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Flat knot 6.245

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,1,2,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.245']
Arrow polynomial of the knot is: 8K13 - 2K1*2 - 6K1K2 - 2K1K3 - 3K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.245', '6.255', '6.478']
Outer characteristic polynomial of the knot is: t^7+76t^5+64t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.245']
2-strand cable arrow polynomial of the knot is: 2272K14K2 - 5680K14 + 1952K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 - 384K12K2*4 + 1056K1*2K2*3 + 544K1*2K2*2K4 - 8592K12K2*2 - 1280K1*2K2K4 + 10040K1*2K2 - 2512K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 3788K1*2 + 1216K1K23K3 - 1984K1K2*2K3 - 608K1K2*2K5 + 256K1K2K33 - 544K1K2K3K4 - 128K1K2K3K6 + 9936K1K2K3 + 2840K1K3K4 + 304K1K4K5 + 16K1K5K6 - 192K2*6 + 416K2*4K4 - 1640K24 + 32K2*3K3K5 - 128K2*3K6 + 64K22K3*2K4 - 1248K22K3*2 - 32K2*2K3K7 - 184K2*2K4*2 + 2168K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4210K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1192K2K3K5 + 168K2K4K6 + 16K2K5K7 + 8K2K6K8 - 160K34 - 32K3*2K4*2 + 160K3*2K6 - 2800K32 + 24K3K4K7 - 1036K42 - 304K5*2 - 70K6*2 - 4K7*2 - 2K8**2 + 4708
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.245']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13902', 'vk6.13997', 'vk6.14171', 'vk6.14410', 'vk6.14973', 'vk6.15094', 'vk6.15639', 'vk6.16093', 'vk6.16723', 'vk6.16755', 'vk6.16843', 'vk6.18818', 'vk6.19292', 'vk6.19584', 'vk6.23162', 'vk6.23224', 'vk6.25412', 'vk6.26477', 'vk6.33713', 'vk6.33788', 'vk6.34267', 'vk6.35150', 'vk6.37537', 'vk6.42724', 'vk6.44701', 'vk6.54122', 'vk6.54928', 'vk6.54958', 'vk6.56404', 'vk6.56615', 'vk6.59353', 'vk6.64595']
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,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,1,2,0,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.245']

Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 6*K1*K2 - 2*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.245', '6.255', '6.478']

Outer characteristic polynomial of the knot is: t^7+76t^5+64t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.245']

2-strand cable arrow polynomial of the knot is: 2272*K1**4*K2 - 5680*K1**4 + 1952*K1**3*K2*K3 + 32*K1**3*K3*K4 - 1504*K1**3*K3 - 384*K1**2*K2**4 + 1056*K1**2*K2**3 + 544*K1**2*K2**2*K4 - 8592*K1**2*K2**2 - 1280*K1**2*K2*K4 + 10040*K1**2*K2 - 2512*K1**2*K3**2 - 96*K1**2*K3*K5 - 128*K1**2*K4**2 - 3788*K1**2 + 1216*K1*K2**3*K3 - 1984*K1*K2**2*K3 - 608*K1*K2**2*K5 + 256*K1*K2*K3**3 - 544*K1*K2*K3*K4 - 128*K1*K2*K3*K6 + 9936*K1*K2*K3 + 2840*K1*K3*K4 + 304*K1*K4*K5 + 16*K1*K5*K6 - 192*K2**6 + 416*K2**4*K4 - 1640*K2**4 + 32*K2**3*K3*K5 - 128*K2**3*K6 + 64*K2**2*K3**2*K4 - 1248*K2**2*K3**2 - 32*K2**2*K3*K7 - 184*K2**2*K4**2 + 2168*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4210*K2**2 - 96*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1192*K2*K3*K5 + 168*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 160*K3**4 - 32*K3**2*K4**2 + 160*K3**2*K6 - 2800*K3**2 + 24*K3*K4*K7 - 1036*K4**2 - 304*K5**2 - 70*K6**2 - 4*K7**2 - 2*K8**2 + 4708

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.245']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13902', 'vk6.13997', 'vk6.14171', 'vk6.14410', 'vk6.14973', 'vk6.15094', 'vk6.15639', 'vk6.16093', 'vk6.16723', 'vk6.16755', 'vk6.16843', 'vk6.18818', 'vk6.19292', 'vk6.19584', 'vk6.23162', 'vk6.23224', 'vk6.25412', 'vk6.26477', 'vk6.33713', 'vk6.33788', 'vk6.34267', 'vk6.35150', 'vk6.37537', 'vk6.42724', 'vk6.44701', 'vk6.54122', 'vk6.54928', 'vk6.54958', 'vk6.56404', 'vk6.56615', 'vk6.59353', 'vk6.64595']

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	O1O2O3O4O5U4O6U1U6U3U5U2
R3 orbit	{'O1O2O3O4O5U4O6U1U6U3U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U1U3U6U5O6U2
Gauss code of K*	O1O2O3O4O5U1U5U3U6U4O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U2U6U3U1U5
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 -4 1 0 -1 3 1],[ 4 0 4 2 0 4 1],[-1 -4 0 -1 -1 2 0],[ 0 -2 1 0 0 2 0],[ 1 0 1 0 0 1 0],[-3 -4 -2 -2 -1 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 0 -2 -2 -1 -4],[-1 0 0 0 0 0 -1],[-1 2 0 0 -1 -1 -4],[ 0 2 0 1 0 0 -2],[ 1 1 0 1 0 0 0],[ 4 4 1 4 2 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,0,2,2,1,4,0,0,0,1,1,1,4,0,2,0]
Phi over symmetry	[-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,1,2,0,0,2]
Phi of -K	[-4,-1,0,1,1,3,3,2,1,4,3,1,1,2,3,0,1,1,0,0,2]
Phi of K*	[-3,-1,-1,0,1,4,0,2,1,3,3,0,0,1,1,1,2,4,1,2,3]
Phi of -K*	[-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,1,2,0,0,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+48t^4+20t^2+1
Outer characteristic polynomial	t^7+76t^5+64t^3+4t
Flat arrow polynomial	8K13 - 2K1*2 - 6K1K2 - 2K1K3 - 3K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial	2272K14K2 - 5680K14 + 1952K1*3K2K3 + 32K1*3K3K4 - 1504K1*3K3 - 384K12K2*4 + 1056K1*2K2*3 + 544K1*2K2*2K4 - 8592K12K2*2 - 1280K1*2K2K4 + 10040K1*2K2 - 2512K12K3*2 - 96K1*2K3K5 - 128K1*2K4*2 - 3788K1*2 + 1216K1K23K3 - 1984K1K2*2K3 - 608K1K2*2K5 + 256K1K2K33 - 544K1K2K3K4 - 128K1K2K3K6 + 9936K1K2K3 + 2840K1K3K4 + 304K1K4K5 + 16K1K5K6 - 192K2*6 + 416K2*4K4 - 1640K24 + 32K2*3K3K5 - 128K2*3K6 + 64K22K3*2K4 - 1248K22K3*2 - 32K2*2K3K7 - 184K2*2K4*2 + 2168K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4210K2*2 - 96K2K32K4 - 32K2K3K4K5 + 1192K2K3K5 + 168K2K4K6 + 16K2K5K7 + 8K2K6K8 - 160K34 - 32K3*2K4*2 + 160K3*2K6 - 2800K32 + 24K3K4K7 - 1036K42 - 304K5*2 - 70K6*2 - 4K7*2 - 2K8**2 + 4708
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}], [{3, 6}, {5}, {4}, {2}, {1}], [{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}, {1, 5}, {4}, {3}, {2}], [{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}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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