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]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{4,7\}$ 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 $1$ Topological 4 genus $1$ Concordance genus $2$ 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	$2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
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)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
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	$-q^{7}+2q^{5}-q^{3}+2q+q^{-5}-2q^{-7}+q^{-9}-q^{-11}+q^{-13}$
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	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	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	$q^{36}-2q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-4q^{22}-4q^{20}+2q^{18}+6q^{16}+2q^{14}-5q^{12}-4q^{10}+3q^{8}+7q^{6}-4q^{2}+6q^{-2}+3q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-5q^{-16}+3q^{-20}-q^{-22}-5q^{-24}-q^{-26}+6q^{-28}+4q^{-30}-3q^{-32}-6q^{-34}+2q^{-36}+7q^{-38}+3q^{-40}-5q^{-42}-5q^{-44}+2q^{-46}+5q^{-48}-4q^{-52}-2q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}+q^{-66}$

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]=

$2z^{4}-z^{2}+1$

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]=

$\{1\}$

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]=

$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {K11n53, ...}

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} $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {128}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {10529}{30}}$ ${\frac {1502}{15}}$ ${\frac {4978}{45}}$ $-{\frac {833}{18}}$ ${\frac {449}{30}}$

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 $j-2r=s+1$ or $j-2r=s-1$, where $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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of 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 sl(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 n}	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_n}
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^{18}-2 q^{17}-q^{16}+6 q^{15}-5 q^{14}-6 q^{13}+15 q^{12}-5 q^{11}-16 q^{10}+23 q^9-2 q^8-27 q^7+28 q^6+4 q^5-34 q^4+27 q^3+9 q^2-33 q+21+9 q^{-1} -23 q^{-2} +13 q^{-3} +5 q^{-4} -11 q^{-5} +6 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }
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^{36}-2 q^{35}-q^{34}+2 q^{33}+5 q^{32}-4 q^{31}-9 q^{30}+3 q^{29}+17 q^{28}-q^{27}-23 q^{26}-7 q^{25}+30 q^{24}+16 q^{23}-34 q^{22}-27 q^{21}+32 q^{20}+41 q^{19}-30 q^{18}-49 q^{17}+21 q^{16}+60 q^{15}-14 q^{14}-65 q^{13}+3 q^{12}+70 q^{11}+8 q^{10}-73 q^9-18 q^8+72 q^7+29 q^6-71 q^5-33 q^4+61 q^3+41 q^2-58 q-34+42 q^{-1} +34 q^{-2} -37 q^{-3} -20 q^{-4} +23 q^{-5} +17 q^{-6} -21 q^{-7} -5 q^{-8} +12 q^{-9} +4 q^{-10} -11 q^{-11} + q^{-12} +6 q^{-13} - q^{-14} -3 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} }
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^{60}-2 q^{59}-q^{58}+2 q^{57}+q^{56}+6 q^{55}-8 q^{54}-7 q^{53}+2 q^{52}+3 q^{51}+26 q^{50}-12 q^{49}-23 q^{48}-11 q^{47}-5 q^{46}+63 q^{45}+3 q^{44}-29 q^{43}-38 q^{42}-46 q^{41}+93 q^{40}+35 q^{39}-50 q^{37}-112 q^{36}+86 q^{35}+48 q^{34}+58 q^{33}-18 q^{32}-166 q^{31}+49 q^{30}+14 q^{29}+107 q^{28}+52 q^{27}-177 q^{26}+12 q^{25}-58 q^{24}+124 q^{23}+128 q^{22}-152 q^{21}-8 q^{20}-140 q^{19}+115 q^{18}+193 q^{17}-109 q^{16}-20 q^{15}-215 q^{14}+98 q^{13}+243 q^{12}-59 q^{11}-26 q^{10}-273 q^9+67 q^8+266 q^7-4 q^6-15 q^5-295 q^4+22 q^3+240 q^2+34 q+23-256 q^{-1} -20 q^{-2} +164 q^{-3} +35 q^{-4} +61 q^{-5} -170 q^{-6} -30 q^{-7} +80 q^{-8} +3 q^{-9} +69 q^{-10} -82 q^{-11} -14 q^{-12} +28 q^{-13} -23 q^{-14} +48 q^{-15} -29 q^{-16} +2 q^{-17} +8 q^{-18} -25 q^{-19} +23 q^{-20} -8 q^{-21} +5 q^{-22} +3 q^{-23} -12 q^{-24} +6 q^{-25} -2 q^{-26} +3 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} }
