9 14

9_13

9_15

Visit 9 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 14's page at Knotilus! Visit 9 14's page at the original Knot Atlas! 9 14 Quick Notes

9 14 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X9,15,10,14 X7,17,8,16 X15,9,16,8 X17,7,18,6
Gauss code	-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 12 16 14 2 18 8 6
Conway Notation	[41112]

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,7\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-7] Hyperbolic Volume 8.95499 A-Polynomial See Data:9 14/A-polynomial

[edit Notes for 9 14's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant 0

[edit Notes for 9 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 37, 0 }
Jones polynomial	$q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}
Kauffman polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+4 a z^5-4 z^5 a^{-1} -16 z^5 a^{-3} -8 z^5 a^{-5} +3 a^2 z^4-12 z^4 a^{-2} -9 z^4 a^{-4} -4 z^4 a^{-6} -4 z^4+a^3 z^3-3 a z^3+2 z^3 a^{-1} +15 z^3 a^{-3} +9 z^3 a^{-5} -2 a^2 z^2+8 z^2 a^{-2} +10 z^2 a^{-4} +4 z^2 a^{-6} -2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1}
The A2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{10}+q^8+q^6-q^4+2 q^2+ q^{-2} + q^{-4} + q^{-8} -2 q^{-10} - q^{-12} - q^{-16} + q^{-18} + q^{-20} }
The G2 invariant	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{52}-2 q^{50}+3 q^{48}-4 q^{46}+q^{44}-3 q^{40}+8 q^{38}-10 q^{36}+11 q^{34}-8 q^{32}+2 q^{30}+4 q^{28}-10 q^{26}+16 q^{24}-16 q^{22}+14 q^{20}-8 q^{18}-q^{16}+11 q^{14}-15 q^{12}+17 q^{10}-12 q^8+3 q^6+6 q^4-10 q^2+10-2 q^{-2} -6 q^{-4} +15 q^{-6} -16 q^{-8} +9 q^{-10} +5 q^{-12} -19 q^{-14} +30 q^{-16} -28 q^{-18} +18 q^{-20} -16 q^{-24} +28 q^{-26} -30 q^{-28} +23 q^{-30} -10 q^{-32} -6 q^{-34} +16 q^{-36} -19 q^{-38} +15 q^{-40} -5 q^{-42} -7 q^{-44} +11 q^{-46} -13 q^{-48} +5 q^{-50} +5 q^{-52} -16 q^{-54} +21 q^{-56} -19 q^{-58} +6 q^{-60} +8 q^{-62} -20 q^{-64} +25 q^{-66} -21 q^{-68} +11 q^{-70} + q^{-72} -10 q^{-74} +16 q^{-76} -14 q^{-78} +10 q^{-80} -2 q^{-82} -2 q^{-84} +3 q^{-86} -4 q^{-88} +3 q^{-90} - q^{-92} + q^{-94} }

Further Quantum Invariants

Further quantum knot invariants for 9_14.

A1 Invariants.

