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Flat knot 6.197

Min(phi) over symmetries of the knot is: [-4,-3,1,1,2,3,0,1,4,3,5,0,2,1,3,0,0,0,0,1,0]
Flat knots (up to 7 crossings) with same phi are :['6.197']
Arrow polynomial of the knot is: -8K14 + 4K1*3 + 4K1*2K2 - 6K12 - 4K1K2 - K1 + 5K2 + K3 + 6
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.197']
Outer characteristic polynomial of the knot is: t^7+106t^5+194t^3+10t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.197']
2-strand cable arrow polynomial of the knot is: -128K16 - 256K1*4K2*2 + 512K1*4K2 - 2768K14 + 320K1*3K2K3 - 352K1*3K3 + 128K12K2*5 - 1472K1*2K2*4 + 2048K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 9056K12K2*2 + 64K1*2K2K32 - 544K1*2K2K4 + 10544K1*2K2 - 496K12K3*2 - 32K1*2K4*2 - 5724K1*2 + 256K1K25K3 + 2304K1K2*3K3 + 160K1K2*2K3K4 - 1760K1K22K3 - 384K1K2*2K5 - 192K1K2K3K4 + 9096K1K2K3 + 1032K1K3K4 + 72K1K4K5 - 128K2*8 + 128K2*6K4 - 928K26 - 256K2*4K3*2 - 32K2*4K4*2 + 672K2*4K4 - 3464K24 + 96K2*3K3K5 - 96K2*3K6 - 1488K22K3*2 - 240K2*2K4*2 + 3144K2*2K4 - 32K22K5*2 - 3398K2*2 + 752K2K3K5 + 88K2K4K6 + 8K2K5K7 - 2276K32 - 714K4*2 - 112K5*2 - 10K6**2 + 4976
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.197']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17018', 'vk6.17261', 'vk6.20234', 'vk6.21530', 'vk6.23434', 'vk6.23737', 'vk6.27445', 'vk6.29051', 'vk6.35508', 'vk6.35957', 'vk6.38862', 'vk6.41050', 'vk6.42928', 'vk6.43225', 'vk6.45613', 'vk6.47368', 'vk6.55199', 'vk6.55436', 'vk6.57070', 'vk6.58208', 'vk6.59586', 'vk6.59910', 'vk6.61599', 'vk6.62777', 'vk6.64998', 'vk6.65204', 'vk6.66695', 'vk6.67542', 'vk6.68278', 'vk6.68431', 'vk6.69344', 'vk6.70092']
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,-3,1,1,2,3,0,1,4,3,5,0,2,1,3,0,0,0,0,1,0]

Flat knots (up to 7 crossings) with same phi are :['6.197']

Arrow polynomial of the knot is: -8*K1**4 + 4*K1**3 + 4*K1**2*K2 - 6*K1**2 - 4*K1*K2 - K1 + 5*K2 + K3 + 6

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.197']

Outer characteristic polynomial of the knot is: t^7+106t^5+194t^3+10t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.197']

2-strand cable arrow polynomial of the knot is: -128*K1**6 - 256*K1**4*K2**2 + 512*K1**4*K2 - 2768*K1**4 + 320*K1**3*K2*K3 - 352*K1**3*K3 + 128*K1**2*K2**5 - 1472*K1**2*K2**4 + 2048*K1**2*K2**3 - 128*K1**2*K2**2*K3**2 + 64*K1**2*K2**2*K4 - 9056*K1**2*K2**2 + 64*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 10544*K1**2*K2 - 496*K1**2*K3**2 - 32*K1**2*K4**2 - 5724*K1**2 + 256*K1*K2**5*K3 + 2304*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1760*K1*K2**2*K3 - 384*K1*K2**2*K5 - 192*K1*K2*K3*K4 + 9096*K1*K2*K3 + 1032*K1*K3*K4 + 72*K1*K4*K5 - 128*K2**8 + 128*K2**6*K4 - 928*K2**6 - 256*K2**4*K3**2 - 32*K2**4*K4**2 + 672*K2**4*K4 - 3464*K2**4 + 96*K2**3*K3*K5 - 96*K2**3*K6 - 1488*K2**2*K3**2 - 240*K2**2*K4**2 + 3144*K2**2*K4 - 32*K2**2*K5**2 - 3398*K2**2 + 752*K2*K3*K5 + 88*K2*K4*K6 + 8*K2*K5*K7 - 2276*K3**2 - 714*K4**2 - 112*K5**2 - 10*K6**2 + 4976

