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Flat knot 6.843

Min(phi) over symmetries of the knot is: [-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,1,1,1,2,2,1,-2]
Flat knots (up to 7 crossings) with same phi are :['6.843']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 - 4K12 - 6K1K2 - 4K1K3 - 3K1 + 2*K2 + K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.404', '6.843', '6.1141']
Outer characteristic polynomial of the knot is: t^7+52t^5+70t^3+4t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.843']
2-strand cable arrow polynomial of the knot is: 128K14K2*3 - 960K1*4K2*2 + 1152K1*4K2 - 1424K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 1376K13K2K3 - 288K1*3K3 - 576K12K2*4 + 2496K1*2K2*3 - 256K1*2K2*2K3*2 + 128K1*2K2*2K4 - 9136K12K2*2 + 32K1*2K2K32 - 544K1*2K2K4 + 7912K1*2K2 - 624K12K3*2 - 5028K1*2 + 1856K1K23K3 + 288K1K2*2K3K4 - 2464K1K22K3 - 224K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 8984K1K2K3 - 128K1K2K4K5 + 1128K1K3K4 + 128K1K4K5 + 40K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 384K2*4K4 - 3152K24 + 256K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 2048K22K3*2 - 64K2*2K3K7 - 408K2*2K4*2 - 32K2*2K4K8 + 2776K2*2K4 - 80K22K5*2 - 16K2*2K6*2 - 3462K2*2 - 128K2K32K4 + 1296K2K3K5 + 384K2K4K6 + 32K2K5K7 + 16K2K6K8 + 24K32K6 - 2548K32 - 860K4*2 - 284K5*2 - 106K6*2 - 4K7*2 - 2K8**2 + 4604
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.843']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4717', 'vk6.5032', 'vk6.6242', 'vk6.6694', 'vk6.8214', 'vk6.8650', 'vk6.9594', 'vk6.9923', 'vk6.20303', 'vk6.21636', 'vk6.27595', 'vk6.29147', 'vk6.39021', 'vk6.41269', 'vk6.45785', 'vk6.47462', 'vk6.48749', 'vk6.48946', 'vk6.49546', 'vk6.49764', 'vk6.50759', 'vk6.50959', 'vk6.51234', 'vk6.51445', 'vk6.57166', 'vk6.58354', 'vk6.61788', 'vk6.62907', 'vk6.66783', 'vk6.67659', 'vk6.69427', 'vk6.70149']
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,-1,1,2,2,0,1,2,1,3,0,0,1,1,1,1,2,2,1,-2]

Flat knots (up to 7 crossings) with same phi are :['6.843']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 - 4*K1**2 - 6*K1*K2 - 4*K1*K3 - 3*K1 + 2*K2 + K3 + K4 + 2

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.404', '6.843', '6.1141']

Outer characteristic polynomial of the knot is: t^7+52t^5+70t^3+4t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.843']

2-strand cable arrow polynomial of the knot is: 128*K1**4*K2**3 - 960*K1**4*K2**2 + 1152*K1**4*K2 - 1424*K1**4 + 128*K1**3*K2**3*K3 - 128*K1**3*K2**2*K3 + 1376*K1**3*K2*K3 - 288*K1**3*K3 - 576*K1**2*K2**4 + 2496*K1**2*K2**3 - 256*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 9136*K1**2*K2**2 + 32*K1**2*K2*K3**2 - 544*K1**2*K2*K4 + 7912*K1**2*K2 - 624*K1**2*K3**2 - 5028*K1**2 + 1856*K1*K2**3*K3 + 288*K1*K2**2*K3*K4 - 2464*K1*K2**2*K3 - 224*K1*K2**2*K5 + 32*K1*K2*K3**3 - 352*K1*K2*K3*K4 + 8984*K1*K2*K3 - 128*K1*K2*K4*K5 + 1128*K1*K3*K4 + 128*K1*K4*K5 + 40*K1*K5*K6 - 64*K2**6 - 128*K2**4*K3**2 - 32*K2**4*K4**2 + 384*K2**4*K4 - 3152*K2**4 + 256*K2**3*K3*K5 + 64*K2**3*K4*K6 - 96*K2**3*K6 - 2048*K2**2*K3**2 - 64*K2**2*K3*K7 - 408*K2**2*K4**2 - 32*K2**2*K4*K8 + 2776*K2**2*K4 - 80*K2**2*K5**2 - 16*K2**2*K6**2 - 3462*K2**2 - 128*K2*K3**2*K4 + 1296*K2*K3*K5 + 384*K2*K4*K6 + 32*K2*K5*K7 + 16*K2*K6*K8 + 24*K3**2*K6 - 2548*K3**2 - 860*K4**2 - 284*K5**2 - 106*K6**2 - 4*K7**2 - 2*K8**2 + 4604

