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Flat knot 6.180

Min(phi) over symmetries of the knot is: [-4,-1,0,1,2,2,1,3,4,2,3,2,1,1,2,0,1,1,2,2,0]
Flat knots (up to 7 crossings) with same phi are :['6.180']
Arrow polynomial of the knot is: -2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']
Outer characteristic polynomial of the knot is: t^7+69t^5+140t^3+9t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.180']
2-strand cable arrow polynomial of the knot is: 32K14K2 - 464K14 + 448K1*3K2K3 - 736K1*3K3 - 256K12K2*2K3*2 - 1808K1*2K2*2 + 512K1*2K2K32 + 32K1*2K2K3K5 - 160K12K2K4 + 5192K1*2K2 - 1712K12K3*2 - 32K1*2K3K5 - 5412K1*2 + 480K1K23K3 + 160K1K2*2K3K4 - 1952K1K22K3 - 128K1K2*2K5 - 736K1K2K3K4 - 64K1K2K3K6 + 7352K1K2K3 + 2408K1K3K4 + 112K1K4K5 + 32K1K5K6 - 520K24 - 32K2*3K6 - 1440K22K3*2 - 48K2*2K4*2 + 1600K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4432K2*2 + 1240K2K3K5 + 104K2K4K6 + 24K2K5K7 + 8K2K6K8 - 192K3*4 + 144K3*2K6 - 2840K32 + 8K3K4K7 - 976K42 - 216K5*2 - 64K6*2 - 12K7*2 - 2K8**2 + 4368
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.180']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4658', 'vk6.4945', 'vk6.6108', 'vk6.6595', 'vk6.8117', 'vk6.8519', 'vk6.9507', 'vk6.9862', 'vk6.20378', 'vk6.21721', 'vk6.27694', 'vk6.29240', 'vk6.39130', 'vk6.41386', 'vk6.45870', 'vk6.47533', 'vk6.48698', 'vk6.48901', 'vk6.49458', 'vk6.49677', 'vk6.50714', 'vk6.50913', 'vk6.51197', 'vk6.51398', 'vk6.57239', 'vk6.58466', 'vk6.61861', 'vk6.62998', 'vk6.66858', 'vk6.67728', 'vk6.69486', 'vk6.70210']
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,-1,0,1,2,2,1,3,4,2,3,2,1,1,2,0,1,1,2,2,0]

Flat knots (up to 7 crossings) with same phi are :['6.180']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.180', '6.263', '6.295', '6.317', '6.350', '6.473', '6.504']

Outer characteristic polynomial of the knot is: t^7+69t^5+140t^3+9t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.180']

2-strand cable arrow polynomial of the knot is: 32*K1**4*K2 - 464*K1**4 + 448*K1**3*K2*K3 - 736*K1**3*K3 - 256*K1**2*K2**2*K3**2 - 1808*K1**2*K2**2 + 512*K1**2*K2*K3**2 + 32*K1**2*K2*K3*K5 - 160*K1**2*K2*K4 + 5192*K1**2*K2 - 1712*K1**2*K3**2 - 32*K1**2*K3*K5 - 5412*K1**2 + 480*K1*K2**3*K3 + 160*K1*K2**2*K3*K4 - 1952*K1*K2**2*K3 - 128*K1*K2**2*K5 - 736*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 7352*K1*K2*K3 + 2408*K1*K3*K4 + 112*K1*K4*K5 + 32*K1*K5*K6 - 520*K2**4 - 32*K2**3*K6 - 1440*K2**2*K3**2 - 48*K2**2*K4**2 + 1600*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 4432*K2**2 + 1240*K2*K3*K5 + 104*K2*K4*K6 + 24*K2*K5*K7 + 8*K2*K6*K8 - 192*K3**4 + 144*K3**2*K6 - 2840*K3**2 + 8*K3*K4*K7 - 976*K4**2 - 216*K5**2 - 64*K6**2 - 12*K7**2 - 2*K8**2 + 4368