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.197']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17018', 'vk6.17261', 'vk6.20234', 'vk6.21530', 'vk6.23434', 'vk6.23737', 'vk6.27445', 'vk6.29051', 'vk6.35508', 'vk6.35957', 'vk6.38862', 'vk6.41050', 'vk6.42928', 'vk6.43225', 'vk6.45613', 'vk6.47368', 'vk6.55199', 'vk6.55436', 'vk6.57070', 'vk6.58208', 'vk6.59586', 'vk6.59910', 'vk6.61599', 'vk6.62777', 'vk6.64998', 'vk6.65204', 'vk6.66695', 'vk6.67542', 'vk6.68278', 'vk6.68431', 'vk6.69344', 'vk6.70092']

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	O1O2O3O4O5U2O6U1U6U5U3U4
R3 orbit	{'O1O2O3O4O5U2O6U1U6U5U3U4'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U2U3U1U6U5O6U4
Gauss code of K*	O1O2O3O4O5U1U6U4U5U3O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U3U1U2U6U5
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 -3 1 3 2 1],[ 4 0 0 4 5 3 1],[ 3 0 0 2 3 1 0],[-1 -4 -2 0 1 0 0],[-3 -5 -3 -1 0 0 0],[-2 -3 -1 0 0 0 0],[-1 -1 0 0 0 0 0]]
Primitive based matrix	[[ 0 3 2 1 1 -3 -4],[-3 0 0 0 -1 -3 -5],[-2 0 0 0 0 -1 -3],[-1 0 0 0 0 0 -1],[-1 1 0 0 0 -2 -4],[ 3 3 1 0 2 0 0],[ 4 5 3 1 4 0 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-1,-1,3,4,0,0,1,3,5,0,0,1,3,0,0,1,2,4,0]
Phi over symmetry	[-4,-3,1,1,2,3,0,1,4,3,5,0,2,1,3,0,0,0,0,1,0]
Phi of -K	[-4,-3,1,1,2,3,1,1,4,3,2,2,4,4,3,0,1,1,1,2,1]
Phi of K*	[-3,-2,-1,-1,3,4,1,1,2,3,2,1,1,4,3,0,2,1,4,4,1]
Phi of -K*	[-4,-3,1,1,2,3,0,1,4,3,5,0,2,1,3,0,0,0,0,1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	-2w^4z^2+7w^3z^2+24w^2z+29w
Inner characteristic polynomial	t^6+66t^4+49t^2+1
Outer characteristic polynomial	t^7+106t^5+194t^3+10t
Flat arrow polynomial	-8K14 + 4K1*3 + 4K1*2K2 - 6K12 - 4K1K2 - K1 + 5K2 + K3 + 6
2-strand cable arrow polynomial	-128K16 - 256K1*4K2*2 + 512K1*4K2 - 2768K14 + 320K1*3K2K3 - 352K1*3K3 + 128K12K2*5 - 1472K1*2K2*4 + 2048K1*2K2*3 - 128K1*2K2*2K3*2 + 64K1*2K2*2K4 - 9056K12K2*2 + 64K1*2K2K32 - 544K1*2K2K4 + 10544K1*2K2 - 496K12K3*2 - 32K1*2K4*2 - 5724K1*2 + 256K1K25K3 + 2304K1K2*3K3 + 160K1K2*2K3K4 - 1760K1K22K3 - 384K1K2*2K5 - 192K1K2K3K4 + 9096K1K2K3 + 1032K1K3K4 + 72K1K4K5 - 128K2*8 + 128K2*6K4 - 928K26 - 256K2*4K3*2 - 32K2*4K4*2 + 672K2*4K4 - 3464K24 + 96K2*3K3K5 - 96K2*3K6 - 1488K22K3*2 - 240K2*2K4*2 + 3144K2*2K4 - 32K22K5*2 - 3398K2*2 + 752K2K3K5 + 88K2K4K6 + 8K2K5K7 - 2276K32 - 714K4*2 - 112K5*2 - 10K6**2 + 4976
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}], [{1, 6}, {5}, {4}, {3}, {2}], [{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}], [{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}], [{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}, {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}], [{6}, {5}, {3, 4}, {2}, {1}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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