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Flat knot 6.288

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,1,1,4,4,3,0,1,2,2,0,1,1,0,2,2]
Flat knots (up to 7 crossings) with same phi are :['6.288']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']
Outer characteristic polynomial of the knot is: t^7+70t^5+65t^3+3t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.288']
2-strand cable arrow polynomial of the knot is: -64K16 + 512K1*4K2 - 1360K14 + 192K1*3K2K3 - 608K1*3K3 - 128K12K2*4 + 320K1*2K2*3 + 128K1*2K2*2K4 - 2032K12K2*2 + 192K1*2K2K32 + 64K1*2K2K42 - 192K1*2K2K4 + 3624K1*2K2 - 624K12K3*2 - 144K1*2K4*2 - 2236K1*2 + 224K1K23K3 - 576K1K2*2K3 - 96K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 448K1K2K3K4 - 64K1K2K3K6 + 3320K1K2K3 - 64K1K3*2K5 + 912K1K3K4 + 152K1K4K5 + 16K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 416K24 + 32K2*3K3K5 - 32K2*3K6 + 64K22K3*2K4 - 336K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 152K2*2K4*2 + 728K2*2K4 - 1900K22 + 448K2K3K5 + 64K2K4K6 - 64K3*4 - 48K3*2K4*2 + 80K3*2K6 - 1108K32 + 16K3K4K7 - 8K44 - 372K4*2 - 104K5*2 - 20K6**2 + 2010
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.288']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13866', 'vk6.13961', 'vk6.14139', 'vk6.14362', 'vk6.14937', 'vk6.15060', 'vk6.15591', 'vk6.16061', 'vk6.16289', 'vk6.16314', 'vk6.16355', 'vk6.16397', 'vk6.17439', 'vk6.22600', 'vk6.22633', 'vk6.22774', 'vk6.23951', 'vk6.25974', 'vk6.26364', 'vk6.33685', 'vk6.34145', 'vk6.34646', 'vk6.34719', 'vk6.34737', 'vk6.36212', 'vk6.36243', 'vk6.38071', 'vk6.38091', 'vk6.42278', 'vk6.44555', 'vk6.44577', 'vk6.53856', 'vk6.54397', 'vk6.54614', 'vk6.55580', 'vk6.56529', 'vk6.56538', 'vk6.59044', 'vk6.59142', 'vk6.64568']
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,-1,0,1,1,3,1,1,4,4,3,0,1,2,2,0,1,1,0,2,2]

Flat knots (up to 7 crossings) with same phi are :['6.288']

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

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.168', '6.222', '6.288']

Outer characteristic polynomial of the knot is: t^7+70t^5+65t^3+3t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.288']

2-strand cable arrow polynomial of the knot is: -64*K1**6 + 512*K1**4*K2 - 1360*K1**4 + 192*K1**3*K2*K3 - 608*K1**3*K3 - 128*K1**2*K2**4 + 320*K1**2*K2**3 + 128*K1**2*K2**2*K4 - 2032*K1**2*K2**2 + 192*K1**2*K2*K3**2 + 64*K1**2*K2*K4**2 - 192*K1**2*K2*K4 + 3624*K1**2*K2 - 624*K1**2*K3**2 - 144*K1**2*K4**2 - 2236*K1**2 + 224*K1*K2**3*K3 - 576*K1*K2**2*K3 - 96*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 448*K1*K2*K3*K4 - 64*K1*K2*K3*K6 + 3320*K1*K2*K3 - 64*K1*K3**2*K5 + 912*K1*K3*K4 + 152*K1*K4*K5 + 16*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 128*K2**4*K4 - 416*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 + 64*K2**2*K3**2*K4 - 336*K2**2*K3**2 - 32*K2**2*K3*K7 + 32*K2**2*K4**3 - 152*K2**2*K4**2 + 728*K2**2*K4 - 1900*K2**2 + 448*K2*K3*K5 + 64*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 80*K3**2*K6 - 1108*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 372*K4**2 - 104*K5**2 - 20*K6**2 + 2010

