9 49

9_48

10_1

Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 49's page at Knotilus! Visit 9 49's page at the original Knot Atlas!

9 49 Quick Notes

9 49 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X12,8,13,7 X5,15,6,14 X3,11,4,10 X11,3,12,2 X15,5,16,4 X17,9,18,8 X9,17,10,16 X18,14,1,13
Gauss code	1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9
Dowker-Thistlethwaite code	6 -10 -14 12 -16 -2 18 -4 -8
Conway Notation	[-20:-20:-20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 2 Bridge index 3 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 \{4,5\}} Nakanishi index 2 Maximal Thurston-Bennequin number [3][-12] Hyperbolic Volume 9.42707 A-Polynomial See Data:9 49/A-polynomial

[edit Notes for 9 49's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ 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 2} 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 -4

[edit Notes for 9 49'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 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }
Conway polynomial	$3z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{5,t+1\}$
Determinant and Signature	{ 25, 4 }
Jones polynomial	$-2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+4a^{-6}-3a^{-8}$
Kauffman polynomial (db, data sources)	$z^{7}a^{-7}+z^{7}a^{-9}+3z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}a^{-10}+2z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}+z^{4}a^{-4}-8z^{4}a^{-6}-9z^{4}a^{-8}-3z^{3}a^{-5}-3z^{3}a^{-7}+3z^{3}a^{-9}+3z^{3}a^{-11}-2z^{2}a^{-4}+9z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+2za^{-7}-2za^{-9}-4za^{-11}-4a^{-6}-3a^{-8}$
The A2 invariant	$q^{-6}-q^{-8}+q^{-10}+q^{-14}+3q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-24}-q^{-26}-2q^{-28}$
The G2 invariant	$q^{-30}-q^{-32}+2q^{-34}-3q^{-36}+2q^{-38}-q^{-40}-2q^{-42}+8q^{-44}-10q^{-46}+12q^{-48}-7q^{-50}-q^{-52}+10q^{-54}-16q^{-56}+19q^{-58}-11q^{-60}+q^{-62}+10q^{-64}-14q^{-66}+13q^{-68}-2q^{-70}-6q^{-72}+14q^{-74}-12q^{-76}+4q^{-78}+9q^{-80}-15q^{-82}+21q^{-84}-16q^{-86}+9q^{-88}+5q^{-90}-13q^{-92}+22q^{-94}-22q^{-96}+16q^{-98}-4q^{-100}-7q^{-102}+13q^{-104}-16q^{-106}+9q^{-108}-q^{-110}-10q^{-112}+10q^{-114}-11q^{-116}-2q^{-118}+10q^{-120}-20q^{-122}+16q^{-124}-9q^{-126}-4q^{-128}+11q^{-130}-16q^{-132}+15q^{-134}-7q^{-136}+q^{-138}+3q^{-140}-7q^{-142}+7q^{-144}-2q^{-146}+q^{-148}+q^{-150}$

Further Quantum Invariants

Further quantum knot invariants for 9_49.

A1 Invariants.

