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Flat knot 6.40

Min(phi) over symmetries of the knot is: [-5,-1,0,1,2,3,1,3,5,2,4,1,2,1,2,1,1,2,1,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.40']
Arrow polynomial of the knot is: -6K12 - 4K1K2 - 2K1K4 + 2K1 + 3K2 + 3K3 + K5 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.40']
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: -64K16 + 224K1*4K2 - 896K14 + 32K1*3K2K3 - 512K1*2K2*2 + 1712K1*2K2 - 608K12K3*2 - 144K1*2K4*2 - 1532K1*2 + 96K1K2K3*3 + 1488K1K2K3 + 32K1K3*3K4 + 824K1K3K4 + 304K1K4K5 + 8K1K5K6 + 16K1K6K7 + 16K1K7K8 - 2K10*2 + 8K10K2K8 - 56K24 - 192K2*2K3*2 - 16K2*2K4*2 + 192K2*2K4 - 32K22K5*2 - 8K2*2K8*2 - 1360K2*2 + 320K2K3K5 + 80K2K4K6 + 48K2K5K7 + 24K2K6K8 - 128K3*4 - 32K3*2K4*2 + 56K3*2K6 - 768K32 + 8K3K4K7 + 16K3K5K8 - 470K4*2 - 260K5*2 - 70K6*2 - 32K7*2 - 28K8**2 + 1704
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.40']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20029', 'vk6.20075', 'vk6.21299', 'vk6.21355', 'vk6.27076', 'vk6.27136', 'vk6.28779', 'vk6.28823', 'vk6.38469', 'vk6.38533', 'vk6.40656', 'vk6.40728', 'vk6.45349', 'vk6.45429', 'vk6.47116', 'vk6.47169', 'vk6.56828', 'vk6.56880', 'vk6.57960', 'vk6.58016', 'vk6.61342', 'vk6.61406', 'vk6.62516', 'vk6.62561', 'vk6.66540', 'vk6.66580', 'vk6.67327', 'vk6.67369', 'vk6.69182', 'vk6.69228', 'vk6.69931', 'vk6.69967']
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,-1,0,1,2,3,1,3,5,2,4,1,2,1,2,1,1,2,1,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.40']

Arrow polynomial of the knot is: -6*K1**2 - 4*K1*K2 - 2*K1*K4 + 2*K1 + 3*K2 + 3*K3 + K5 + 4

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.40']

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 + 224*K1**4*K2 - 896*K1**4 + 32*K1**3*K2*K3 - 512*K1**2*K2**2 + 1712*K1**2*K2 - 608*K1**2*K3**2 - 144*K1**2*K4**2 - 1532*K1**2 + 96*K1*K2*K3**3 + 1488*K1*K2*K3 + 32*K1*K3**3*K4 + 824*K1*K3*K4 + 304*K1*K4*K5 + 8*K1*K5*K6 + 16*K1*K6*K7 + 16*K1*K7*K8 - 2*K10**2 + 8*K10*K2*K8 - 56*K2**4 - 192*K2**2*K3**2 - 16*K2**2*K4**2 + 192*K2**2*K4 - 32*K2**2*K5**2 - 8*K2**2*K8**2 - 1360*K2**2 + 320*K2*K3*K5 + 80*K2*K4*K6 + 48*K2*K5*K7 + 24*K2*K6*K8 - 128*K3**4 - 32*K3**2*K4**2 + 56*K3**2*K6 - 768*K3**2 + 8*K3*K4*K7 + 16*K3*K5*K8 - 470*K4**2 - 260*K5**2 - 70*K6**2 - 32*K7**2 - 28*K8**2 + 1704

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.40']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.20029', 'vk6.20075', 'vk6.21299', 'vk6.21355', 'vk6.27076', 'vk6.27136', 'vk6.28779', 'vk6.28823', 'vk6.38469', 'vk6.38533', 'vk6.40656', 'vk6.40728', 'vk6.45349', 'vk6.45429', 'vk6.47116', 'vk6.47169', 'vk6.56828', 'vk6.56880', 'vk6.57960', 'vk6.58016', 'vk6.61342', 'vk6.61406', 'vk6.62516', 'vk6.62561', 'vk6.66540', 'vk6.66580', 'vk6.67327', 'vk6.67369', 'vk6.69182', 'vk6.69228', 'vk6.69931', 'vk6.69967']

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	O1O2O3O4O5O6U1U4U6U3U5U2
R3 orbit	{'O1O2O3O4O5O6U1U4U6U3U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U2U4U1U3U6
Gauss code of K*	O1O2O3O4O5O6U1U6U4U2U5U3
Gauss code of -K*	O1O2O3O4O5O6U4U2U5U3U1U6
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 0 -1 3 2],[ 5 0 5 3 1 4 2],[-1 -5 0 -1 -2 2 1],[ 0 -3 1 0 -1 2 1],[ 1 -1 2 1 0 2 1],[-3 -4 -2 -2 -2 0 0],[-2 -2 -1 -1 -1 0 0]]
Primitive based matrix	[[ 0 3 2 1 0 -1 -5],[-3 0 0 -2 -2 -2 -4],[-2 0 0 -1 -1 -1 -2],[-1 2 1 0 -1 -2 -5],[ 0 2 1 1 0 -1 -3],[ 1 2 1 2 1 0 -1],[ 5 4 2 5 3 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,0,1,5,0,2,2,2,4,1,1,1,2,1,2,5,1,3,1]
Phi over symmetry	[-5,-1,0,1,2,3,1,3,5,2,4,1,2,1,2,1,1,2,1,2,0]
Phi of -K	[-5,-1,0,1,2,3,3,2,1,5,4,0,0,2,2,0,1,1,0,0,1]
Phi of K*	[-3,-2,-1,0,1,5,1,0,1,2,4,0,1,2,5,0,0,1,0,2,3]
Phi of -K*	[-5,-1,0,1,2,3,1,3,5,2,4,1,2,1,2,1,1,2,1,2,0]
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	-6K12 - 4K1K2 - 2K1K4 + 2K1 + 3K2 + 3K3 + K5 + 4
2-strand cable arrow polynomial	-64K16 + 224K1*4K2 - 896K14 + 32K1*3K2K3 - 512K1*2K2*2 + 1712K1*2K2 - 608K12K3*2 - 144K1*2K4*2 - 1532K1*2 + 96K1K2K3*3 + 1488K1K2K3 + 32K1K3*3K4 + 824K1K3K4 + 304K1K4K5 + 8K1K5K6 + 16K1K6K7 + 16K1K7K8 - 2K10*2 + 8K10K2K8 - 56K24 - 192K2*2K3*2 - 16K2*2K4*2 + 192K2*2K4 - 32K22K5*2 - 8K2*2K8*2 - 1360K2*2 + 320K2K3K5 + 80K2K4K6 + 48K2K5K7 + 24K2K6K8 - 128K3*4 - 32K3*2K4*2 + 56K3*2K6 - 768K32 + 8K3K4K7 + 16K3K5K8 - 470K4*2 - 260K5*2 - 70K6*2 - 32K7*2 - 28K8**2 + 1704
Genus of based matrix	1
Fillings of based matrix	[[{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {5}, {2, 4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {5}, {1, 4}, {3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {4, 5}, {1, 2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {4, 5}, {3}, {1, 2}]]
If K is slice	False

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