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.180']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.4658', 'vk6.4945', 'vk6.6108', 'vk6.6595', 'vk6.8117', 'vk6.8519', 'vk6.9507', 'vk6.9862', 'vk6.20378', 'vk6.21721', 'vk6.27694', 'vk6.29240', 'vk6.39130', 'vk6.41386', 'vk6.45870', 'vk6.47533', 'vk6.48698', 'vk6.48901', 'vk6.49458', 'vk6.49677', 'vk6.50714', 'vk6.50913', 'vk6.51197', 'vk6.51398', 'vk6.57239', 'vk6.58466', 'vk6.61861', 'vk6.62998', 'vk6.66858', 'vk6.67728', 'vk6.69486', 'vk6.70210']

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	O1O2O3O4O5U1O6U5U6U2U4U3
R3 orbit	{'O1O2O3O4O5U1O6U5U6U2U4U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U2U4U6U1O6U5
Gauss code of K*	O1O2O3O4O5U6U3U5U4U1O6U2
Gauss code of -K*	O1O2O3O4O5U4O6U5U2U1U3U6
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 2 2 0 1],[ 4 0 2 4 3 1 1],[ 1 -2 0 2 1 -1 1],[-2 -4 -2 0 0 -1 1],[-2 -3 -1 0 0 -1 1],[ 0 -1 1 1 1 0 1],[-1 -1 -1 -1 -1 -1 0]]
Primitive based matrix	[[ 0 2 2 1 0 -1 -4],[-2 0 0 1 -1 -1 -3],[-2 0 0 1 -1 -2 -4],[-1 -1 -1 0 -1 -1 -1],[ 0 1 1 1 0 1 -1],[ 1 1 2 1 -1 0 -2],[ 4 3 4 1 1 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-2,-1,0,1,4,0,-1,1,1,3,-1,1,2,4,1,1,1,-1,1,2]
Phi over symmetry	[-4,-1,0,1,2,2,1,3,4,2,3,2,1,1,2,0,1,1,2,2,0]
Phi of -K	[-4,-1,0,1,2,2,1,3,4,2,3,2,1,1,2,0,1,1,2,2,0]
Phi of K*	[-2,-2,-1,0,1,4,0,2,1,1,2,2,1,2,3,0,1,4,2,3,1]
Phi of -K*	[-4,-1,0,1,2,2,2,1,1,3,4,-1,1,1,2,1,1,1,-1,-1,0]
Symmetry type of based matrix	c
u-polynomial	t^4-2t^2
Normalized Jones-Krushkal polynomial	5z^2+24z+29
Enhanced Jones-Krushkal polynomial	5w^3z^2-2w^3z+26w^2z+29w
Inner characteristic polynomial	t^6+43t^4+38t^2
Outer characteristic polynomial	t^7+69t^5+140t^3+9t
Flat arrow polynomial	-2K12 - 4K1K2 - 2K1K3 + 2K1 + 2K2 + 2K3 + K4 + 2
2-strand cable arrow polynomial	32K14K2 - 464K14 + 448K1*3K2K3 - 736K1*3K3 - 256K12K2*2K3*2 - 1808K1*2K2*2 + 512K1*2K2K32 + 32K1*2K2K3K5 - 160K12K2K4 + 5192K1*2K2 - 1712K12K3*2 - 32K1*2K3K5 - 5412K1*2 + 480K1K23K3 + 160K1K2*2K3K4 - 1952K1K22K3 - 128K1K2*2K5 - 736K1K2K3K4 - 64K1K2K3K6 + 7352K1K2K3 + 2408K1K3K4 + 112K1K4K5 + 32K1K5K6 - 520K24 - 32K2*3K6 - 1440K22K3*2 - 48K2*2K4*2 + 1600K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 4432K2*2 + 1240K2K3K5 + 104K2K4K6 + 24K2K5K7 + 8K2K6K8 - 192K3*4 + 144K3*2K6 - 2840K32 + 8K3K4K7 - 976K42 - 216K5*2 - 64K6*2 - 12K7*2 - 2K8**2 + 4368
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}], [{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}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {2, 5}, {4}, {3}, {1}], [{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}, {2, 3}, {1}]]
If K is slice	False

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