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.843']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4717', 'vk6.5032', 'vk6.6242', 'vk6.6694', 'vk6.8214', 'vk6.8650', 'vk6.9594', 'vk6.9923', 'vk6.20303', 'vk6.21636', 'vk6.27595', 'vk6.29147', 'vk6.39021', 'vk6.41269', 'vk6.45785', 'vk6.47462', 'vk6.48749', 'vk6.48946', 'vk6.49546', 'vk6.49764', 'vk6.50759', 'vk6.50959', 'vk6.51234', 'vk6.51445', 'vk6.57166', 'vk6.58354', 'vk6.61788', 'vk6.62907', 'vk6.66783', 'vk6.67659', 'vk6.69427', 'vk6.70149']

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	O1O2O3O4U1U2O5O6U5U4U6U3
R3 orbit	{'O1O2O3O4U1U2O5O6U5U4U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4U2U5U1U6O5O6U3U4
Gauss code of K*	O1O2O3O4U5U6U4U2O5O6U1U3
Gauss code of -K*	O1O2O3O4U2U4O5O6U3U1U5U6
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 2 1 -1 2],[ 3 0 1 3 2 0 1],[ 1 -1 0 2 1 0 1],[-2 -3 -2 0 -1 -1 2],[-1 -2 -1 1 0 0 2],[ 1 0 0 1 0 0 1],[-2 -1 -1 -2 -2 -1 0]]
Primitive based matrix	[[ 0 2 2 1 -1 -1 -3],[-2 0 2 -1 -1 -2 -3],[-2 -2 0 -2 -1 -1 -1],[-1 1 2 0 0 -1 -2],[ 1 1 1 0 0 0 0],[ 1 2 1 1 0 0 -1],[ 3 3 1 2 0 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,1,1,3,-2,1,1,2,3,2,1,1,1,0,1,2,0,0,1]
Phi over symmetry	[-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,1,1,1,2,2,1,-2]
Phi of -K	[-3,-1,-1,1,2,2,1,2,2,2,4,0,1,1,2,2,2,2,0,-1,-2]
Phi of K*	[-2,-2,-1,1,1,3,-2,-1,2,2,4,0,1,2,2,1,2,2,0,1,2]
Phi of -K*	[-3,-1,-1,1,2,2,0,1,2,1,3,0,0,1,1,1,1,2,2,1,-2]
Symmetry type of based matrix	c
u-polynomial	t^3-2t^2+t
Normalized Jones-Krushkal polynomial	8z^2+29z+27
Enhanced Jones-Krushkal polynomial	8w^3z^2+29w^2z+27w
Inner characteristic polynomial	t^6+32t^4+22t^2+1
Outer characteristic polynomial	t^7+52t^5+70t^3+4t
Flat arrow polynomial	8K13 + 4K1*2K2 - 4K12 - 6K1K2 - 4K1K3 - 3K1 + 2*K2 + K3 + K4 + 2
2-strand cable arrow polynomial	128K14K2*3 - 960K1*4K2*2 + 1152K1*4K2 - 1424K14 + 128K1*3K2*3K3 - 128K13K2*2K3 + 1376K13K2K3 - 288K1*3K3 - 576K12K2*4 + 2496K1*2K2*3 - 256K1*2K2*2K3*2 + 128K1*2K2*2K4 - 9136K12K2*2 + 32K1*2K2K32 - 544K1*2K2K4 + 7912K1*2K2 - 624K12K3*2 - 5028K1*2 + 1856K1K23K3 + 288K1K2*2K3K4 - 2464K1K22K3 - 224K1K2*2K5 + 32K1K2K33 - 352K1K2K3K4 + 8984K1K2K3 - 128K1K2K4K5 + 1128K1K3K4 + 128K1K4K5 + 40K1K5K6 - 64K2*6 - 128K2*4K3*2 - 32K2*4K4*2 + 384K2*4K4 - 3152K24 + 256K2*3K3K5 + 64K2*3K4K6 - 96K2*3K6 - 2048K22K3*2 - 64K2*2K3K7 - 408K2*2K4*2 - 32K2*2K4K8 + 2776K2*2K4 - 80K22K5*2 - 16K2*2K6*2 - 3462K2*2 - 128K2K32K4 + 1296K2K3K5 + 384K2K4K6 + 32K2K5K7 + 16K2K6K8 + 24K32K6 - 2548K32 - 860K4*2 - 284K5*2 - 106K6*2 - 4K7*2 - 2K8**2 + 4604
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}], [{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}, {2, 4}, {1, 3}], [{6}, {5}, {2, 4}, {3}, {1}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]]
If K is slice	False

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