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Flat knot 6.713

Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,0,0,1,2,2,0,1,2,3,0,0,1,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.713']
Arrow polynomial of the knot is: 4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']
Outer characteristic polynomial of the knot is: t^7+39t^5+58t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.713']
2-strand cable arrow polynomial of the knot is: 96K14K2 - 736K14 + 32K1*3K2K3 - 96K1*3K3 + 128K12K2*3 - 2064K1*2K2*2 + 32K1*2K2K32 - 192K1*2K2K4 + 4000K1*2K2 - 96K12K3*2 - 16K1*2K4*2 - 3188K1*2 + 96K1K23K3 - 576K1K2*2K3 - 192K1K2*2K5 - 32K1K2K3K4 + 3280K1K2K3 + 552K1K3K4 + 80K1K4K5 - 32K2*6 + 64K2*4K4 - 424K24 - 32K2*3K6 - 128K22K3*2 - 16K2*2K4*2 + 752K2*2K4 - 2374K22 + 176K2K3K5 + 16K2K4K6 - 1104K3*2 - 342K4*2 - 52K5*2 - 2K6**2 + 2388
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.713']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71384', 'vk6.71443', 'vk6.71906', 'vk6.71965', 'vk6.72440', 'vk6.72591', 'vk6.72710', 'vk6.72803', 'vk6.72866', 'vk6.73019', 'vk6.73365', 'vk6.73526', 'vk6.74251', 'vk6.74380', 'vk6.74446', 'vk6.75059', 'vk6.75533', 'vk6.75836', 'vk6.76424', 'vk6.76635', 'vk6.77041', 'vk6.77742', 'vk6.77793', 'vk6.78255', 'vk6.78506', 'vk6.78639', 'vk6.78832', 'vk6.79299', 'vk6.79418', 'vk6.79834', 'vk6.79894', 'vk6.80267', 'vk6.80760', 'vk6.80866', 'vk6.85147', 'vk6.86524', 'vk6.87211', 'vk6.87348', 'vk6.89252', 'vk6.89440']
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: [-2,-2,0,1,1,2,0,0,1,2,2,0,1,2,3,0,0,1,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.713']

Arrow polynomial of the knot is: 4*K1**3 - 10*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.241', '6.341', '6.542', '6.567', '6.699', '6.713', '6.771', '6.791', '6.1025', '6.1039', '6.1041', '6.1072', '6.1077', '6.1121', '6.1123', '6.1499', '6.1502', '6.1531', '6.1645', '6.1648', '6.1726', '6.1727', '6.1761', '6.1784', '6.1807', '6.1823', '6.1832', '6.1869', '6.1873', '6.1874']

Outer characteristic polynomial of the knot is: t^7+39t^5+58t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.713']

2-strand cable arrow polynomial of the knot is: 96*K1**4*K2 - 736*K1**4 + 32*K1**3*K2*K3 - 96*K1**3*K3 + 128*K1**2*K2**3 - 2064*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 192*K1**2*K2*K4 + 4000*K1**2*K2 - 96*K1**2*K3**2 - 16*K1**2*K4**2 - 3188*K1**2 + 96*K1*K2**3*K3 - 576*K1*K2**2*K3 - 192*K1*K2**2*K5 - 32*K1*K2*K3*K4 + 3280*K1*K2*K3 + 552*K1*K3*K4 + 80*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 424*K2**4 - 32*K2**3*K6 - 128*K2**2*K3**2 - 16*K2**2*K4**2 + 752*K2**2*K4 - 2374*K2**2 + 176*K2*K3*K5 + 16*K2*K4*K6 - 1104*K3**2 - 342*K4**2 - 52*K5**2 - 2*K6**2 + 2388

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.713']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71384', 'vk6.71443', 'vk6.71906', 'vk6.71965', 'vk6.72440', 'vk6.72591', 'vk6.72710', 'vk6.72803', 'vk6.72866', 'vk6.73019', 'vk6.73365', 'vk6.73526', 'vk6.74251', 'vk6.74380', 'vk6.74446', 'vk6.75059', 'vk6.75533', 'vk6.75836', 'vk6.76424', 'vk6.76635', 'vk6.77041', 'vk6.77742', 'vk6.77793', 'vk6.78255', 'vk6.78506', 'vk6.78639', 'vk6.78832', 'vk6.79299', 'vk6.79418', 'vk6.79834', 'vk6.79894', 'vk6.80267', 'vk6.80760', 'vk6.80866', 'vk6.85147', 'vk6.86524', 'vk6.87211', 'vk6.87348', 'vk6.89252', 'vk6.89440']

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	O1O2O3O4U5O6U4U2O5U1U6U3
R3 orbit	{'O1O2O3O4U5O6U4U2O5U1U6U3', 'O1O2O3O4U5U3O6U2O5U1U4U6'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4U2U5U4O6U3U1O5U6
Gauss code of K*	O1O2O3U1U4U3U5O6U2O5O4U6
Gauss code of -K*	O1O2O3U4O5O6U2O4U6U1U5U3
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 -2 -1 2 0 -1 2],[ 2 0 1 3 1 0 2],[ 1 -1 0 1 0 0 1],[-2 -3 -1 0 0 -2 0],[ 0 -1 0 0 0 0 0],[ 1 0 0 2 0 0 2],[-2 -2 -1 0 0 -2 0]]
Primitive based matrix	[[ 0 2 2 0 -1 -1 -2],[-2 0 0 0 -1 -2 -2],[-2 0 0 0 -1 -2 -3],[ 0 0 0 0 0 0 -1],[ 1 1 1 0 0 0 -1],[ 1 2 2 0 0 0 0],[ 2 2 3 1 1 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,0,1,1,2,0,0,1,2,2,0,1,2,3,0,0,1,0,1,0]
Phi over symmetry	[-2,-2,0,1,1,2,0,0,1,2,2,0,1,2,3,0,0,1,0,1,0]
Phi of -K	[-2,-1,-1,0,2,2,0,1,1,1,2,0,1,2,2,1,1,1,2,2,0]
Phi of K*	[-2,-2,0,1,1,2,0,2,1,2,1,2,1,2,2,1,1,1,0,1,0]
Phi of -K*	[-2,-1,-1,0,2,2,0,1,1,2,3,0,0,2,2,0,1,1,0,0,0]
Symmetry type of based matrix	c
u-polynomial	-t^2+2t
Normalized Jones-Krushkal polynomial	z^2+14z+25
Enhanced Jones-Krushkal polynomial	w^3z^2+14w^2z+25w
Inner characteristic polynomial	t^6+25t^4+23t^2
Outer characteristic polynomial	t^7+39t^5+58t^3+3t
Flat arrow polynomial	4K13 - 10K1*2 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	96K14K2 - 736K14 + 32K1*3K2K3 - 96K1*3K3 + 128K12K2*3 - 2064K1*2K2*2 + 32K1*2K2K32 - 192K1*2K2K4 + 4000K1*2K2 - 96K12K3*2 - 16K1*2K4*2 - 3188K1*2 + 96K1K23K3 - 576K1K2*2K3 - 192K1K2*2K5 - 32K1K2K3K4 + 3280K1K2K3 + 552K1K3K4 + 80K1K4K5 - 32K2*6 + 64K2*4K4 - 424K24 - 32K2*3K6 - 128K22K3*2 - 16K2*2K4*2 + 752K2*2K4 - 2374K22 + 176K2K3K5 + 16K2K4K6 - 1104K3*2 - 342K4*2 - 52K5*2 - 2K6**2 + 2388
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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {3}, {1, 2}]]
If K is slice	False

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