Weight	Invariant
1	$q^{-3}-q^{-5}+2q^{-7}+q^{-11}+q^{-13}-q^{-15}+q^{-17}-2q^{-19}$
2	$q^{-6}-q^{-8}+5q^{-12}-q^{-14}-5q^{-16}+5q^{-18}+2q^{-20}-5q^{-22}+4q^{-24}+4q^{-26}-3q^{-28}-q^{-30}+q^{-32}+q^{-34}-5q^{-36}+q^{-38}+5q^{-40}-6q^{-42}-2q^{-44}+5q^{-46}-3q^{-48}-3q^{-50}+3q^{-52}+q^{-54}$
3	$q^{-9}-q^{-11}+3q^{-15}+3q^{-17}-3q^{-19}-7q^{-21}+3q^{-23}+14q^{-25}+4q^{-27}-14q^{-29}-13q^{-31}+14q^{-33}+18q^{-35}-9q^{-37}-21q^{-39}+4q^{-41}+23q^{-43}+q^{-45}-19q^{-47}-3q^{-49}+14q^{-51}+6q^{-53}-9q^{-55}-9q^{-57}+2q^{-59}+8q^{-61}+q^{-63}-14q^{-65}-7q^{-67}+14q^{-69}+14q^{-71}-16q^{-73}-17q^{-75}+12q^{-77}+21q^{-79}-6q^{-81}-21q^{-83}-q^{-85}+18q^{-87}+5q^{-89}-10q^{-91}-7q^{-93}+5q^{-95}+8q^{-97}-q^{-99}-2q^{-101}-2q^{-103}$
4	$q^{-12}-q^{-14}+3q^{-18}+q^{-20}+q^{-22}-6q^{-24}-4q^{-26}+8q^{-28}+10q^{-30}+14q^{-32}-12q^{-34}-28q^{-36}-13q^{-38}+12q^{-40}+51q^{-42}+23q^{-44}-29q^{-46}-58q^{-48}-37q^{-50}+56q^{-52}+76q^{-54}+19q^{-56}-66q^{-58}-93q^{-60}+10q^{-62}+84q^{-64}+69q^{-66}-28q^{-68}-97q^{-70}-28q^{-72}+51q^{-74}+71q^{-76}+4q^{-78}-62q^{-80}-38q^{-82}+14q^{-84}+45q^{-86}+17q^{-88}-25q^{-90}-34q^{-92}-9q^{-94}+25q^{-96}+32q^{-98}+7q^{-100}-43q^{-102}-38q^{-104}+13q^{-106}+56q^{-108}+43q^{-110}-49q^{-112}-73q^{-114}-13q^{-116}+71q^{-118}+87q^{-120}-25q^{-122}-88q^{-124}-57q^{-126}+39q^{-128}+99q^{-130}+24q^{-132}-50q^{-134}-75q^{-136}-17q^{-138}+58q^{-140}+45q^{-142}+7q^{-144}-37q^{-146}-35q^{-148}+5q^{-150}+19q^{-152}+20q^{-154}-q^{-156}-13q^{-158}-7q^{-160}-3q^{-162}+3q^{-164}+3q^{-166}+q^{-168}$
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^{-15} - q^{-17} +3 q^{-21} + q^{-23} - q^{-25} -2 q^{-27} -3 q^{-29} +9 q^{-33} +14 q^{-35} +3 q^{-37} -10 q^{-39} -24 q^{-41} -23 q^{-43} +40 q^{-47} +59 q^{-49} +34 q^{-51} -28 q^{-53} -92 q^{-55} -97 q^{-57} -27 q^{-59} +96 q^{-61} +171 q^{-63} +117 q^{-65} -44 q^{-67} -206 q^{-69} -231 q^{-71} -64 q^{-73} +197 q^{-75} +325 q^{-77} +194 q^{-79} -114 q^{-81} -363 q^{-83} -326 q^{-85} -8 q^{-87} +339 q^{-89} +402 q^{-91} +127 q^{-93} -257 q^{-95} -424 q^{-97} -219 q^{-99} +159 q^{-101} +387 q^{-103} +263 q^{-105} -72 q^{-107} -310 q^{-109} -263 q^{-111} +5 q^{-113} +228 q^{-115} +229 q^{-117} +33 q^{-119} -156 q^{-121} -183 q^{-123} -56 q^{-125} +92 q^{-127} +141 q^{-129} +66 q^{-131} -52 q^{-133} -117 q^{-135} -81 q^{-137} +20 q^{-139} +108 q^{-141} +112 q^{-143} +16 q^{-145} -116 q^{-147} -156 q^{-149} -50 q^{-151} +121 q^{-153} +216 q^{-155} +116 q^{-157} -128 q^{-159} -281 q^{-161} -184 q^{-163} +100 q^{-165} +334 q^{-167} +279 q^{-169} -42 q^{-171} -353 q^{-173} -359 q^{-175} -46 q^{-177} +317 q^{-179} +417 q^{-181} +151 q^{-183} -229 q^{-185} -414 q^{-187} -254 q^{-189} +106 q^{-191} +349 q^{-193} +299 q^{-195} +25 q^{-197} -232 q^{-199} -292 q^{-201} -124 q^{-203} +107 q^{-205} +219 q^{-207} +160 q^{-209} +3 q^{-211} -120 q^{-213} -141 q^{-215} -61 q^{-217} +42 q^{-219} +82 q^{-221} +63 q^{-223} +12 q^{-225} -31 q^{-227} -46 q^{-229} -21 q^{-231} +3 q^{-233} +12 q^{-235} +17 q^{-237} +8 q^{-239} - q^{-241} -4 q^{-243} -2 q^{-245} -2 q^{-247} }

