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Flat knot 6.94

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,0,3,2,2,4,1,1,1,1,0,1,1,1,0,-1]
Flat knots (up to 7 crossings) with same phi are :['6.94']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']
Outer characteristic polynomial of the knot is: t^7+79t^5+56t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.94']
2-strand cable arrow polynomial of the knot is: -192K14K2*2 + 320K1*4K2 - 736K14 + 160K1*3K2K3 - 64K1*3K3 - 256K12K2*4 + 640K1*2K2*3 + 64K1*2K2*2K4 - 2576K12K2*2 - 320K1*2K2K4 + 2792K1*2K2 - 64K12K3*2 - 1772K1*2 + 512K1K23K3 + 224K1K2*2K3K4 - 416K1K22K3 + 32K1K2*2K4K5 - 128K1K22K5 - 96K1K2K3K4 + 2544K1K2K3 + 312K1K3K4 + 40K1K4K5 + 8K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 744K24 + 96K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 528K22K3*2 - 304K2*2K4*2 + 944K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 1386K2*2 + 464K2K3K5 + 96K2K4K6 + 32K2K5K7 - 776K32 - 332K4*2 - 140K5*2 - 14K6**2 + 1682
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.94']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4678', 'vk6.4973', 'vk6.6148', 'vk6.6629', 'vk6.8151', 'vk6.8559', 'vk6.9531', 'vk6.9882', 'vk6.17674', 'vk6.17723', 'vk6.22143', 'vk6.24237', 'vk6.28238', 'vk6.29661', 'vk6.29906', 'vk6.29939', 'vk6.30002', 'vk6.30065', 'vk6.36507', 'vk6.39694', 'vk6.41933', 'vk6.43606', 'vk6.46266', 'vk6.47871', 'vk6.48710', 'vk6.48917', 'vk6.49482', 'vk6.49699', 'vk6.51617', 'vk6.51650', 'vk6.51695', 'vk6.51720', 'vk6.55708', 'vk6.58788', 'vk6.60278', 'vk6.63247', 'vk6.63349', 'vk6.63395', 'vk6.65412', 'vk6.68550']
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: [-4,-1,0,1,2,2,0,3,2,2,4,1,1,1,1,0,1,1,1,0,-1]

Flat knots (up to 7 crossings) with same phi are :['6.94']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.94', '6.482', '6.492']

Outer characteristic polynomial of the knot is: t^7+79t^5+56t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.94']

2-strand cable arrow polynomial of the knot is: -192*K1**4*K2**2 + 320*K1**4*K2 - 736*K1**4 + 160*K1**3*K2*K3 - 64*K1**3*K3 - 256*K1**2*K2**4 + 640*K1**2*K2**3 + 64*K1**2*K2**2*K4 - 2576*K1**2*K2**2 - 320*K1**2*K2*K4 + 2792*K1**2*K2 - 64*K1**2*K3**2 - 1772*K1**2 + 512*K1*K2**3*K3 + 224*K1*K2**2*K3*K4 - 416*K1*K2**2*K3 + 32*K1*K2**2*K4*K5 - 128*K1*K2**2*K5 - 96*K1*K2*K3*K4 + 2544*K1*K2*K3 + 312*K1*K3*K4 + 40*K1*K4*K5 + 8*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 744*K2**4 + 96*K2**3*K3*K5 + 32*K2**3*K4*K6 - 32*K2**3*K6 - 528*K2**2*K3**2 - 304*K2**2*K4**2 + 944*K2**2*K4 - 112*K2**2*K5**2 - 8*K2**2*K6**2 - 1386*K2**2 + 464*K2*K3*K5 + 96*K2*K4*K6 + 32*K2*K5*K7 - 776*K3**2 - 332*K4**2 - 140*K5**2 - 14*K6**2 + 1682

