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Flat knot 6.7

Min(phi) over symmetries of the knot is: [-5,-3,0,2,3,3,1,2,5,3,4,1,4,2,3,2,1,2,1,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.7']
Arrow polynomial of the knot is: -2K12 - 2K1*K4 + K2 + K3 + K5 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.7', '6.28']
Outer characteristic polynomial of the knot is: t^7+152t^5+227t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.7']
2-strand cable arrow polynomial of the knot is: -1088K12K2*2 + 2032K1*2K2 - 544K12K3*2 - 64K1*2K3K5 - 2616K1*2 + 704K1K23K3 + 64K1K2*2K3K4 - 736K1K22K3 - 192K1K2*2K5 + 480K1K2K33 - 384K1K2K3K4 - 160K1K2K3K6 + 4184K1K2K3 - 96K1K3*2K5 + 992K1K3K4 + 192K1K4K5 + 8K1K7K8 - 2K10*2 + 8K10K2K8 - 1496K24 + 320K2*3K3K5 - 256K2*2K3*4 + 160K2*2K3*2K6 - 2464K22K3*2 - 64K2*2K3K7 - 32K2*2K4*2 + 1696K2*2K4 - 256K22K5*2 - 32K2*2K6*2 - 8K2*2K8*2 - 2196K2*2 + 32K2K33K5 - 160K2K3*2K4 - 64K2K3K4K5 + 2072K2K3K5 + 64K2K4K6 + 176K2K5K7 + 8K2K6K8 - 320K34 + 272K3*2K6 - 1744K32 + 40K3K4K7 + 8K3K5K8 - 694K4*2 - 472K5*2 - 42K6*2 - 56K7*2 - 12K8**2 + 2736
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.7']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73982', 'vk6.73986', 'vk6.74503', 'vk6.74507', 'vk6.75959', 'vk6.75967', 'vk6.76713', 'vk6.76719', 'vk6.78949', 'vk6.78955', 'vk6.79496', 'vk6.79502', 'vk6.80478', 'vk6.80485', 'vk6.80953', 'vk6.80957', 'vk6.83006', 'vk6.83094', 'vk6.83644', 'vk6.83787', 'vk6.83939', 'vk6.84104', 'vk6.84255', 'vk6.85181', 'vk6.85539', 'vk6.85863', 'vk6.86262', 'vk6.86574', 'vk6.86741', 'vk6.87454', 'vk6.88295', 'vk6.89740']
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

Min(phi) over symmetries of the knot is: [-5,-3,0,2,3,3,1,2,5,3,4,1,4,2,3,2,1,2,1,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.7']

Arrow polynomial of the knot is: -2*K1**2 - 2*K1*K4 + K2 + K3 + K5 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.7', '6.28']

Outer characteristic polynomial of the knot is: t^7+152t^5+227t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.7']

2-strand cable arrow polynomial of the knot is: -1088*K1**2*K2**2 + 2032*K1**2*K2 - 544*K1**2*K3**2 - 64*K1**2*K3*K5 - 2616*K1**2 + 704*K1*K2**3*K3 + 64*K1*K2**2*K3*K4 - 736*K1*K2**2*K3 - 192*K1*K2**2*K5 + 480*K1*K2*K3**3 - 384*K1*K2*K3*K4 - 160*K1*K2*K3*K6 + 4184*K1*K2*K3 - 96*K1*K3**2*K5 + 992*K1*K3*K4 + 192*K1*K4*K5 + 8*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 1496*K2**4 + 320*K2**3*K3*K5 - 256*K2**2*K3**4 + 160*K2**2*K3**2*K6 - 2464*K2**2*K3**2 - 64*K2**2*K3*K7 - 32*K2**2*K4**2 + 1696*K2**2*K4 - 256*K2**2*K5**2 - 32*K2**2*K6**2 - 8*K2**2*K8**2 - 2196*K2**2 + 32*K2*K3**3*K5 - 160*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 2072*K2*K3*K5 + 64*K2*K4*K6 + 176*K2*K5*K7 + 8*K2*K6*K8 - 320*K3**4 + 272*K3**2*K6 - 1744*K3**2 + 40*K3*K4*K7 + 8*K3*K5*K8 - 694*K4**2 - 472*K5**2 - 42*K6**2 - 56*K7**2 - 12*K8**2 + 2736