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^{-6} - q^{-8} + q^{-10} + q^{-14} +3 q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} - q^{-26} -2 q^{-28} }
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^{-12} -2 q^{-14} +4 q^{-16} -8 q^{-18} +19 q^{-20} -22 q^{-22} +36 q^{-24} -42 q^{-26} +48 q^{-28} -40 q^{-30} +34 q^{-32} -10 q^{-34} -6 q^{-36} +38 q^{-38} -56 q^{-40} +70 q^{-42} -87 q^{-44} +78 q^{-46} -84 q^{-48} +58 q^{-50} -45 q^{-52} +18 q^{-54} +10 q^{-56} -22 q^{-58} +45 q^{-60} -48 q^{-62} +48 q^{-64} -42 q^{-66} +25 q^{-68} -22 q^{-70} +10 q^{-72} -2 q^{-74} +2 q^{-76} +2 q^{-78} }
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^{-12} - q^{-14} - q^{-16} +3 q^{-18} +3 q^{-20} - q^{-22} - q^{-24} +2 q^{-26} +2 q^{-28} -2 q^{-30} + q^{-32} +4 q^{-34} +2 q^{-36} +3 q^{-38} +5 q^{-40} +2 q^{-42} - q^{-44} - q^{-46} -2 q^{-48} -6 q^{-50} -4 q^{-52} - q^{-54} - q^{-56} -5 q^{-58} - q^{-60} + q^{-62} +3 q^{-70} + q^{-72} + q^{-74} }

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^{-12} - q^{-14} +2 q^{-18} -3 q^{-20} +2 q^{-22} +6 q^{-24} -2 q^{-26} +5 q^{-28} +6 q^{-30} + q^{-34} +2 q^{-36} - q^{-38} -2 q^{-40} -3 q^{-42} -3 q^{-46} -6 q^{-48} +2 q^{-50} -2 q^{-52} -5 q^{-54} +4 q^{-56} + q^{-58} - q^{-60} +3 q^{-62} }
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^{-9} - q^{-11} + q^{-13} - q^{-15} + q^{-17} + q^{-19} +3 q^{-21} +3 q^{-23} +2 q^{-25} +2 q^{-27} - q^{-29} - q^{-31} -3 q^{-33} - q^{-35} -2 q^{-37} }

A4 Invariants.

Weight	Invariant
0,1,0,0	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^{-18} - q^{-20} + q^{-24} - q^{-26} - q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} +4 q^{-36} +7 q^{-38} +3 q^{-40} +2 q^{-42} +8 q^{-44} +8 q^{-46} +3 q^{-50} +5 q^{-52} -3 q^{-54} -6 q^{-56} -2 q^{-58} -8 q^{-60} -13 q^{-62} -8 q^{-64} -5 q^{-66} -6 q^{-68} -3 q^{-70} +6 q^{-72} +6 q^{-74} +2 q^{-76} +3 q^{-78} +4 q^{-80} - q^{-84} }
1,0,0,0	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^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-22} + q^{-24} +3 q^{-26} +3 q^{-28} +4 q^{-30} +2 q^{-32} +2 q^{-34} - q^{-36} - q^{-38} -3 q^{-40} -3 q^{-42} - q^{-44} -2 q^{-46} }

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}-q^{-14}+2q^{-16}-4q^{-18}+5q^{-20}-4q^{-22}+4q^{-24}+3q^{-28}+2q^{-30}-2q^{-32}+7q^{-34}-8q^{-36}+7q^{-38}-8q^{-40}+5q^{-42}-6q^{-44}+3q^{-46}-2q^{-50}+4q^{-52}-3q^{-54}+4q^{-56}-5q^{-58}+3q^{-60}-3q^{-62}$
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^{-18} - q^{-22} - q^{-24} + q^{-26} +3 q^{-28} -4 q^{-32} - q^{-34} +5 q^{-36} +5 q^{-38} -2 q^{-40} -2 q^{-42} + q^{-44} +6 q^{-46} +2 q^{-48} -2 q^{-50} - q^{-52} +4 q^{-54} +2 q^{-56} -2 q^{-58} -2 q^{-60} + q^{-62} +3 q^{-64} - q^{-66} -4 q^{-68} -2 q^{-70} +2 q^{-72} -5 q^{-76} -4 q^{-78} +2 q^{-80} +3 q^{-82} -3 q^{-84} -5 q^{-86} - q^{-88} +4 q^{-90} +3 q^{-92} -2 q^{-94} -2 q^{-96} + q^{-98} +3 q^{-100} }

D4 Invariants.

