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Flat knot 6.508

Min(phi) over symmetries of the knot is: [-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,0,1,0,0,2]
Flat knots (up to 7 crossings) with same phi are :['6.508']
Arrow polynomial of the knot is: 4K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K2*2 + 4K2 + 2*K3 + 7
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.508']
Outer characteristic polynomial of the knot is: t^7+92t^5+88t^3+11t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.508']
2-strand cable arrow polynomial of the knot is: -192K16 - 384K1*4K2*2 + 2528K1*4K2 - 4912K14 - 256K1*3K2*2K3 + 1312K13K2K3 + 64K1*3K3K4 - 1184K1*3K3 - 128K12K2*4 + 1504K1*2K2*3 + 416K1*2K2*2K4 - 8336K12K2*2 + 320K1*2K2K32 + 128K1*2K2K42 - 1312K1*2K2K4 + 11264K1*2K2 - 1264K12K3*2 - 368K1*2K4*2 - 6020K1*2 + 608K1K23K3 + 32K1K2*2K3K4 - 2176K1K22K3 - 416K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 704K1K2K3K4 + 9808K1K2K3 - 160K1K2K4K5 + 2496K1K3K4 + 528K1K4K5 + 32K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1392K24 + 32K2*3K3K5 + 64K2*2K3*2K4 - 816K22K3*2 + 32K2*2K4*3 - 376K2*2K4*2 + 2520K2*2K4 - 5748K22 - 160K2K32K4 - 32K2K3K4K5 + 968K2K3K5 + 328K2K4K6 - 64K34 - 48K3*2K4*2 + 88K3*2K6 - 2956K32 + 16K3K4K7 - 8K44 - 1320K4*2 - 352K5*2 - 84K6**2 + 5950
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.508']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11480', 'vk6.11785', 'vk6.12802', 'vk6.13139', 'vk6.17055', 'vk6.17297', 'vk6.20858', 'vk6.20938', 'vk6.22265', 'vk6.22348', 'vk6.23775', 'vk6.28328', 'vk6.31245', 'vk6.31596', 'vk6.32820', 'vk6.35563', 'vk6.36014', 'vk6.39948', 'vk6.40104', 'vk6.42025', 'vk6.42966', 'vk6.43263', 'vk6.46493', 'vk6.46626', 'vk6.52233', 'vk6.53070', 'vk6.53388', 'vk6.55463', 'vk6.58853', 'vk6.59943', 'vk6.64410', 'vk6.69721']
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,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,0,1,0,0,2]

Flat knots (up to 7 crossings) with same phi are :['6.508']

Arrow polynomial of the knot is: 4*K1**3 + 4*K1**2*K2 - 12*K1**2 - 6*K1*K2 - 2*K2**2 + 4*K2 + 2*K3 + 7

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.508']

Outer characteristic polynomial of the knot is: t^7+92t^5+88t^3+11t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.508']

2-strand cable arrow polynomial of the knot is: -192*K1**6 - 384*K1**4*K2**2 + 2528*K1**4*K2 - 4912*K1**4 - 256*K1**3*K2**2*K3 + 1312*K1**3*K2*K3 + 64*K1**3*K3*K4 - 1184*K1**3*K3 - 128*K1**2*K2**4 + 1504*K1**2*K2**3 + 416*K1**2*K2**2*K4 - 8336*K1**2*K2**2 + 320*K1**2*K2*K3**2 + 128*K1**2*K2*K4**2 - 1312*K1**2*K2*K4 + 11264*K1**2*K2 - 1264*K1**2*K3**2 - 368*K1**2*K4**2 - 6020*K1**2 + 608*K1*K2**3*K3 + 32*K1*K2**2*K3*K4 - 2176*K1*K2**2*K3 - 416*K1*K2**2*K5 + 64*K1*K2*K3**3 + 32*K1*K2*K3*K4**2 - 704*K1*K2*K3*K4 + 9808*K1*K2*K3 - 160*K1*K2*K4*K5 + 2496*K1*K3*K4 + 528*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 - 64*K2**4*K3**2 - 32*K2**4*K4**2 + 160*K2**4*K4 - 1392*K2**4 + 32*K2**3*K3*K5 + 64*K2**2*K3**2*K4 - 816*K2**2*K3**2 + 32*K2**2*K4**3 - 376*K2**2*K4**2 + 2520*K2**2*K4 - 5748*K2**2 - 160*K2*K3**2*K4 - 32*K2*K3*K4*K5 + 968*K2*K3*K5 + 328*K2*K4*K6 - 64*K3**4 - 48*K3**2*K4**2 + 88*K3**2*K6 - 2956*K3**2 + 16*K3*K4*K7 - 8*K4**4 - 1320*K4**2 - 352*K5**2 - 84*K6**2 + 5950

