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Flat knot 6.87

Min(phi) over symmetries of the knot is: [-4,-3,0,2,2,3,0,3,2,3,5,2,1,2,3,1,1,2,0,0,0]
Flat knots (up to 7 crossings) with same phi are :['6.87']
Arrow polynomial of the knot is: -4K1K2 + 2K1 - 2K2*2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']
Outer characteristic polynomial of the knot is: t^7+113t^5+83t^3+6t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.87']
2-strand cable arrow polynomial of the knot is: -320K13K3 - 1088K12K2*2 - 800K1*2K2K4 + 3040K1*2K2 - 752K12K3*2 - 64K1*2K3K5 - 112K1*2K4*2 - 4712K1*2 + 128K1K23K3 + 256K1K2*2K3K4 - 448K1K22K3 + 224K1K2K3K4*2 - 576K1K2K3K4 - 64K1K2K3K6 + 5256K1K2K3 - 32K1K2K4K7 + 96K1K3*3K4 - 64K1K3*2K5 - 32K1K3K4K6 + 2808K1K3K4 + 408K1K4K5 + 64K1K5K6 + 8K1K6K7 - 64K24 - 416K2*2K3*2 - 432K2*2K4*2 + 1320K2*2K4 - 3528K22 - 288K2K32K4 - 64K2K3K4K5 + 608K2K3K5 + 328K2K4K6 - 144K34 - 224K3*2K4*2 + 224K3*2K6 - 2756K32 + 120K3K4K7 - 8K44 + 8K4*2K8 - 1468K42 - 268K5*2 - 120K6*2 - 16K7*2 - 2K8**2 + 3964
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.87']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71632', 'vk6.71803', 'vk6.72223', 'vk6.72359', 'vk6.73402', 'vk6.73595', 'vk6.73871', 'vk6.74289', 'vk6.74913', 'vk6.75368', 'vk6.75672', 'vk6.75874', 'vk6.76465', 'vk6.77250', 'vk6.77338', 'vk6.77586', 'vk6.77688', 'vk6.78329', 'vk6.78870', 'vk6.79328', 'vk6.80114', 'vk6.80293', 'vk6.80420', 'vk6.80789', 'vk6.82031', 'vk6.82765', 'vk6.85355', 'vk6.86694', 'vk6.86933', 'vk6.87030', 'vk6.87606', 'vk6.89465']
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,-3,0,2,2,3,0,3,2,3,5,2,1,2,3,1,1,2,0,0,0]

Flat knots (up to 7 crossings) with same phi are :['6.87']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.87', '6.88', '6.184', '6.302', '6.459', '6.467', '6.506']

Outer characteristic polynomial of the knot is: t^7+113t^5+83t^3+6t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.87']

2-strand cable arrow polynomial of the knot is: -320*K1**3*K3 - 1088*K1**2*K2**2 - 800*K1**2*K2*K4 + 3040*K1**2*K2 - 752*K1**2*K3**2 - 64*K1**2*K3*K5 - 112*K1**2*K4**2 - 4712*K1**2 + 128*K1*K2**3*K3 + 256*K1*K2**2*K3*K4 - 448*K1*K2**2*K3 + 224*K1*K2*K3*K4**2 - 576*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 5256*K1*K2*K3 - 32*K1*K2*K4*K7 + 96*K1*K3**3*K4 - 64*K1*K3**2*K5 - 32*K1*K3*K4*K6 + 2808*K1*K3*K4 + 408*K1*K4*K5 + 64*K1*K5*K6 + 8*K1*K6*K7 - 64*K2**4 - 416*K2**2*K3**2 - 432*K2**2*K4**2 + 1320*K2**2*K4 - 3528*K2**2 - 288*K2*K3**2*K4 - 64*K2*K3*K4*K5 + 608*K2*K3*K5 + 328*K2*K4*K6 - 144*K3**4 - 224*K3**2*K4**2 + 224*K3**2*K6 - 2756*K3**2 + 120*K3*K4*K7 - 8*K4**4 + 8*K4**2*K8 - 1468*K4**2 - 268*K5**2 - 120*K6**2 - 16*K7**2 - 2*K8**2 + 3964