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.7']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.73982', 'vk6.73986', 'vk6.74503', 'vk6.74507', 'vk6.75959', 'vk6.75967', 'vk6.76713', 'vk6.76719', 'vk6.78949', 'vk6.78955', 'vk6.79496', 'vk6.79502', 'vk6.80478', 'vk6.80485', 'vk6.80953', 'vk6.80957', 'vk6.83006', 'vk6.83094', 'vk6.83644', 'vk6.83787', 'vk6.83939', 'vk6.84104', 'vk6.84255', 'vk6.85181', 'vk6.85539', 'vk6.85863', 'vk6.86262', 'vk6.86574', 'vk6.86741', 'vk6.87454', 'vk6.88295', 'vk6.89740']

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	O1O2O3O4O5O6U1U2U4U6U5U3
R3 orbit	{'O1O2O3O4O5O6U1U2U4U6U5U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U2U1U3U5U6
Gauss code of K*	O1O2O3O4O5O6U1U2U6U3U5U4
Gauss code of -K*	O1O2O3O4O5O6U3U2U4U1U5U6
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 -5 -3 2 0 3 3],[ 5 0 1 5 2 4 3],[ 3 -1 0 4 1 3 2],[-2 -5 -4 0 -2 1 1],[ 0 -2 -1 2 0 2 1],[-3 -4 -3 -1 -2 0 0],[-3 -3 -2 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 3 2 0 -3 -5],[-3 0 0 -1 -1 -2 -3],[-3 0 0 -1 -2 -3 -4],[-2 1 1 0 -2 -4 -5],[ 0 1 2 2 0 -1 -2],[ 3 2 3 4 1 0 -1],[ 5 3 4 5 2 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-3,-2,0,3,5,0,1,1,2,3,1,2,3,4,2,4,5,1,2,1]
Phi over symmetry	[-5,-3,0,2,3,3,1,2,5,3,4,1,4,2,3,2,1,2,1,1,0]
Phi of -K	[-5,-3,0,2,3,3,1,3,2,4,5,2,1,3,4,0,1,2,0,0,0]
Phi of K*	[-3,-3,-2,0,3,5,0,0,1,3,4,0,2,4,5,0,1,2,2,3,1]
Phi of -K*	[-5,-3,0,2,3,3,1,2,5,3,4,1,4,2,3,2,1,2,1,1,0]
Symmetry type of based matrix	c
u-polynomial	t^5-t^3-t^2
Normalized Jones-Krushkal polynomial	7z^2+24z+21
Enhanced Jones-Krushkal polynomial	-2w^4z^2+9w^3z^2-2w^3z+26w^2z+21w
Inner characteristic polynomial	t^6+96t^4+34t^2+1
Outer characteristic polynomial	t^7+152t^5+227t^3+9t
Flat arrow polynomial	-2K12 - 2K1*K4 + K2 + K3 + K5 + 2
2-strand cable arrow polynomial	-1088K12K2*2 + 2032K1*2K2 - 544K12K3*2 - 64K1*2K3K5 - 2616K1*2 + 704K1K23K3 + 64K1K2*2K3K4 - 736K1K22K3 - 192K1K2*2K5 + 480K1K2K33 - 384K1K2K3K4 - 160K1K2K3K6 + 4184K1K2K3 - 96K1K3*2K5 + 992K1K3K4 + 192K1K4K5 + 8K1K7K8 - 2K10*2 + 8K10K2K8 - 1496K24 + 320K2*3K3K5 - 256K2*2K3*4 + 160K2*2K3*2K6 - 2464K22K3*2 - 64K2*2K3K7 - 32K2*2K4*2 + 1696K2*2K4 - 256K22K5*2 - 32K2*2K6*2 - 8K2*2K8*2 - 2196K2*2 + 32K2K33K5 - 160K2K3*2K4 - 64K2K3K4K5 + 2072K2K3K5 + 64K2K4K6 + 176K2K5K7 + 8K2K6K8 - 320K34 + 272K3*2K6 - 1744K32 + 40K3K4K7 + 8K3K5K8 - 694K4*2 - 472K5*2 - 42K6*2 - 56K7*2 - 12K8**2 + 2736
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}], [{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}, {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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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