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Flat knot 6.43

Min(phi) over symmetries of the knot is: [-5,-1,0,1,1,4,2,1,3,5,4,0,1,2,3,0,1,1,0,2,3]
Flat knots (up to 7 crossings) with same phi are :['6.43']
Arrow polynomial of the knot is: 8K13 + 4K1*2K2 + 4K12K3 - 14K12 - 10K1K2 - 2K1K3 - 2K1K4 - 2K1 + 6K2 + 2K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.43']
Outer characteristic polynomial of the knot is: t^7+128t^5+94t^3+8t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.43']
2-strand cable arrow polynomial of the knot is: -320K14K2*2 + 1568K1*4K2 - 3360K14 + 128K1*3K2*3K3 - 768K13K2*2K3 + 1376K13K2K3 - 704K1*3K3 - 384K12K2*4 + 256K1*2K2*3K3*2 + 3200K1*2K2*3 - 640K1*2K2*2K3*2 + 128K1*2K2*2K4 - 11952K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 1088K1*2K2K4 + 11672K1*2K2 - 544K12K3*2 - 128K1*2K4*2 - 6768K1*2 - 256K1K24K3 - 128K1K2*3K3K4 + 3488K1K23K3 + 768K1K2*2K3K4 - 2912K1K22K3 + 192K1K2*2K4K5 + 32K1K22K5K6 - 480K1K22K5 + 96K1K2K33 - 704K1K2K3K4 - 96K1K2K3K6 + 10784K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 - 32K1K2K5K6 + 1824K1K3K4 + 472K1K4K5 + 24K1K5K6 - 64K2*6 - 448K2*4K3*2 - 32K2*4K4*2 + 512K2*4K4 - 32K24K6*2 - 4200K2*4 + 416K2*3K3K5 + 160K2*3K4K6 + 32K2*3K5K7 + 32K2*3K6K8 - 128K2*3K6 + 64K22K3*2K6 - 2560K22K3*2 + 32K2*2K3K4K7 - 64K22K3K7 - 904K2*2K4*2 - 32K2*2K4K8 + 3792K2*2K4 - 272K22K5*2 - 136K2*2K6*2 - 32K2*2K7*2 - 8K2*2K8*2 - 4264K2*2 - 160K2K32K4 - 32K2K3K4K5 + 1456K2K3K5 + 504K2K4K6 + 136K2K5K7 + 32K2K6K8 + 56K32K6 - 3088K32 + 16K3K4K7 - 1360K42 - 408K5*2 - 80K6*2 - 16K7*2 - 4K8**2 + 6210
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.43']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19944', 'vk6.20059', 'vk6.21189', 'vk6.21340', 'vk6.26909', 'vk6.27120', 'vk6.28663', 'vk6.28809', 'vk6.38329', 'vk6.38517', 'vk6.40469', 'vk6.40715', 'vk6.45202', 'vk6.45413', 'vk6.47025', 'vk6.47157', 'vk6.56733', 'vk6.56865', 'vk6.57833', 'vk6.58005', 'vk6.61158', 'vk6.61390', 'vk6.62399', 'vk6.62552', 'vk6.66428', 'vk6.66572', 'vk6.67199', 'vk6.67364', 'vk6.69081', 'vk6.69220', 'vk6.69862', 'vk6.69963']
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: [-5,-1,0,1,1,4,2,1,3,5,4,0,1,2,3,0,1,1,0,2,3]

Flat knots (up to 7 crossings) with same phi are :['6.43']

Arrow polynomial of the knot is: 8*K1**3 + 4*K1**2*K2 + 4*K1**2*K3 - 14*K1**2 - 10*K1*K2 - 2*K1*K3 - 2*K1*K4 - 2*K1 + 6*K2 + 2*K3 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.43']

Outer characteristic polynomial of the knot is: t^7+128t^5+94t^3+8t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.43']

2-strand cable arrow polynomial of the knot is: -320*K1**4*K2**2 + 1568*K1**4*K2 - 3360*K1**4 + 128*K1**3*K2**3*K3 - 768*K1**3*K2**2*K3 + 1376*K1**3*K2*K3 - 704*K1**3*K3 - 384*K1**2*K2**4 + 256*K1**2*K2**3*K3**2 + 3200*K1**2*K2**3 - 640*K1**2*K2**2*K3**2 + 128*K1**2*K2**2*K4 - 11952*K1**2*K2**2 + 256*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 1088*K1**2*K2*K4 + 11672*K1**2*K2 - 544*K1**2*K3**2 - 128*K1**2*K4**2 - 6768*K1**2 - 256*K1*K2**4*K3 - 128*K1*K2**3*K3*K4 + 3488*K1*K2**3*K3 + 768*K1*K2**2*K3*K4 - 2912*K1*K2**2*K3 + 192*K1*K2**2*K4*K5 + 32*K1*K2**2*K5*K6 - 480*K1*K2**2*K5 + 96*K1*K2*K3**3 - 704*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 10784*K1*K2*K3 - 128*K1*K2*K4*K5 - 64*K1*K2*K4*K7 - 32*K1*K2*K5*K6 + 1824*K1*K3*K4 + 472*K1*K4*K5 + 24*K1*K5*K6 - 64*K2**6 - 448*K2**4*K3**2 - 32*K2**4*K4**2 + 512*K2**4*K4 - 32*K2**4*K6**2 - 4200*K2**4 + 416*K2**3*K3*K5 + 160*K2**3*K4*K6 + 32*K2**3*K5*K7 + 32*K2**3*K6*K8 - 128*K2**3*K6 + 64*K2**2*K3**2*K6 - 2560*K2**2*K3**2 + 32*K2**2*K3*K4*K7 - 64*K2**2*K3*K7 - 904*K2**2*K4**2 - 32*K2**2*K4*K8 + 3792*K2**2*K4 - 272*K2**2*K5**2 - 136*K2**2*K6**2 - 32*K2**2*K7**2 - 8*K2**2*K8**2 - 4264*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 1456*K2*K3*K5 + 504*K2*K4*K6 + 136*K2*K5*K7 + 32*K2*K6*K8 + 56*K3**2*K6 - 3088*K3**2 + 16*K3*K4*K7 - 1360*K4**2 - 408*K5**2 - 80*K6**2 - 16*K7**2 - 4*K8**2 + 6210

