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	$2t^{2}-9t+15-9t^{-1}+2t^{-2}$
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	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^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 q^{-1} +3 q^{-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	$q^{20}-2q^{18}-q^{16}+4q^{14}-4q^{12}+7q^{8}-5q^{6}-q^{4}+7q^{2}-3-3q^{-2}+3q^{-4}+2q^{-6}-3q^{-8}-2q^{-10}+5q^{-12}-q^{-14}-6q^{-16}+5q^{-18}+2q^{-20}-6q^{-22}+4q^{-24}+4q^{-26}-5q^{-28}+3q^{-32}-2q^{-34}-q^{-36}+q^{-38}$
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 -q^{39}+2 q^{37}+q^{35}-2 q^{33}-2 q^{31}+q^{29}+3 q^{27}-5 q^{25}+6 q^{21}-10 q^{17}+3 q^{15}+14 q^{13}-q^{11}-17 q^9+19 q^5+5 q^3-16 q-9 q^{-1} +10 q^{-3} +11 q^{-5} -2 q^{-7} -14 q^{-9} -3 q^{-11} +12 q^{-13} +10 q^{-15} -11 q^{-17} -13 q^{-19} +8 q^{-21} +16 q^{-23} -6 q^{-25} -16 q^{-27} +2 q^{-29} +18 q^{-31} +2 q^{-33} -17 q^{-35} -6 q^{-37} +16 q^{-39} +12 q^{-41} -13 q^{-43} -15 q^{-45} +5 q^{-47} +16 q^{-49} - q^{-51} -14 q^{-53} -4 q^{-55} +10 q^{-57} +7 q^{-59} -5 q^{-61} -6 q^{-63} +2 q^{-65} +4 q^{-67} -2 q^{-71} - q^{-73} + q^{-75} }
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 q^{64}-2 q^{62}-q^{60}+2 q^{58}+5 q^{54}-4 q^{52}-2 q^{50}-6 q^{46}+11 q^{44}-2 q^{42}+3 q^{40}-21 q^{36}+4 q^{34}+6 q^{32}+26 q^{30}+10 q^{28}-43 q^{26}-22 q^{24}+4 q^{22}+56 q^{20}+40 q^{18}-48 q^{16}-56 q^{14}-24 q^{12}+60 q^{10}+70 q^8-16 q^6-54 q^4-55 q^2+21+63 q^{-2} +24 q^{-4} -14 q^{-6} -52 q^{-8} -26 q^{-10} +19 q^{-12} +41 q^{-14} +30 q^{-16} -25 q^{-18} -48 q^{-20} -17 q^{-22} +41 q^{-24} +47 q^{-26} -3 q^{-28} -53 q^{-30} -36 q^{-32} +39 q^{-34} +51 q^{-36} +8 q^{-38} -57 q^{-40} -48 q^{-42} +34 q^{-44} +54 q^{-46} +29 q^{-48} -47 q^{-50} -64 q^{-52} +8 q^{-54} +45 q^{-56} +56 q^{-58} -14 q^{-60} -63 q^{-62} -29 q^{-64} +8 q^{-66} +62 q^{-68} +30 q^{-70} -28 q^{-72} -41 q^{-74} -34 q^{-76} +32 q^{-78} +44 q^{-80} +15 q^{-82} -17 q^{-84} -47 q^{-86} -6 q^{-88} +21 q^{-90} +27 q^{-92} +12 q^{-94} -25 q^{-96} -17 q^{-98} -4 q^{-100} +12 q^{-102} +16 q^{-104} -4 q^{-106} -6 q^{-108} -6 q^{-110} +6 q^{-114} + q^{-116} -2 q^{-120} - q^{-122} + q^{-124} }
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 -q^{95}+2 q^{93}+q^{91}-2 q^{89}-3 q^{85}-2 q^{83}+3 q^{81}+7 q^{79}+3 q^{77}-q^{75}-8 q^{73}-12 q^{71}-4 q^{69}+13 q^{67}+26 q^{65}+10 q^{63}-15 q^{61}-36 q^{59}-38 q^{57}+9 q^{55}+66 q^{53}+62 q^{51}+q^{49}-80 q^{47}-112 q^{45}-33 q^{43}+111 q^{41}+167 q^{39}+71 q^{37}-114 q^{35}-226 q^{33}-137 q^{31}+104 q^{29}+277 q^{27}+209 q^{25}-60 q^{23}-298 q^{21}-276 q^{19}-8 q^{17}+276 q^{15}+324 q^{13}+90 q^{11}-216 q^9-326 q^7-162 q^5+119 q^3+282 q+208 q^{-1} -17 q^{-3} -202 q^{-5} -216 q^{-7} -66 q^{-9} +103 q^{-11} +184 q^{-13} +132 q^{-15} -13 q^{-17} -142 q^{-19} -158 q^{-21} -53 q^{-23} +90 q^{-25} +172 q^{-27} +104 q^{-29} -65 q^{-31} -169 q^{-33} -121 q^{-35} +46 q^{-37} +177 q^{-39} +134 q^{-41} -48 q^{-43} -189 q^{-45} -148 q^{-47} +46 q^{-49} +209 q^{-51} +168 q^{-53} -43 q^{-55} -223 q^{-57} -200 q^{-59} +18 q^{-61} +233 q^{-63} +234 q^{-65} +20 q^{-67} -213 q^{-69} -260 q^{-71} -80 q^{-73} +172 q^{-75} +272 q^{-77} +137 q^{-79} -105 q^{-81} -250 q^{-83} -190 q^{-85} +19 q^{-87} +202 q^{-89} +217 q^{-91} +59 q^{-93} -122 q^{-95} -199 q^{-97} -131 q^{-99} +32 q^{-101} +157 q^{-103} +159 q^{-105} +50 q^{-107} -79 q^{-109} -149 q^{-111} -112 q^{-113} +3 q^{-115} +104 q^{-117} +127 q^{-119} +61 q^{-121} -42 q^{-123} -107 q^{-125} -95 q^{-127} -16 q^{-129} +65 q^{-131} +93 q^{-133} +51 q^{-135} -18 q^{-137} -65 q^{-139} -62 q^{-141} -15 q^{-143} +34 q^{-145} +51 q^{-147} +27 q^{-149} -7 q^{-151} -29 q^{-153} -28 q^{-155} -6 q^{-157} +14 q^{-159} +17 q^{-161} +7 q^{-163} -2 q^{-165} -8 q^{-167} -8 q^{-169} +4 q^{-173} +3 q^{-175} + q^{-177} -2 q^{-181} - q^{-183} + q^{-185} }

A2 Invariants.

Weight	Invariant
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^{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} }
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	$-q^{22}+2q^{20}-3q^{18}+4q^{16}-5q^{14}+6q^{12}-6q^{10}+6q^{8}-4q^{6}+3q^{4}+q^{2}-3+7q^{-2}-8q^{-4}+12q^{-6}-11q^{-8}+12q^{-10}-10q^{-12}+8q^{-14}-6q^{-16}+q^{-18}-4q^{-22}+4q^{-24}-6q^{-26}+6q^{-28}-6q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}-q^{-38}+q^{-40}$
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

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} }

.

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]=

$2t^{2}-9t+15-9t^{-1}+2t^{-2}$

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]=

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^6-2 q^5+3 q^4-5 q^3+6 q^2-6 q+6-4 q^{-1} +3 q^{-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} $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {128}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {10529}{30}}$ ${\frac {1502}{15}}$ ${\frac {4978}{45}}$ $-{\frac {833}{18}}$ ${\frac {449}{30}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 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} , 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

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