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^{90}-2 q^{89}-q^{88}+2 q^{87}+q^{86}+2 q^{85}+2 q^{84}-6 q^{83}-9 q^{82}+2 q^{81}+7 q^{80}+11 q^{79}+12 q^{78}-9 q^{77}-29 q^{76}-20 q^{75}+6 q^{74}+33 q^{73}+46 q^{72}+15 q^{71}-46 q^{70}-69 q^{69}-41 q^{68}+30 q^{67}+93 q^{66}+84 q^{65}-4 q^{64}-97 q^{63}-122 q^{62}-49 q^{61}+81 q^{60}+149 q^{59}+99 q^{58}-31 q^{57}-145 q^{56}-150 q^{55}-34 q^{54}+112 q^{53}+169 q^{52}+98 q^{51}-36 q^{50}-152 q^{49}-159 q^{48}-51 q^{47}+101 q^{46}+175 q^{45}+145 q^{44}+6 q^{43}-174 q^{42}-234 q^{41}-108 q^{40}+115 q^{39}+290 q^{38}+248 q^{37}-39 q^{36}-334 q^{35}-360 q^{34}-65 q^{33}+337 q^{32}+481 q^{31}+175 q^{30}-335 q^{29}-575 q^{28}-283 q^{27}+314 q^{26}+661 q^{25}+386 q^{24}-294 q^{23}-738 q^{22}-477 q^{21}+273 q^{20}+802 q^{19}+568 q^{18}-251 q^{17}-869 q^{16}-644 q^{15}+225 q^{14}+906 q^{13}+737 q^{12}-183 q^{11}-951 q^{10}-787 q^9+120 q^8+922 q^7+866 q^6-38 q^5-899 q^4-868 q^3-49 q^2+772 q+880+147 q^{-1} -674 q^{-2} -794 q^{-3} -212 q^{-4} +491 q^{-5} +716 q^{-6} +257 q^{-7} -368 q^{-8} -560 q^{-9} -260 q^{-10} +207 q^{-11} +448 q^{-12} +235 q^{-13} -130 q^{-14} -291 q^{-15} -192 q^{-16} +34 q^{-17} +207 q^{-18} +146 q^{-19} -18 q^{-20} -106 q^{-21} -96 q^{-22} -22 q^{-23} +63 q^{-24} +67 q^{-25} +14 q^{-26} -25 q^{-27} -35 q^{-28} -18 q^{-29} +6 q^{-30} +20 q^{-31} +16 q^{-32} -4 q^{-33} -10 q^{-34} -2 q^{-35} -7 q^{-36} +3 q^{-37} +8 q^{-38} -3 q^{-40} +2 q^{-41} -3 q^{-42} - q^{-43} +3 q^{-44} - q^{-45} }
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 q^{126}-2 q^{125}-q^{124}+2 q^{123}+q^{122}+2 q^{121}-2 q^{120}+4 q^{119}-8 q^{118}-9 q^{117}+5 q^{116}+6 q^{115}+12 q^{114}+16 q^{112}-22 q^{111}-35 q^{110}-9 q^{109}+5 q^{108}+36 q^{107}+21 q^{106}+70 q^{105}-21 q^{104}-79 q^{103}-68 q^{102}-48 q^{101}+30 q^{100}+43 q^{99}+193 q^{98}+62 q^{97}-60 q^{96}-133 q^{95}-165 q^{94}-86 q^{93}-49 q^{92}+295 q^{91}+209 q^{90}+108 q^{89}-57 q^{88}-200 q^{87}-245 q^{86}-311 q^{85}+201 q^{84}+214 q^{83}+285 q^{82}+170 q^{81}+27 q^{80}-189 q^{79}-516 q^{78}-37 q^{77}-79 q^{76}+170 q^{75}+243 q^{74}+381 q^{73}+195 q^{72}-341 q^{71}-51 q^{70}-447 q^{69}-301 q^{68}-148 q^{67}+447 q^{66}+615 q^{65}+212 q^{64}+436 q^{63}-453 q^{62}-784 q^{61}-930 q^{60}-15 q^{59}+645 q^{58}+765 q^{57}+1296 q^{56}+92 q^{55}-879 q^{54}-1711 q^{53}-848 q^{52}+151 q^{51}+960 q^{50}+2146 q^{49}+1001 q^{48}-494 q^{47}-2172 q^{46}-1702 q^{45}-668 q^{44}+748 q^{43}+2720 q^{42}+1943 q^{41}+167 q^{40}-2283 q^{39}-2349 q^{38}-1514 q^{37}+326 q^{36}+3007 q^{35}+2713 q^{34}+832 q^{33}-2217 q^{32}-2784 q^{31}-2212 q^{30}-69 q^{29}+3160 q^{28}+3299 q^{27}+1356 q^{26}-2150 q^{25}-3128 q^{24}-2763 q^{23}-352 q^{22}+3297 q^{21}+3795 q^{20}+1786 q^{19}-2085 q^{18}-3443 q^{17}-3262 q^{16}-644 q^{15}+3332 q^{14}+4210 q^{13}+2258 q^{12}-1817 q^{11}-3570 q^{10}-3697 q^9-1116 q^8+3007 q^7+4328 q^6+2736 q^5-1174 q^4-3219 q^3-3810 q^2-1684 q+2192+3857 q^{-1} +2909 q^{-2} -334 q^{-3} -2323 q^{-4} -3321 q^{-5} -1975 q^{-6} +1165 q^{-7} +2814 q^{-8} +2515 q^{-9} +269 q^{-10} -1243 q^{-11} -2327 q^{-12} -1750 q^{-13} +391 q^{-14} +1636 q^{-15} +1707 q^{-16} +415 q^{-17} -438 q^{-18} -1290 q^{-19} -1195 q^{-20} +47 q^{-21} +765 q^{-22} +919 q^{-23} +276 q^{-24} -49 q^{-25} -574 q^{-26} -664 q^{-27} -20 q^{-28} +291 q^{-29} +409 q^{-30} +117 q^{-31} +64 q^{-32} -207 q^{-33} -323 q^{-34} -9 q^{-35} +87 q^{-36} +155 q^{-37} +33 q^{-38} +67 q^{-39} -58 q^{-40} -139 q^{-41} + q^{-42} +14 q^{-43} +50 q^{-44} +2 q^{-45} +41 q^{-46} -11 q^{-47} -49 q^{-48} +4 q^{-49} -3 q^{-50} +14 q^{-51} -5 q^{-52} +17 q^{-53} - q^{-54} -14 q^{-55} +4 q^{-56} -3 q^{-57} +3 q^{-58} -2 q^{-59} +3 q^{-60} + q^{-61} -3 q^{-62} + q^{-63} }
7	Not Available

