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Flat knot 6.978

Min(phi) over symmetries of the knot is: [-3,-1,0,1,1,2,-1,2,2,2,4,2,0,1,1,0,0,2,1,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.978']
Arrow polynomial of the knot is: -6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']
Outer characteristic polynomial of the knot is: t^7+50t^5+98t^3+14t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.978']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 608K1*4K2 - 1216K14 + 512K1*3K2K3 - 640K1*3K3 + 608K12K2*3 - 4976K1*2K2*2 + 32K1*2K2K32 - 640K1*2K2K4 + 7248K1*2K2 - 672K12K3*2 - 16K1*2K4*2 - 6256K1*2 + 160K1K23K3 - 800K1K2*2K3 - 128K1K2*2K5 - 128K1K2K3K4 + 7632K1K2K3 + 1368K1K3K4 + 176K1K4K5 + 24K1K5K6 - 648K24 - 272K2*2K3*2 - 8K2*2K4*2 + 1008K2*2K4 - 4446K22 + 408K2K3K5 + 16K2K4K6 - 2648K3*2 - 674K4*2 - 192K5*2 - 18K6**2 + 4768
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.978']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71639', 'vk6.71648', 'vk6.71650', 'vk6.71661', 'vk6.71818', 'vk6.71828', 'vk6.72240', 'vk6.72249', 'vk6.72252', 'vk6.72264', 'vk6.72370', 'vk6.72375', 'vk6.77263', 'vk6.77271', 'vk6.77361', 'vk6.77365', 'vk6.77367', 'vk6.77381', 'vk6.77611', 'vk6.77614', 'vk6.77705', 'vk6.77708', 'vk6.77709', 'vk6.77724', 'vk6.81405', 'vk6.81416', 'vk6.81440', 'vk6.86947', 'vk6.87155', 'vk6.87160', 'vk6.87999', 'vk6.89553']
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: [-3,-1,0,1,1,2,-1,2,2,2,4,2,0,1,1,0,0,2,1,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.978']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.323', '6.380', '6.444', '6.472', '6.523', '6.579', '6.592', '6.595', '6.609', '6.614', '6.620', '6.644', '6.648', '6.669', '6.671', '6.681', '6.693', '6.724', '6.725', '6.757', '6.766', '6.785', '6.786', '6.797', '6.798', '6.816', '6.833', '6.972', '6.978', '6.1056', '6.1064', '6.1066', '6.1087', '6.1094', '6.1273', '6.1277', '6.1282', '6.1295', '6.1300', '6.1313', '6.1344', '6.1353', '6.1354']

Outer characteristic polynomial of the knot is: t^7+50t^5+98t^3+14t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.978']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 608*K1**4*K2 - 1216*K1**4 + 512*K1**3*K2*K3 - 640*K1**3*K3 + 608*K1**2*K2**3 - 4976*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 640*K1**2*K2*K4 + 7248*K1**2*K2 - 672*K1**2*K3**2 - 16*K1**2*K4**2 - 6256*K1**2 + 160*K1*K2**3*K3 - 800*K1*K2**2*K3 - 128*K1*K2**2*K5 - 128*K1*K2*K3*K4 + 7632*K1*K2*K3 + 1368*K1*K3*K4 + 176*K1*K4*K5 + 24*K1*K5*K6 - 648*K2**4 - 272*K2**2*K3**2 - 8*K2**2*K4**2 + 1008*K2**2*K4 - 4446*K2**2 + 408*K2*K3*K5 + 16*K2*K4*K6 - 2648*K3**2 - 674*K4**2 - 192*K5**2 - 18*K6**2 + 4768

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.978']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71639', 'vk6.71648', 'vk6.71650', 'vk6.71661', 'vk6.71818', 'vk6.71828', 'vk6.72240', 'vk6.72249', 'vk6.72252', 'vk6.72264', 'vk6.72370', 'vk6.72375', 'vk6.77263', 'vk6.77271', 'vk6.77361', 'vk6.77365', 'vk6.77367', 'vk6.77381', 'vk6.77611', 'vk6.77614', 'vk6.77705', 'vk6.77708', 'vk6.77709', 'vk6.77724', 'vk6.81405', 'vk6.81416', 'vk6.81440', 'vk6.86947', 'vk6.87155', 'vk6.87160', 'vk6.87999', 'vk6.89553']

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	O1O2O3O4U1U3O5U4O6U2U5U6
R3 orbit	{'O1O2O3O4U1U3O5U4O6U2U5U6'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U5U6U3O5U1O6U2U4
Gauss code of K*	O1O2O3U4U1U5U6O4O5U2O6U3
Gauss code of -K*	O1O2O3U1O4U2O5O6U4U5U3U6
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 -3 -1 0 1 1 2],[ 3 0 3 1 2 2 1],[ 1 -3 0 -1 1 2 2],[ 0 -1 1 0 1 1 0],[-1 -2 -1 -1 0 1 1],[-1 -2 -2 -1 -1 0 1],[-2 -1 -2 0 -1 -1 0]]
Primitive based matrix	[[ 0 2 1 1 0 -1 -3],[-2 0 -1 -1 0 -2 -1],[-1 1 0 1 -1 -1 -2],[-1 1 -1 0 -1 -2 -2],[ 0 0 1 1 0 1 -1],[ 1 2 1 2 -1 0 -3],[ 3 1 2 2 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,0,1,3,1,1,0,2,1,-1,1,1,2,1,2,2,-1,1,3]
Phi over symmetry	[-3,-1,0,1,1,2,-1,2,2,2,4,2,0,1,1,0,0,2,1,0,0]
Phi of -K	[-3,-1,0,1,1,2,-1,2,2,2,4,2,0,1,1,0,0,2,1,0,0]
Phi of K*	[-2,-1,-1,0,1,3,0,0,2,1,4,-1,0,0,2,0,1,2,2,2,-1]
Phi of -K*	[-3,-1,0,1,1,2,3,1,2,2,1,-1,1,2,2,1,1,0,1,1,1]
Symmetry type of based matrix	c
u-polynomial	t^3-t^2-t
Normalized Jones-Krushkal polynomial	5z^2+26z+33
Enhanced Jones-Krushkal polynomial	5w^3z^2+26w^2z+33w
Inner characteristic polynomial	t^6+34t^4+25t^2+1
Outer characteristic polynomial	t^7+50t^5+98t^3+14t
Flat arrow polynomial	-6K12 - 2K1K2 + K1 + 3K2 + K3 + 4
2-strand cable arrow polynomial	-192K14K2*2 + 608K1*4K2 - 1216K14 + 512K1*3K2K3 - 640K1*3K3 + 608K12K2*3 - 4976K1*2K2*2 + 32K1*2K2K32 - 640K1*2K2K4 + 7248K1*2K2 - 672K12K3*2 - 16K1*2K4*2 - 6256K1*2 + 160K1K23K3 - 800K1K2*2K3 - 128K1K2*2K5 - 128K1K2K3K4 + 7632K1K2K3 + 1368K1K3K4 + 176K1K4K5 + 24K1K5K6 - 648K24 - 272K2*2K3*2 - 8K2*2K4*2 + 1008K2*2K4 - 4446K22 + 408K2K3K5 + 16K2K4K6 - 2648K3*2 - 674K4*2 - 192K5*2 - 18K6**2 + 4768
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}], [{2, 6}, {5}, {4}, {3}, {1}], [{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}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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