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.43']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.19944', 'vk6.20059', 'vk6.21189', 'vk6.21340', 'vk6.26909', 'vk6.27120', 'vk6.28663', 'vk6.28809', 'vk6.38329', 'vk6.38517', 'vk6.40469', 'vk6.40715', 'vk6.45202', 'vk6.45413', 'vk6.47025', 'vk6.47157', 'vk6.56733', 'vk6.56865', 'vk6.57833', 'vk6.58005', 'vk6.61158', 'vk6.61390', 'vk6.62399', 'vk6.62552', 'vk6.66428', 'vk6.66572', 'vk6.67199', 'vk6.67364', 'vk6.69081', 'vk6.69220', 'vk6.69862', 'vk6.69963']

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	O1O2O3O4O5O6U1U5U3U4U6U2
R3 orbit	{'O1O2O3O4O5O6U1U5U3U4U6U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U5U1U3U4U2U6
Gauss code of K*	O1O2O3O4O5O6U1U6U3U4U2U5
Gauss code of -K*	O1O2O3O4O5O6U2U5U3U4U1U6
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 -5 1 -1 1 0 4],[ 5 0 5 2 3 1 4],[-1 -5 0 -2 0 -1 3],[ 1 -2 2 0 1 0 3],[-1 -3 0 -1 0 0 2],[ 0 -1 1 0 0 0 1],[-4 -4 -3 -3 -2 -1 0]]
Primitive based matrix	[[ 0 4 1 1 0 -1 -5],[-4 0 -2 -3 -1 -3 -4],[-1 2 0 0 0 -1 -3],[-1 3 0 0 -1 -2 -5],[ 0 1 0 1 0 0 -1],[ 1 3 1 2 0 0 -2],[ 5 4 3 5 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-4,-1,-1,0,1,5,2,3,1,3,4,0,0,1,3,1,2,5,0,1,2]
Phi over symmetry	[-5,-1,0,1,1,4,2,1,3,5,4,0,1,2,3,0,1,1,0,2,3]
Phi of -K	[-5,-1,0,1,1,4,2,4,1,3,5,1,0,1,2,0,1,3,0,0,1]
Phi of K*	[-4,-1,-1,0,1,5,0,1,3,2,5,0,0,0,1,1,1,3,1,4,2]
Phi of -K*	[-5,-1,0,1,1,4,2,1,3,5,4,0,1,2,3,0,1,1,0,2,3]
Symmetry type of based matrix	c
u-polynomial	t^5-t^4-t
Normalized Jones-Krushkal polynomial	6z^2+27z+31
Enhanced Jones-Krushkal polynomial	6w^3z^2+27w^2z+31w
Inner characteristic polynomial	t^6+84t^4+28t^2+1
Outer characteristic polynomial	t^7+128t^5+94t^3+8t
Flat arrow polynomial	8K13 + 4K1*2K2 + 4K12K3 - 14K12 - 10K1K2 - 2K1K3 - 2K1K4 - 2K1 + 6K2 + 2K3 + 7
2-strand cable arrow polynomial	-320K14K2*2 + 1568K1*4K2 - 3360K14 + 128K1*3K2*3K3 - 768K13K2*2K3 + 1376K13K2K3 - 704K1*3K3 - 384K12K2*4 + 256K1*2K2*3K3*2 + 3200K1*2K2*3 - 640K1*2K2*2K3*2 + 128K1*2K2*2K4 - 11952K12K2*2 + 256K1*2K2K32 + 64K1*2K2K42 - 1088K1*2K2K4 + 11672K1*2K2 - 544K12K3*2 - 128K1*2K4*2 - 6768K1*2 - 256K1K24K3 - 128K1K2*3K3K4 + 3488K1K23K3 + 768K1K2*2K3K4 - 2912K1K22K3 + 192K1K2*2K4K5 + 32K1K22K5K6 - 480K1K22K5 + 96K1K2K33 - 704K1K2K3K4 - 96K1K2K3K6 + 10784K1K2K3 - 128K1K2K4K5 - 64K1K2K4K7 - 32K1K2K5K6 + 1824K1K3K4 + 472K1K4K5 + 24K1K5K6 - 64K2*6 - 448K2*4K3*2 - 32K2*4K4*2 + 512K2*4K4 - 32K24K6*2 - 4200K2*4 + 416K2*3K3K5 + 160K2*3K4K6 + 32K2*3K5K7 + 32K2*3K6K8 - 128K2*3K6 + 64K22K3*2K6 - 2560K22K3*2 + 32K2*2K3K4K7 - 64K22K3K7 - 904K2*2K4*2 - 32K2*2K4K8 + 3792K2*2K4 - 272K22K5*2 - 136K2*2K6*2 - 32K2*2K7*2 - 8K2*2K8*2 - 4264K2*2 - 160K2K32K4 - 32K2K3K4K5 + 1456K2K3K5 + 504K2K4K6 + 136K2K5K7 + 32K2K6K8 + 56K32K6 - 3088K32 + 16K3K4K7 - 1360K42 - 408K5*2 - 80K6*2 - 16K7*2 - 4K8**2 + 6210
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}], [{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}]]
If K is slice	False

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