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[9, 14]]
Out[2]=	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[3]:=	GaussCode[Knot[9, 14]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[4]:=	DTCode[Knot[9, 14]]
Out[4]=	DTCode[4, 10, 12, 16, 14, 2, 18, 8, 6]
In[5]:=	br = BR[Knot[9, 14]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, 2, -3, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[9, 14]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[9, 14]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[9, 14]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 2, 2, {4, 7}, 1}
In[10]:=	alex = Alexander[Knot[9, 14]][t]
Out[10]=	2 9 2 15 + -- - - - 9 t + 2 t 2 t t
In[11]:=	Conway[Knot[9, 14]][z]
Out[11]=	2 4 1 - z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[9, 14]}
In[13]:=	{KnotDet[Knot[9, 14]], KnotSignature[Knot[9, 14]]}
Out[13]=	{37, 0}
In[14]:=	Jones[Knot[9, 14]][q]
Out[14]=	-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[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[9, 14], Knot[11, NonAlternating, 53]}
In[16]:=	A2Invariant[Knot[9, 14]][q]
Out[16]=	-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[17]:=	HOMFLYPT[Knot[9, 14]][a, z]
Out[17]=	2 2 4 -6 2 -2 2 2 z z 2 2 4 z 1 + a - -- + a + z - ---- + -- - a z + z + -- 4 4 2 2 a a a a
In[18]:=	Kauffman[Knot[9, 14]][a, z]
Out[18]=	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[19]:=	{Vassiliev[2][Knot[9, 14]], Vassiliev[3][Knot[9, 14]]}
Out[19]=	{-1, -2}
In[20]:=	Kh[Knot[9, 14]][q, t]
Out[20]=	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
In[21]:=	ColouredJones[Knot[9, 14], 2][q]
Out[21]=	-9 3 -7 6 11 5 13 23 9 2 21 + q - -- + q + -- - -- + -- + -- - -- + - - 33 q + 9 q + 8 6 5 4 3 2 q q q q q q q 3 4 5 6 7 8 9 10 27 q - 34 q + 4 q + 28 q - 27 q - 2 q + 23 q - 16 q - 11 12 13 14 15 16 17 18 5 q + 15 q - 6 q - 5 q + 6 q - q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

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