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Flat knot 6.499

Min(phi) over symmetries of the knot is: [-4,-1,1,1,1,2,0,3,4,4,2,2,1,2,1,0,0,1,0,1,1]
Flat knots (up to 7 crossings) with same phi are :['6.499']
Arrow polynomial of the knot is: 4K13 - 4K1K2 - 2K1*K3 - K1 + K2 + K3 + K4 + 1
Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']
Outer characteristic polynomial of the knot is: t^7+60t^5+94t^3+13t
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.499']
2-strand cable arrow polynomial of the knot is: -592K14 + 640K1*3K2K3 - 448K1*3K3 + 96K12K2*3 - 3168K1*2K2*2 - 160K1*2K2K4 + 3976K1*2K2 - 976K12K3*2 - 3672K1*2 + 864K1K23K3 - 800K1K2*2K3 - 256K1K2*2K5 + 192K1K2K33 - 320K1K2K3K4 - 96K1K2K3K6 + 6600K1K2K3 + 1240K1K3K4 + 88K1K4K5 + 32K1K5K6 - 32K2*6 + 64K2*4K4 - 1232K24 + 32K2*3K3K5 - 32K2*3K6 - 1728K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 1512K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3234K2*2 - 224K2K32K4 + 1512K2K3K5 + 144K2K4K6 + 16K2K5K7 + 8K2K6K8 - 128K34 + 200K3*2K6 - 2532K32 + 8K3K4K7 - 686K42 - 344K5*2 - 94K6*2 - 4K7*2 - 2K8**2 + 3606
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.499']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10148', 'vk6.10217', 'vk6.10362', 'vk6.10437', 'vk6.16685', 'vk6.19069', 'vk6.19116', 'vk6.19264', 'vk6.19556', 'vk6.22996', 'vk6.23115', 'vk6.25698', 'vk6.25745', 'vk6.26075', 'vk6.26449', 'vk6.29935', 'vk6.29998', 'vk6.30094', 'vk6.34987', 'vk6.35110', 'vk6.37792', 'vk6.37852', 'vk6.42556', 'vk6.44673', 'vk6.51632', 'vk6.51737', 'vk6.54905', 'vk6.56594', 'vk6.59328', 'vk6.64867', 'vk6.66186', 'vk6.66217']
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,-1,1,1,1,2,0,3,4,4,2,2,1,2,1,0,0,1,0,1,1]

Flat knots (up to 7 crossings) with same phi are :['6.499']

Arrow polynomial of the knot is: 4*K1**3 - 4*K1*K2 - 2*K1*K3 - K1 + K2 + K3 + K4 + 1

Flat knots (up to 7 crossings) with same arrow polynomial are :['6.71', '6.148', '6.170', '6.253', '6.259', '6.298', '6.439', '6.453', '6.499', '6.503']

Outer characteristic polynomial of the knot is: t^7+60t^5+94t^3+13t

Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.499']

2-strand cable arrow polynomial of the knot is: -592*K1**4 + 640*K1**3*K2*K3 - 448*K1**3*K3 + 96*K1**2*K2**3 - 3168*K1**2*K2**2 - 160*K1**2*K2*K4 + 3976*K1**2*K2 - 976*K1**2*K3**2 - 3672*K1**2 + 864*K1*K2**3*K3 - 800*K1*K2**2*K3 - 256*K1*K2**2*K5 + 192*K1*K2*K3**3 - 320*K1*K2*K3*K4 - 96*K1*K2*K3*K6 + 6600*K1*K2*K3 + 1240*K1*K3*K4 + 88*K1*K4*K5 + 32*K1*K5*K6 - 32*K2**6 + 64*K2**4*K4 - 1232*K2**4 + 32*K2**3*K3*K5 - 32*K2**3*K6 - 1728*K2**2*K3**2 - 32*K2**2*K3*K7 - 16*K2**2*K4**2 + 1512*K2**2*K4 - 16*K2**2*K5**2 - 8*K2**2*K6**2 - 3234*K2**2 - 224*K2*K3**2*K4 + 1512*K2*K3*K5 + 144*K2*K4*K6 + 16*K2*K5*K7 + 8*K2*K6*K8 - 128*K3**4 + 200*K3**2*K6 - 2532*K3**2 + 8*K3*K4*K7 - 686*K4**2 - 344*K5**2 - 94*K6**2 - 4*K7**2 - 2*K8**2 + 3606

Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.499']

Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.10148', 'vk6.10217', 'vk6.10362', 'vk6.10437', 'vk6.16685', 'vk6.19069', 'vk6.19116', 'vk6.19264', 'vk6.19556', 'vk6.22996', 'vk6.23115', 'vk6.25698', 'vk6.25745', 'vk6.26075', 'vk6.26449', 'vk6.29935', 'vk6.29998', 'vk6.30094', 'vk6.34987', 'vk6.35110', 'vk6.37792', 'vk6.37852', 'vk6.42556', 'vk6.44673', 'vk6.51632', 'vk6.51737', 'vk6.54905', 'vk6.56594', 'vk6.59328', 'vk6.64867', 'vk6.66186', 'vk6.66217']

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	O1O2O3O4O5U1U5U4O6U2U6U3
R3 orbit	{'O1O2O3O4O5U1U5U4O6U2U6U3'}
R3 orbit length	1
Gauss code of -K	O1O2O3O4O5U3U6U4O6U2U1U5
Gauss code of K*	O1O2O3U4O5O4O6U1U5U6U3U2
Gauss code of -K*	O1O2O3U2O4O5O6U5U4U1U3U6
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 1 1 1],[ 4 0 3 4 2 1 1],[ 1 -3 0 2 0 0 1],[-2 -4 -2 0 0 0 0],[-1 -2 0 0 0 0 0],[-1 -1 0 0 0 0 0],[-1 -1 -1 0 0 0 0]]
Primitive based matrix	[[ 0 2 1 1 1 -1 -4],[-2 0 0 0 0 -2 -4],[-1 0 0 0 0 0 -1],[-1 0 0 0 0 0 -2],[-1 0 0 0 0 -1 -1],[ 1 2 0 0 1 0 -3],[ 4 4 1 2 1 3 0]]
If based matrix primitive	True
Phi of primitive based matrix	[-2,-1,-1,-1,1,4,0,0,0,2,4,0,0,0,1,0,0,2,1,1,3]
Phi over symmetry	[-4,-1,1,1,1,2,0,3,4,4,2,2,1,2,1,0,0,1,0,1,1]
Phi of -K	[-4,-1,1,1,1,2,0,3,4,4,2,2,1,2,1,0,0,1,0,1,1]
Phi of K*	[-2,-1,-1,-1,1,4,1,1,1,1,2,0,0,1,4,0,2,3,2,4,0]
Phi of -K*	[-4,-1,1,1,1,2,3,1,1,2,4,0,1,0,2,0,0,0,0,0,0]
Symmetry type of based matrix	c
u-polynomial	t^4-t^2-2t
Normalized Jones-Krushkal polynomial	7z^2+28z+29
Enhanced Jones-Krushkal polynomial	7w^3z^2+28w^2z+29w
Inner characteristic polynomial	t^6+36t^4+29t^2
Outer characteristic polynomial	t^7+60t^5+94t^3+13t
Flat arrow polynomial	4K13 - 4K1K2 - 2K1*K3 - K1 + K2 + K3 + K4 + 1
2-strand cable arrow polynomial	-592K14 + 640K1*3K2K3 - 448K1*3K3 + 96K12K2*3 - 3168K1*2K2*2 - 160K1*2K2K4 + 3976K1*2K2 - 976K12K3*2 - 3672K1*2 + 864K1K23K3 - 800K1K2*2K3 - 256K1K2*2K5 + 192K1K2K33 - 320K1K2K3K4 - 96K1K2K3K6 + 6600K1K2K3 + 1240K1K3K4 + 88K1K4K5 + 32K1K5K6 - 32K2*6 + 64K2*4K4 - 1232K24 + 32K2*3K3K5 - 32K2*3K6 - 1728K22K3*2 - 32K2*2K3K7 - 16K2*2K4*2 + 1512K2*2K4 - 16K22K5*2 - 8K2*2K6*2 - 3234K2*2 - 224K2K32K4 + 1512K2K3K5 + 144K2K4K6 + 16K2K5K7 + 8K2K6K8 - 128K34 + 200K3*2K6 - 2532K32 + 8K3K4K7 - 686K42 - 344K5*2 - 94K6*2 - 4K7*2 - 2K8**2 + 3606
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}], [{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}], [{4, 6}, {5}, {3}, {2}, {1}], [{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}, {3}, {1, 2}]]
If K is slice	False

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