Weight	Invariant
1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^7+2 q^5-q^3+2 q+ q^{-5} -2 q^{-7} + q^{-9} - q^{-11} + q^{-13} }
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{20}-2 q^{18}-q^{16}+4 q^{14}-4 q^{12}+7 q^8-5 q^6-q^4+7 q^2-3-3 q^{-2} +3 q^{-4} +2 q^{-6} -3 q^{-8} -2 q^{-10} +5 q^{-12} - q^{-14} -6 q^{-16} +5 q^{-18} +2 q^{-20} -6 q^{-22} +4 q^{-24} +4 q^{-26} -5 q^{-28} +3 q^{-32} -2 q^{-34} - q^{-36} + q^{-38} }
3	$-q^{39}+2q^{37}+q^{35}-2q^{33}-2q^{31}+q^{29}+3q^{27}-5q^{25}+6q^{21}-10q^{17}+3q^{15}+14q^{13}-q^{11}-17q^{9}+19q^{5}+5q^{3}-16q-9q^{-1}+10q^{-3}+11q^{-5}-2q^{-7}-14q^{-9}-3q^{-11}+12q^{-13}+10q^{-15}-11q^{-17}-13q^{-19}+8q^{-21}+16q^{-23}-6q^{-25}-16q^{-27}+2q^{-29}+18q^{-31}+2q^{-33}-17q^{-35}-6q^{-37}+16q^{-39}+12q^{-41}-13q^{-43}-15q^{-45}+5q^{-47}+16q^{-49}-q^{-51}-14q^{-53}-4q^{-55}+10q^{-57}+7q^{-59}-5q^{-61}-6q^{-63}+2q^{-65}+4q^{-67}-2q^{-71}-q^{-73}+q^{-75}$
4	$q^{64}-2q^{62}-q^{60}+2q^{58}+5q^{54}-4q^{52}-2q^{50}-6q^{46}+11q^{44}-2q^{42}+3q^{40}-21q^{36}+4q^{34}+6q^{32}+26q^{30}+10q^{28}-43q^{26}-22q^{24}+4q^{22}+56q^{20}+40q^{18}-48q^{16}-56q^{14}-24q^{12}+60q^{10}+70q^{8}-16q^{6}-54q^{4}-55q^{2}+21+63q^{-2}+24q^{-4}-14q^{-6}-52q^{-8}-26q^{-10}+19q^{-12}+41q^{-14}+30q^{-16}-25q^{-18}-48q^{-20}-17q^{-22}+41q^{-24}+47q^{-26}-3q^{-28}-53q^{-30}-36q^{-32}+39q^{-34}+51q^{-36}+8q^{-38}-57q^{-40}-48q^{-42}+34q^{-44}+54q^{-46}+29q^{-48}-47q^{-50}-64q^{-52}+8q^{-54}+45q^{-56}+56q^{-58}-14q^{-60}-63q^{-62}-29q^{-64}+8q^{-66}+62q^{-68}+30q^{-70}-28q^{-72}-41q^{-74}-34q^{-76}+32q^{-78}+44q^{-80}+15q^{-82}-17q^{-84}-47q^{-86}-6q^{-88}+21q^{-90}+27q^{-92}+12q^{-94}-25q^{-96}-17q^{-98}-4q^{-100}+12q^{-102}+16q^{-104}-4q^{-106}-6q^{-108}-6q^{-110}+6q^{-114}+q^{-116}-2q^{-120}-q^{-122}+q^{-124}$
5	$-q^{95}+2q^{93}+q^{91}-2q^{89}-3q^{85}-2q^{83}+3q^{81}+7q^{79}+3q^{77}-q^{75}-8q^{73}-12q^{71}-4q^{69}+13q^{67}+26q^{65}+10q^{63}-15q^{61}-36q^{59}-38q^{57}+9q^{55}+66q^{53}+62q^{51}+q^{49}-80q^{47}-112q^{45}-33q^{43}+111q^{41}+167q^{39}+71q^{37}-114q^{35}-226q^{33}-137q^{31}+104q^{29}+277q^{27}+209q^{25}-60q^{23}-298q^{21}-276q^{19}-8q^{17}+276q^{15}+324q^{13}+90q^{11}-216q^{9}-326q^{7}-162q^{5}+119q^{3}+282q+208q^{-1}-17q^{-3}-202q^{-5}-216q^{-7}-66q^{-9}+103q^{-11}+184q^{-13}+132q^{-15}-13q^{-17}-142q^{-19}-158q^{-21}-53q^{-23}+90q^{-25}+172q^{-27}+104q^{-29}-65q^{-31}-169q^{-33}-121q^{-35}+46q^{-37}+177q^{-39}+134q^{-41}-48q^{-43}-189q^{-45}-148q^{-47}+46q^{-49}+209q^{-51}+168q^{-53}-43q^{-55}-223q^{-57}-200q^{-59}+18q^{-61}+233q^{-63}+234q^{-65}+20q^{-67}-213q^{-69}-260q^{-71}-80q^{-73}+172q^{-75}+272q^{-77}+137q^{-79}-105q^{-81}-250q^{-83}-190q^{-85}+19q^{-87}+202q^{-89}+217q^{-91}+59q^{-93}-122q^{-95}-199q^{-97}-131q^{-99}+32q^{-101}+157q^{-103}+159q^{-105}+50q^{-107}-79q^{-109}-149q^{-111}-112q^{-113}+3q^{-115}+104q^{-117}+127q^{-119}+61q^{-121}-42q^{-123}-107q^{-125}-95q^{-127}-16q^{-129}+65q^{-131}+93q^{-133}+51q^{-135}-18q^{-137}-65q^{-139}-62q^{-141}-15q^{-143}+34q^{-145}+51q^{-147}+27q^{-149}-7q^{-151}-29q^{-153}-28q^{-155}-6q^{-157}+14q^{-159}+17q^{-161}+7q^{-163}-2q^{-165}-8q^{-167}-8q^{-169}+4q^{-173}+3q^{-175}+q^{-177}-2q^{-181}-q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{28}-4 q^{26}+8 q^{24}-12 q^{22}+18 q^{20}-28 q^{18}+34 q^{16}-38 q^{14}+43 q^{12}-48 q^{10}+50 q^8-44 q^6+46 q^4-32 q^2+22-24 q^{-4} +50 q^{-6} -80 q^{-8} +102 q^{-10} -118 q^{-12} +126 q^{-14} -126 q^{-16} +108 q^{-18} -91 q^{-20} +62 q^{-22} -30 q^{-24} +34 q^{-28} -50 q^{-30} +70 q^{-32} -76 q^{-34} +71 q^{-36} -62 q^{-38} +48 q^{-40} -36 q^{-42} +21 q^{-44} -12 q^{-46} +6 q^{-48} -2 q^{-50} + q^{-52} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{26}-q^{24}-2 q^{22}+q^{20}+2 q^{18}-q^{16}-4 q^{14}+3 q^{12}+5 q^{10}-3 q^8-2 q^6+6 q^4+4 q^2-2-2 q^{-2} +2 q^{-4} - q^{-8} + q^{-10} -3 q^{-14} +2 q^{-16} + q^{-18} -5 q^{-20} -3 q^{-22} +2 q^{-24} +3 q^{-26} -2 q^{-28} +5 q^{-32} +4 q^{-34} -2 q^{-36} -2 q^{-38} + q^{-40} + q^{-42} -2 q^{-44} -3 q^{-46} + q^{-50} + q^{-52} }

