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Flat knot 5.23

Min(phi) over symmetries of the knot is: [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Flat knots (up to 7 crossings) with same phi are :['5.23', '6.970']
Arrow polynomial of the knot is: -4*K1**3 + 2*K1**2 + 4*K1*K2 + K1 - K2 - K3
Flat knots (up to 7 crossings) with same arrow polynomial are :['5.23', '5.54', '5.58', '5.84', '5.86', '5.91', '5.93', '7.343', '7.862', '7.952', '7.2246', '7.2258', '7.2600', '7.2609', '7.3002', '7.3006', '7.3345', '7.3391', '7.3719', '7.3739', '7.3767', '7.4199', '7.4245', '7.5100', '7.5658', '7.6743', '7.6906', '7.7950', '7.9676', '7.9682', '7.9714', '7.9938', '7.9953', '7.9968', '7.10026', '7.10213', '7.10241', '7.10425', '7.11694', '7.11701', '7.12028', '7.12052', '7.12068', '7.12075', '7.12223', '7.12351', '7.12576', '7.12749', '7.12974', '7.13452', '7.13468', '7.13484', '7.13494', '7.13612', '7.13825', '7.14007', '7.14155', '7.14216', '7.14249', '7.14263', '7.14509', '7.14667', '7.14888', '7.14930', '7.14946', '7.15030', '7.15056', '7.15088', '7.15089', '7.15112', '7.15136', '7.15168', '7.16066', '7.16078', '7.16193', '7.16269', '7.16665', '7.16724', '7.16903', '7.16965', '7.17023', '7.17044', '7.17062', '7.17100', '7.17169', '7.17181', '7.17202', '7.17260', '7.17289', '7.17374', '7.17565', '7.17719', '7.20216', '7.20217', '7.20450', '7.20451', '7.20754', '7.20976', '7.21245', '7.21369', '7.21371', '7.21454', '7.21455', '7.21743', '7.21777', '7.22139', '7.22141', '7.22181', '7.22230', '7.22553', '7.22755', '7.22867', '7.23547', '7.24141', '7.24144', '7.24163', '7.24303', '7.24374', '7.24385', '7.24534', '7.24597', '7.24601', '7.24603', '7.24816', '7.24820', '7.24951', '7.24994', '7.25332', '7.25334', '7.25439', '7.25467', '7.25476', '7.25707', '7.25759', '7.25760', '7.25793', '7.25840', '7.25966', '7.25995', '7.26126', '7.26160', '7.26240', '7.26448', '7.26449', '7.26458', '7.26628', '7.26677', '7.26770', '7.26785', '7.26839', '7.26949', '7.26951', '7.27167', '7.27352', '7.27546', '7.27585', '7.27587', '7.27592', '7.27645', '7.27653', '7.27684', '7.27785', '7.27865', '7.27883', '7.27894', '7.28174', '7.28177', '7.28185', '7.28368', '7.28379', '7.28385', '7.28403', '7.28409', '7.28418', '7.28553', '7.28660', '7.28720', '7.28981', '7.28984', '7.29094', '7.29460', '7.29462', '7.29489', '7.29638', '7.29749', '7.29995', '7.30005', '7.30009', '7.30170', '7.30179', '7.30186', '7.30389', '7.30392', '7.30404', '7.30412', '7.30472', '7.30486', '7.30556', '7.30559', '7.30561', '7.30571', '7.30575', '7.30579', '7.30617', '7.30641', '7.30656', '7.30811', '7.30860', '7.30882', '7.30920', '7.30927', '7.30931', '7.30934', '7.31006', '7.31028', '7.31115', '7.31199', '7.31253', '7.31753', '7.32124', '7.32148', '7.32167', '7.32255', '7.32385', '7.32489', '7.32490', '7.32494', '7.32510', '7.32511', '7.32617', '7.32631', '7.32632', '7.32804', '7.33038', '7.33562', '7.33581', '7.33592', '7.33615', '7.33626', '7.33736', '7.33805', '7.33817', '7.33824', '7.33983', '7.34153', '7.34166', '7.34192', '7.34204', '7.34205', '7.34261', '7.34303', '7.34309', '7.34312', '7.34342', '7.34408', '7.34537', '7.34587', '7.35187', '7.35763', '7.35785', '7.36802', '7.36922', '7.36968', '7.37132', '7.37164', '7.37173', '7.37186', '7.37210', '7.37222', '7.37279', '7.37342', '7.37441', '7.37444', '7.37468', '7.37924', '7.37948', '7.37949', '7.37963', '7.37964', '7.38100', '7.38129', '7.38180', '7.38198', '7.38213', '7.38231', '7.38250', '7.38267', '7.38321', '7.38350', '7.38369', '7.38378', '7.38411', '7.38454', '7.38456', '7.38470', '7.38473', '7.38489', '7.38517', '7.38539', '7.38551', '7.38612', '7.38664', '7.38666', '7.38686', '7.38716', '7.38727', '7.38739', '7.38778', '7.38794', '7.38795', '7.38796', '7.38818', '7.38860', '7.38883', '7.38884', '7.38960', '7.39276', '7.39277', '7.39279', '7.39285', '7.39287', '7.39293', '7.39489', '7.39524', '7.39583', '7.39610', '7.39611', '7.39650', '7.39680', '7.39936', '7.40104', '7.40335', '7.40341', '7.40