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.508']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.11480', 'vk6.11785', 'vk6.12802', 'vk6.13139', 'vk6.17055', 'vk6.17297', 'vk6.20858', 'vk6.20938', 'vk6.22265', 'vk6.22348', 'vk6.23775', 'vk6.28328', 'vk6.31245', 'vk6.31596', 'vk6.32820', 'vk6.35563', 'vk6.36014', 'vk6.39948', 'vk6.40104', 'vk6.42025', 'vk6.42966', 'vk6.43263', 'vk6.46493', 'vk6.46626', 'vk6.52233', 'vk6.53070', 'vk6.53388', 'vk6.55463', 'vk6.58853', 'vk6.59943', 'vk6.64410', 'vk6.69721']

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	O1O2O3O4O5U1U6U4O6U3U5U2
R3 orbit	{'O1O2O3O4O5U1U6U4O6U3U5U2'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U4U1U3O6U2U6U5
Gauss code of K*	O1O2O3U2O4O5O6U1U6U4U3U5
Gauss code of -K*	O1O2O3U4O5O4O6U2U5U3U1U6
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 0 1 3 -1],[ 4 0 4 2 1 3 3],[-1 -4 0 -1 0 2 -2],[ 0 -2 1 0 1 2 -1],[-1 -1 0 -1 0 0 -1],[-3 -3 -2 -2 0 0 -3],[ 1 -3 2 1 1 3 0]]
Primitive based matrix	[[ 0 3 1 1 0 -1 -4],[-3 0 0 -2 -2 -3 -3],[-1 0 0 0 -1 -1 -1],[-1 2 0 0 -1 -2 -4],[ 0 2 1 1 0 -1 -2],[ 1 3 1 2 1 0 -3],[ 4 3 1 4 2 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-3,-1,-1,0,1,4,0,2,2,3,3,0,1,1,1,1,2,4,1,2,3]
Phi over symmetry	[-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,0,1,0,0,2]
Phi of -K	[-4,-1,0,1,1,3,0,2,1,4,4,0,0,1,1,0,0,1,0,0,2]
Phi of K*	[-3,-1,-1,0,1,4,0,2,1,1,4,0,0,0,1,0,1,4,0,2,0]
Phi of -K*	[-4,-1,0,1,1,3,3,2,1,4,3,1,1,2,3,1,1,2,0,0,2]
Symmetry type of based matrix	c
u-polynomial	t^4-t^3-t
Normalized Jones-Krushkal polynomial	4z^2+25z+35
Enhanced Jones-Krushkal polynomial	4w^3z^2+25w^2z+35w
Inner characteristic polynomial	t^6+64t^4+52t^2+4
Outer characteristic polynomial	t^7+92t^5+88t^3+11t
Flat arrow polynomial	4K13 + 4K1*2K2 - 12K12 - 6K1K2 - 2K2*2 + 4K2 + 2*K3 + 7
2-strand cable arrow polynomial	-192K16 - 384K1*4K2*2 + 2528K1*4K2 - 4912K14 - 256K1*3K2*2K3 + 1312K13K2K3 + 64K1*3K3K4 - 1184K1*3K3 - 128K12K2*4 + 1504K1*2K2*3 + 416K1*2K2*2K4 - 8336K12K2*2 + 320K1*2K2K32 + 128K1*2K2K42 - 1312K1*2K2K4 + 11264K1*2K2 - 1264K12K3*2 - 368K1*2K4*2 - 6020K1*2 + 608K1K23K3 + 32K1K2*2K3K4 - 2176K1K22K3 - 416K1K2*2K5 + 64K1K2K33 + 32K1K2K3K42 - 704K1K2K3K4 + 9808K1K2K3 - 160K1K2K4K5 + 2496K1K3K4 + 528K1K4K5 + 32K1K5K6 - 32K2*6 - 64K2*4K3*2 - 32K2*4K4*2 + 160K2*4K4 - 1392K24 + 32K2*3K3K5 + 64K2*2K3*2K4 - 816K22K3*2 + 32K2*2K4*3 - 376K2*2K4*2 + 2520K2*2K4 - 5748K22 - 160K2K32K4 - 32K2K3K4K5 + 968K2K3K5 + 328K2K4K6 - 64K34 - 48K3*2K4*2 + 88K3*2K6 - 2956K32 + 16K3K4K7 - 8K44 - 1320K4*2 - 352K5*2 - 84K6**2 + 5950
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}, {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}], [{6}, {5}, {4}, {2, 3}, {1}]]
If K is slice	False

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