A3 Invariants.

Weight	Invariant
0,1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{22}-2 q^{20}-q^{18}+4 q^{16}-3 q^{14}-2 q^{12}+6 q^{10}-2 q^8-4 q^6+7 q^4+q^2-1+5 q^{-2} +2 q^{-4} -2 q^{-6} -3 q^{-8} -6 q^{-14} +2 q^{-16} +5 q^{-18} -4 q^{-20} +2 q^{-22} +4 q^{-24} -4 q^{-26} +2 q^{-30} -3 q^{-32} + q^{-34} + q^{-36} - q^{-38} + q^{-40} }
1,0,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{13}+q^{11}+q^7-q^5+2 q^3+ q^{-1} + q^{-3} + q^{-5} + q^{-7} + q^{-11} -2 q^{-13} - q^{-15} -2 q^{-17} - q^{-21} + q^{-23} + q^{-25} + q^{-27} }

B2 Invariants.

Weight	Invariant
0,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -q^{22}+2 q^{20}-3 q^{18}+4 q^{16}-5 q^{14}+6 q^{12}-6 q^{10}+6 q^8-4 q^6+3 q^4+q^2-3+7 q^{-2} -8 q^{-4} +12 q^{-6} -11 q^{-8} +12 q^{-10} -10 q^{-12} +8 q^{-14} -6 q^{-16} + q^{-18} -4 q^{-22} +4 q^{-24} -6 q^{-26} +6 q^{-28} -6 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} - q^{-38} + q^{-40} }
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{32}-2 q^{30}+q^{28}+4 q^{26}+q^{24}-4 q^{22}-4 q^{20}+2 q^{18}+6 q^{16}+2 q^{14}-5 q^{12}-4 q^{10}+3 q^8+7 q^6-4 q^2+6 q^{-2} +3 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +3 q^{-12} -2 q^{-14} -5 q^{-16} +3 q^{-20} - q^{-22} -5 q^{-24} - q^{-26} +6 q^{-28} +4 q^{-30} -3 q^{-32} -6 q^{-34} +2 q^{-36} +7 q^{-38} +3 q^{-40} -5 q^{-42} -5 q^{-44} +2 q^{-46} +5 q^{-48} -4 q^{-52} -2 q^{-54} +2 q^{-56} +2 q^{-58} - q^{-60} - q^{-62} + q^{-66} }

