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Flat knot 6.45

Min(phi) over symmetries of the knot is: [-5,-1,0,1,2,3,2,1,5,4,3,0,2,2,2,1,1,1,1,2,1]
Flat knots (up to 7 crossings) with same phi are :['6.45']
Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K2 + 2*K1 - 2*K2*K3 + K2 + K3 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.45']
Outer characteristic polynomial of the knot is: t^7+116t^5+65t^3
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.40', '6.45']
2-strand cable arrow polynomial of the knot is: -64*K1**6 + 64*K1**4*K2 - 704*K1**4 - 224*K1**2*K2**2 + 864*K1**2*K2 - 544*K1**2*K3**2 - 144*K1**2*K4**2 - 884*K1**2 + 64*K1*K2*K3**3 + 1152*K1*K2*K3 + 64*K1*K3**3*K4 + 1184*K1*K3*K4 + 144*K1*K4*K5 + 24*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K4*K6 - 8*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 72*K2**2*K4 - 696*K2**2 + 88*K2*K3*K5 + 40*K2*K4*K6 - 128*K3**4 - 160*K3**2*K4**2 + 80*K3**2*K6 - 792*K3**2 + 96*K3*K4*K7 + 8*K3*K5*K8 - 8*K4**2*K6**2 - 506*K4**2 - 68*K5**2 - 54*K6**2 - 32*K7**2 - 8*K8**2 + 1168
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.45']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20013', 'vk6.20091', 'vk6.21283', 'vk6.21371', 'vk6.27060', 'vk6.27152', 'vk6.28763', 'vk6.28839', 'vk6.38453', 'vk6.38549', 'vk6.40640', 'vk6.40744', 'vk6.45333', 'vk6.45445', 'vk6.47100', 'vk6.47185', 'vk6.56812', 'vk6.56896', 'vk6.57944', 'vk6.58032', 'vk6.61326', 'vk6.61422', 'vk6.62500', 'vk6.62577', 'vk6.66524', 'vk6.66596', 'vk6.67311', 'vk6.67385', 'vk6.69166', 'vk6.69244', 'vk6.69915', 'vk6.69983']
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 O1O2O3O4O5O6U1U5U3U6U4U2
R3 orbit {'O1O2O3O4O5O6U1U5U3U6U4U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5O6U5U3U1U4U2U6
Gauss code of K* O1O2O3O4O5O6U1U6U3U5U2U4
Gauss code of -K* O1O2O3O4O5O6U3U5U2U4U1U6
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 -5 1 -1 2 0 3],[ 5 0 5 2 4 1 3],[-1 -5 0 -2 1 -1 2],[ 1 -2 2 0 2 0 2],[-2 -4 -1 -2 0 -1 1],[ 0 -1 1 0 1 0 1],[-3 -3 -2 -2 -1 -1 0]]
Primitive based matrix [[ 0 3 2 1 0 -1 -5],[-3 0 -1 -2 -1 -2 -3],[-2 1 0 -1 -1 -2 -4],[-1 2 1 0 -1 -2 -5],[ 0 1 1 1 0 0 -1],[ 1 2 2 2 0 0 -2],[ 5 3 4 5 1 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-3,-2,-1,0,1,5,1,2,1,2,3,1,1,2,4,1,2,5,0,1,2]
Phi over symmetry [-5,-1,0,1,2,3,2,1,5,4,3,0,2,2,2,1,1,1,1,2,1]
Phi of -K [-5,-1,0,1,2,3,2,4,1,3,5,1,0,1,2,0,1,2,0,0,0]
Phi of K* [-3,-2,-1,0,1,5,0,0,2,2,5,0,1,1,3,0,0,1,1,4,2]
Phi of -K* [-5,-1,0,1,2,3,2,1,5,4,3,0,2,2,2,1,1,1,1,2,1]
Symmetry type of based matrix c
u-polynomial t^5-t^3-t^2
Normalized Jones-Krushkal polynomial 11z+23
Enhanced Jones-Krushkal polynomial 11w^2z+23w
Inner characteristic polynomial t^6+76t^4
Outer characteristic polynomial t^7+116t^5+65t^3
Flat arrow polynomial -2*K1**2 - 2*K1*K2 + 2*K1 - 2*K2*K3 + K2 + K3 + K5 + 2
2-strand cable arrow polynomial -64*K1**6 + 64*K1**4*K2 - 704*K1**4 - 224*K1**2*K2**2 + 864*K1**2*K2 - 544*K1**2*K3**2 - 144*K1**2*K4**2 - 884*K1**2 + 64*K1*K2*K3**3 + 1152*K1*K2*K3 + 64*K1*K3**3*K4 + 1184*K1*K3*K4 + 144*K1*K4*K5 + 24*K1*K6*K7 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K4*K6 - 8*K2**4 - 96*K2**2*K3**2 - 8*K2**2*K4**2 + 72*K2**2*K4 - 696*K2**2 + 88*K2*K3*K5 + 40*K2*K4*K6 - 128*K3**4 - 160*K3**2*K4**2 + 80*K3**2*K6 - 792*K3**2 + 96*K3*K4*K7 + 8*K3*K5*K8 - 8*K4**2*K6**2 - 506*K4**2 - 68*K5**2 - 54*K6**2 - 32*K7**2 - 8*K8**2 + 1168
Genus of based matrix 1
Fillings of based matrix [[{1, 6}, {2, 5}, {3, 4}], [{3, 6}, {4, 5}, {1, 2}], [{6}, {2, 5}, {3, 4}, {1}]]
If K is slice False
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