**3 - 192*K1**2*K2**2*K3**2 - 2208*K1**2*K2**2 + 2200*K1**2*K2 - 320*K1**2*K3**2 - 32*K1**2*K4**2 - 1664*K1**2 + 960*K1*K2**3*K3 + 96*K1*K2*K3**3 + 2944*K1*K2*K3 + 368*K1*K3*K4 + 64*K1*K4*K5 + 24*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 96*K2**4*K4 - 816*K2**4 + 96*K2**3*K3*K5 + 64*K2**3*K4*K6 - 1008*K2**2*K3**2 - 136*K2**2*K4**2 + 416*K2**2*K4 - 48*K2**2*K5**2 - 48*K2**2*K6**2 - 1116*K2**2 + 440*K2*K3*K5 + 120*K2*K4*K6 + 8*K2*K5*K7 + 16*K2*K6*K8 - 32*K3**4 + 48*K3**2*K6 - 1100*K3**2 - 296*K4**2 - 100*K5**2 - 68*K6**2 - 2*K8**2 + 1832
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {4, 5}, {2, 3}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {3, 5}, {1, 2}], [{5, 6}, {1, 4}, {2, 3}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {3, 5}, {4}, {1, 2}]]
If K is slice False
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