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+8q^{38}-10q^{36}+11q^{34}-8q^{32}+2q^{30}+4q^{28}-10q^{26}+16q^{24}-16q^{22}+14q^{20}-8q^{18}-q^{16}+11q^{14}-15q^{12}+17q^{10}-12q^{8}+3q^{6}+6q^{4}-10q^{2}+10-2q^{-2}-6q^{-4}+15q^{-6}-16q^{-8}+9q^{-10}+5q^{-12}-19q^{-14}+30q^{-16}-28q^{-18}+18q^{-20}-16q^{-24}+28q^{-26}-30q^{-28}+23q^{-30}-10q^{-32}-6q^{-34}+16q^{-36}-19q^{-38}+15q^{-40}-5q^{-42}-7q^{-44}+11q^{-46}-13q^{-48}+5q^{-50}+5q^{-52}-16q^{-54}+21q^{-56}-19q^{-58}+6q^{-60}+8q^{-62}-20q^{-64}+25q^{-66}-21q^{-68}+11q^{-70}+q^{-72}-10q^{-74}+16q^{-76}-14q^{-78}+10q^{-80}-2q^{-82}-2q^{-84}+3q^{-86}-4q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 14"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 37, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^8 a^{-2} +z^8 a^{-4} +3 z^7 a^{-1} +5 z^7 a^{-3} +2 z^7 a^{-5} +3 z^6 a^{-2} +z^6 a^{-6} +4 z^6+4 a z^5-4 z^5 a^{-1} -16 z^5 a^{-3} -8 z^5 a^{-5} +3 a^2 z^4-12 z^4 a^{-2} -9 z^4 a^{-4} -4 z^4 a^{-6} -4 z^4+a^3 z^3-3 a z^3+2 z^3 a^{-1} +15 z^3 a^{-3} +9 z^3 a^{-5} -2 a^2 z^2+8 z^2 a^{-2} +10 z^2 a^{-4} +4 z^2 a^{-6} -2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1}

Vassiliev invariants

V2 and V3:

(-1, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -16} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{110}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{34}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 64} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{32}{3}} ${\frac {128}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -48} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{32}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 128} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{440}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{136}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{10529}{30}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1502}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4978}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{833}{18}} ${\frac {449}{30}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or $j-2r=s-1$, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 9 14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 1   -1 9               2 1   1 7             3 1     -2 5           3 2       1 3         3 3         0 1       3 3           0 -1     2 4             2 -3   1 2               -1 -5   2                 2 -7 1                   -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} ${\mathbb {Z} }^{3}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 14]]
Out[2]=	9
In[3]:=	PD[Knot[9, 14]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 18, 14, 1], X[9, 15, 10, 14], X[7, 17, 8, 16], X[15, 9, 16, 8], X[17, 7, 18, 6]]
In[4]:=	GaussCode[Knot[9, 14]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[5]:=	BR[Knot[9, 14]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, 2, -3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 14]][t]
Out[6]=	2 9 2 15 + -- - - - 9 t + 2 t 2 t t
In[7]:=	Conway[Knot[9, 14]][z]
Out[7]=	2 4 1 - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 14]}
In[9]:=	{KnotDet[Knot[9, 14]], KnotSignature[Knot[9, 14]]}
Out[9]=	{37, 0}
In[10]:=	J=Jones[Knot[9, 14]][q]
Out[10]=	-3 3 4 2 3 4 5 6 6 - q + -- - - - 6 q + 6 q - 5 q + 3 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 14], Knot[11, NonAlternating, 53]}
In[12]:=	A2Invariant[Knot[9, 14]][q]
Out[12]=	-10 -8 -6 -4 2 2 4 8 10 12 16 18 -q + q + q - q + -- + q + q + q - 2 q - q - q + q + 2 q 20 q
In[13]:=	Kauffman[Knot[9, 14]][a, z]
Out[13]=	2 2 2 -6 2 -2 3 z 5 z 2 z 4 z 10 z 8 z 2 2 1 - a - -- - a - --- - --- - --- + ---- + ----- + ---- - 2 a z + 4 5 3 a 6 4 2 a a a a a a 3 3 3 4 4 4 9 z 15 z 2 z 3 3 3 4 4 z 9 z 12 z ---- + ----- + ---- - 3 a z + a z - 4 z - ---- - ---- - ----- + 5 3 a 6 4 2 a a a a a 5 5 5 6 6 7 2 4 8 z 16 z 4 z 5 6 z 3 z 2 z 3 a z - ---- - ----- - ---- + 4 a z + 4 z + -- + ---- + ---- + 5 3 a 6 2 5 a a a a a 7 7 8 8 5 z 3 z z z ---- + ---- + -- + -- 3 a 4 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 14]], Vassiliev[3][Knot[9, 14]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 14]][q, t]
Out[15]=	4 1 2 1 2 2 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t + 11 5 13 6 q t + q t