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.87']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.71632', 'vk6.71803', 'vk6.72223', 'vk6.72359', 'vk6.73402', 'vk6.73595', 'vk6.73871', 'vk6.74289', 'vk6.74913', 'vk6.75368', 'vk6.75672', 'vk6.75874', 'vk6.76465', 'vk6.77250', 'vk6.77338', 'vk6.77586', 'vk6.77688', 'vk6.78329', 'vk6.78870', 'vk6.79328', 'vk6.80114', 'vk6.80293', 'vk6.80420', 'vk6.80789', 'vk6.82031', 'vk6.82765', 'vk6.85355', 'vk6.86694', 'vk6.86933', 'vk6.87030', 'vk6.87606', 'vk6.89465']

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	O1O2O3O4O5O6U2U5U1U6U4U3
R3 orbit	{'O1O2O3O4O5O6U2U5U1U6U4U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5O6U4U3U1U6U2U5
Gauss code of K*	O1O2O3O4O5O6U3U1U6U5U2U4
Gauss code of -K*	O1O2O3O4O5O6U3U5U2U1U6U4
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 -4 2 2 0 3],[ 3 0 -1 4 3 1 3],[ 4 1 0 4 3 1 2],[-2 -4 -4 0 0 -1 1],[-2 -3 -3 0 0 -1 1],[ 0 -1 -1 1 1 0 1],[-3 -3 -2 -1 -1 -1 0]]
Primitive based matrix	[[ 0 3 2 2 0 -3 -4],[-3 0 -1 -1 -1 -3 -2],[-2 1 0 0 -1 -3 -3],[-2 1 0 0 -1 -4 -4],[ 0 1 1 1 0 -1 -1],[ 3 3 3 4 1 0 -1],[ 4 2 3 4 1 1 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-2,-2,0,3,4,1,1,1,3,2,0,1,3,3,1,4,4,1,1,1]
Phi over symmetry	[-4,-3,0,2,2,3,0,3,2,3,5,2,1,2,3,1,1,2,0,0,0]
Phi of -K	[-4,-3,0,2,2,3,0,3,2,3,5,2,1,2,3,1,1,2,0,0,0]
Phi of K*	[-3,-2,-2,0,3,4,0,0,2,3,5,0,1,1,2,1,2,3,2,3,0]
Phi of -K*	[-4,-3,0,2,2,3,1,1,3,4,2,1,3,4,3,1,1,1,0,1,1]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	6z^2+25z+27
Enhanced Jones-Krushkal polynomial	6w^3z^2+25w^2z+27w
Inner characteristic polynomial	t^6+71t^4+21t^2+1
Outer characteristic polynomial	t^7+113t^5+83t^3+6t
Flat arrow polynomial	-4K1K2 + 2K1 - 2K2*2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	-320K13K3 - 1088K12K2*2 - 800K1*2K2K4 + 3040K1*2K2 - 752K12K3*2 - 64K1*2K3K5 - 112K1*2K4*2 - 4712K1*2 + 128K1K23K3 + 256K1K2*2K3K4 - 448K1K22K3 + 224K1K2K3K4*2 - 576K1K2K3K4 - 64K1K2K3K6 + 5256K1K2K3 - 32K1K2K4K7 + 96K1K3*3K4 - 64K1K3*2K5 - 32K1K3K4K6 + 2808K1K3K4 + 408K1K4K5 + 64K1K5K6 + 8K1K6K7 - 64K24 - 416K2*2K3*2 - 432K2*2K4*2 + 1320K2*2K4 - 3528K22 - 288K2K32K4 - 64K2K3K4K5 + 608K2K3K5 + 328K2K4K6 - 144K34 - 224K3*2K4*2 + 224K3*2K6 - 2756K32 + 120K3K4K7 - 8K44 + 8K4*2K8 - 1468K42 - 268K5*2 - 120K6*2 - 16K7*2 - 2K8**2 + 3964
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}], [{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}, {3, 5}, {4}, {2}, {1}], [{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}, {1, 3}, {2}]]
If K is slice	False

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