470', '7.40501', '7.40517', '7.40520', '7.40647', '7.40653', '7.40711', '7.40719', '7.40721', '7.40742', '7.40772', '7.40826', '7.40847', '7.40892', '7.40912', '7.40941', '7.40970', '7.41038', '7.41100', '7.41102', '7.41120', '7.41159', '7.41172', '7.41174', '7.41196', '7.41203', '7.41205', '7.41219', '7.41238', '7.41254', '7.41257', '7.41264', '7.41265', '7.41280', '7.41300', '7.41315', '7.41318', '7.41328', '7.41393', '7.41415', '7.41434', '7.41450', '7.41463', '7.41528', '7.41530', '7.41541', '7.41607', '7.41627', '7.41630', '7.41650', '7.41659', '7.41673', '7.41754', '7.41766', '7.41769', '7.41778', '7.41816', '7.41825', '7.41829', '7.41834', '7.41852', '7.41856', '7.41875', '7.41902', '7.41912', '7.42043', '7.42059', '7.42280', '7.42330', '7.42346', '7.42380', '7.42381', '7.42409', '7.42410', '7.42411', '7.42416', '7.42518', '7.42524', '7.42527', '7.42540', '7.42580', '7.42605', '7.42608', '7.42616', '7.42770', '7.42773', '7.42788', '7.42791', '7.42796', '7.42838', '7.42843', '7.42910', '7.42918', '7.43007', '7.43026', '7.43080', '7.43164', '7.43295', '7.43296', '7.43372', '7.43392', '7.43435', '7.43497', '7.43545', '7.43838', '7.43866', '7.43896', '7.43932', '7.43947', '7.44008', '7.44044', '7.44079', '7.44114', '7.44124', '7.44216', '7.44242', '7.44243', '7.44244', '7.44248', '7.44256', '7.44278']
Outer characteristic polynomial of the knot is: t^6+26t^4+24t^2+1
Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['5.23']
2-strand cable arrow polynomial of the knot is: -256*K1**4 + 128*K1**3*K2*K3 + 224*K1**2*K2**3 - 1392*K1**2*K2**2 - 32*K1**2*K2*K4 + 1432*K1**2*K2 - 256*K1**2*K3**2 - 768*K1**2 + 256*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 1536*K1*K2*K3 + 264*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 16*K2**2*K4**2 + 416*K2**2*K4 - 446*K2**2 + 184*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 70*K4**2 - 12*K5**2 - 2*K6**2 + 620
Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['5.23']
Virtual knots (up to 6 crossings) projecting to this knot are :'vk5.20', 'vk5.32', 'vk5.35', 'vk5.68', 'vk5.77', 'vk5.82', 'vk5.99', 'vk5.118', 'vk5.247', 'vk5.339', 'vk5.686', 'vk5.721', 'vk5.836', 'vk5.875', 'vk5.1232', 'vk5.1701']
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 O1O2O3O4O5U3U5U4U1U2
R3 orbit {'O1O2O3O4O5U3U5U4U1U2'}
R3 orbit length 1
Gauss code of -K O1O2O3O4O5U4U5U2U1U3
Gauss code of K* O1O2O3O4O5U4U5U1U3U2
Gauss code of -K* O1O2O3O4O5U4U3U5U1U2
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 -1 1 -2 1 1],[ 1 0 1 -2 1 1],[-1 -1 0 -2 1 1],[ 2 2 2 0 2 1],[-1 -1 -1 -2 0 0],[-1 -1 -1 -1 0 0]]
Primitive based matrix [[ 0 1 1 1 -1 -2],[-1 0 1 1 -1 -2],[-1 -1 0 0 -1 -1],[-1 -1 0 0 -1 -2],[ 1 1 1 1 0 -2],[ 2 2 1 2 2 0]]
If based matrix primitive True
Phi of primitive based matrix [-1,-1,-1,1,2,-1,-1,1,2,0,1,1,1,2,2]
Phi over symmetry [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Phi of -K [-2,-1,1,1,1,-1,1,1,2,1,1,1,-1,-1,0]
Phi of K* [-1,-1,-1,1,2,-1,0,1,1,1,1,1,1,2,-1]
Phi of -K* [-2,-1,1,1,1,2,1,2,2,1,1,1,-1,0,1]
Symmetry type of based matrix c
u-polynomial t^2-2t
Normalized Jones-Krushkal polynomial -3z^2-14z-15
Enhanced Jones-Krushkal polynomial -3w^3z^2-14w^2z-15w
Inner characteristic polynomial t^5+18t^3+7t
Outer characteristic polynomial t^6+26t^4+24t^2+1
Flat arrow polynomial -4*K1**3 + 2*K1**2 + 4*K1*K2 + K1 - K2 - K3
2-strand cable arrow polynomial -256*K1**4 + 128*K1**3*K2*K3 + 224*K1**2*K2**3 - 1392*K1**2*K2**2 - 32*K1**2*K2*K4 + 1432*K1**2*K2 - 256*K1**2*K3**2 - 768*K1**2 + 256*K1*K2**3*K3 - 512*K1*K2**2*K3 - 64*K1*K2**2*K5 + 1536*K1*K2*K3 + 264*K1*K3*K4 + 8*K1*K4*K5 - 32*K2**6 + 64*K2**4*K4 - 520*K2**4 - 32*K2**3*K6 - 400*K2**2*K3**2 - 16*K2**2*K4**2 + 416*K2**2*K4 - 446*K2**2 + 184*K2*K3*K5 + 16*K2*K4*K6 - 340*K3**2 - 70*K4**2 - 12*K5**2 - 2*K6**2 + 620
Genus of based matrix 1
Fillings of based matrix [[{3, 5}, {4}, {1, 2}], [{4, 5}, {3}, {1, 2}], [{5}, {3, 4}, {1, 2}]]
If K is slice False
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