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.94']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4678', 'vk6.4973', 'vk6.6148', 'vk6.6629', 'vk6.8151', 'vk6.8559', 'vk6.9531', 'vk6.9882', 'vk6.17674', 'vk6.17723', 'vk6.22143', 'vk6.24237', 'vk6.28238', 'vk6.29661', 'vk6.29906', 'vk6.29939', 'vk6.30002', 'vk6.30065', 'vk6.36507', 'vk6.39694', 'vk6.41933', 'vk6.43606', 'vk6.46266', 'vk6.47871', 'vk6.48710', 'vk6.48917', 'vk6.49482', 'vk6.49699', 'vk6.51617', 'vk6.51650', 'vk6.51695', 'vk6.51720', 'vk6.55708', 'vk6.58788', 'vk6.60278', 'vk6.63247', 'vk6.63349', 'vk6.63395', 'vk6.65412', 'vk6.68550']

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	O1O2O3O4O5O6U2U5U6U4U1U3
R3 orbit	{'O1O2O3O4O5U1U4U5U6U2O6U3', 'O1O2O3O4O5O6U2U5U6U4U1U3'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5O6U4U6U3U1U2U5
Gauss code of K*	O1O2O3O4O5O6U5U1U6U4U2U3
Gauss code of -K*	O1O2O3O4O5O6U4U5U3U1U6U2
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 -1 -4 2 1 0 2],[ 1 0 -3 2 1 0 2],[ 4 3 0 4 3 1 2],[-2 -2 -4 0 0 -1 1],[-1 -1 -3 0 0 -1 1],[ 0 0 -1 1 1 0 1],[-2 -2 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 1 0 -1 -2 -4],[-2 -1 0 -1 -1 -2 -2],[-1 0 1 0 -1 -1 -3],[ 0 1 1 1 0 0 -1],[ 1 2 2 1 0 0 -3],[ 4 4 2 3 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,-1,0,1,2,4,1,1,2,2,1,1,3,0,1,3]
Phi over symmetry	[-4,-1,0,1,2,2,0,3,2,2,4,1,1,1,1,0,1,1,1,0,-1]
Phi of -K	[-4,-1,0,1,2,2,0,3,2,2,4,1,1,1,1,0,1,1,1,0,-1]
Phi of K*	[-2,-2,-1,0,1,4,-1,0,1,1,4,1,1,1,2,0,1,2,1,3,0]
Phi of -K*	[-4,-1,0,1,2,2,3,1,3,2,4,0,1,2,2,1,1,1,1,0,-1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	3z^2+16z+21
Enhanced Jones-Krushkal polynomial	3w^3z^2+16w^2z+21w
Inner characteristic polynomial	t^6+53t^4+18t^2+1
Outer characteristic polynomial	t^7+79t^5+56t^3+4t
Flat arrow polynomial	4K13 + 4K1*2K2 - 6K12 - 4K1K2 - 2K1K3 - K1 + 2K2 + K3 + 3
2-strand cable arrow polynomial	-192K14K2*2 + 320K1*4K2 - 736K14 + 160K1*3K2K3 - 64K1*3K3 - 256K12K2*4 + 640K1*2K2*3 + 64K1*2K2*2K4 - 2576K12K2*2 - 320K1*2K2K4 + 2792K1*2K2 - 64K12K3*2 - 1772K1*2 + 512K1K23K3 + 224K1K2*2K3K4 - 416K1K22K3 + 32K1K2*2K4K5 - 128K1K22K5 - 96K1K2K3K4 + 2544K1K2K3 + 312K1K3K4 + 40K1K4K5 + 8K1K5K6 - 32K26 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 744K24 + 96K2*3K3K5 + 32K2*3K4K6 - 32K2*3K6 - 528K22K3*2 - 304K2*2K4*2 + 944K2*2K4 - 112K22K5*2 - 8K2*2K6*2 - 1386K2*2 + 464K2K3K5 + 96K2K4K6 + 32K2K5K7 - 776K32 - 332K4*2 - 140K5*2 - 14K6**2 + 1682
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}], [{5, 6}, {4}, {3}, {2}, {1}], [{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}, {2, 5}, {4}, {3}, {1}], [{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}, {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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