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.288']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.13866', 'vk6.13961', 'vk6.14139', 'vk6.14362', 'vk6.14937', 'vk6.15060', 'vk6.15591', 'vk6.16061', 'vk6.16289', 'vk6.16314', 'vk6.16355', 'vk6.16397', 'vk6.17439', 'vk6.22600', 'vk6.22633', 'vk6.22774', 'vk6.23951', 'vk6.25974', 'vk6.26364', 'vk6.33685', 'vk6.34145', 'vk6.34646', 'vk6.34719', 'vk6.34737', 'vk6.36212', 'vk6.36243', 'vk6.38071', 'vk6.38091', 'vk6.42278', 'vk6.44555', 'vk6.44577', 'vk6.53856', 'vk6.54397', 'vk6.54614', 'vk6.55580', 'vk6.56529', 'vk6.56538', 'vk6.59044', 'vk6.59142', 'vk6.64568']

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	O1O2O3O4O5U1U5O6U3U6U2U4
R3 orbit	{'O1O2O3O4O5U1U5U2O6U3U6U4', 'O1O2O3O4O5U1U5O6U3U6U2U4'}
R3 orbit length	2
Gauss code of -K	O1O2O3O4O5U2U4U6U3O6U1U5
Gauss code of K*	O1O2O3O4U5U3U1U4U6O5O6U2
Gauss code of -K*	O1O2O3O4U3O5O6U5U1U4U2U6
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 0 -1 3 1 1],[ 4 0 3 2 4 1 1],[ 0 -3 0 -1 2 0 1],[ 1 -2 1 0 2 0 1],[-3 -4 -2 -2 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 -1 0 0 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 0 0 -2 -2 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 -1 -1 -1],[ 0 2 0 1 0 -1 -3],[ 1 2 0 1 1 0 -2],[ 4 4 1 1 3 2 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,0,0,2,2,4,0,0,0,1,1,1,1,1,3,2]
Phi over symmetry	[-4,-1,0,1,1,3,1,1,4,4,3,0,1,2,2,0,1,1,0,2,2]
Phi of -K	[-4,-1,0,1,1,3,1,1,4,4,3,0,1,2,2,0,1,1,0,2,2]
Phi of K*	[-3,-1,-1,0,1,4,2,2,1,2,3,0,0,1,4,1,2,4,0,1,1]
Phi of -K*	[-4,-1,0,1,1,3,2,3,1,1,4,1,0,1,2,0,1,2,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	2z^2+15z+23
Enhanced Jones-Krushkal polynomial	2w^3z^2+15w^2z+23w
Inner characteristic polynomial	t^6+42t^4+23t^2
Outer characteristic polynomial	t^7+70t^5+65t^3+3t
Flat arrow polynomial	4K13 + 4K1*2K2 - 8K12 - 6K1K2 - 2K2*2 + 2K2 + 2*K3 + 5
2-strand cable arrow polynomial	-64K16 + 512K1*4K2 - 1360K14 + 192K1*3K2K3 - 608K1*3K3 - 128K12K2*4 + 320K1*2K2*3 + 128K1*2K2*2K4 - 2032K12K2*2 + 192K1*2K2K32 + 64K1*2K2K42 - 192K1*2K2K4 + 3624K1*2K2 - 624K12K3*2 - 144K1*2K4*2 - 2236K1*2 + 224K1K23K3 - 576K1K2*2K3 - 96K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 448K1K2K3K4 - 64K1K2K3K6 + 3320K1K2K3 - 64K1K3*2K5 + 912K1K3K4 + 152K1K4K5 + 16K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 128K2*4K4 - 416K24 + 32K2*3K3K5 - 32K2*3K6 + 64K22K3*2K4 - 336K22K3*2 - 32K2*2K3K7 + 32K2*2K4*3 - 152K2*2K4*2 + 728K2*2K4 - 1900K22 + 448K2K3K5 + 64K2K4K6 - 64K3*4 - 48K3*2K4*2 + 80K3*2K6 - 1108K32 + 16K3K4K7 - 8K44 - 372K4*2 - 104K5*2 - 20K6**2 + 2010
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}, {4, 5}, {3}, {2}, {1}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {1, 4}, {3}, {2}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}]]
If K is slice	False

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