Weight

Invariant

1,0,0,0

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^{-18} - q^{-20} + q^{-22} -2 q^{-24} +3 q^{-26} -4 q^{-28} +4 q^{-30} -2 q^{-32} +5 q^{-34} + q^{-36} +4 q^{-38} +4 q^{-40} +5 q^{-42} +6 q^{-44} -2 q^{-46} +6 q^{-48} -5 q^{-50} +5 q^{-52} -8 q^{-54} +2 q^{-56} -8 q^{-58} +3 q^{-60} -5 q^{-62} -4 q^{-66} -2 q^{-68} + q^{-70} -4 q^{-72} + q^{-74} -4 q^{-76} +4 q^{-78} -2 q^{-80} +4 q^{-82} -2 q^{-84} +3 q^{-86} }

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^{-30} - q^{-32} +2 q^{-34} -3 q^{-36} +2 q^{-38} - q^{-40} -2 q^{-42} +8 q^{-44} -10 q^{-46} +12 q^{-48} -7 q^{-50} - q^{-52} +10 q^{-54} -16 q^{-56} +19 q^{-58} -11 q^{-60} + q^{-62} +10 q^{-64} -14 q^{-66} +13 q^{-68} -2 q^{-70} -6 q^{-72} +14 q^{-74} -12 q^{-76} +4 q^{-78} +9 q^{-80} -15 q^{-82} +21 q^{-84} -16 q^{-86} +9 q^{-88} +5 q^{-90} -13 q^{-92} +22 q^{-94} -22 q^{-96} +16 q^{-98} -4 q^{-100} -7 q^{-102} +13 q^{-104} -16 q^{-106} +9 q^{-108} - q^{-110} -10 q^{-112} +10 q^{-114} -11 q^{-116} -2 q^{-118} +10 q^{-120} -20 q^{-122} +16 q^{-124} -9 q^{-126} -4 q^{-128} +11 q^{-130} -16 q^{-132} +15 q^{-134} -7 q^{-136} + q^{-138} +3 q^{-140} -7 q^{-142} +7 q^{-144} -2 q^{-146} + q^{-148} + q^{-150} }

.

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 49"];

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 3 t^2-6 t+7-6 t^{-1} +3 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

$3z^{4}+6z^{2}+1$

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]=

$\{5,t+1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 25, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+2z^{4}a^{-6}+2z^{2}a^{-4}+6z^{2}a^{-6}-2z^{2}a^{-8}+4a^{-6}-3a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{7}a^{-7}+z^{7}a^{-9}+3z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}a^{-10}+2z^{5}a^{-5}+z^{5}a^{-7}-z^{5}a^{-9}+z^{4}a^{-4}-8z^{4}a^{-6}-9z^{4}a^{-8}-3z^{3}a^{-5}-3z^{3}a^{-7}+3z^{3}a^{-9}+3z^{3}a^{-11}-2z^{2}a^{-4}+9z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}+2za^{-7}-2za^{-9}-4za^{-11}-4a^{-6}-3a^{-8}$

Vassiliev invariants

V2 and V3:

(6, 14)

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 24} 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 112} 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 288} 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 700} 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 116} $2688$ 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{14368}{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{2560}{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 688} 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 2304} 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 6272} 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 16800} 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 2784} 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{166191}{5}} ${\frac {1876}{5}}$ 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 \frac{207244}{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{689}{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{9391}{5}}

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 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=} 4 is the signature of 9 49. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	0	1	2	3	4	5	6	7	χ
19								2	-2
17							1		1
15						3	2		-1
13					2	1			1
11				2	3				1
9			2	2					0
7			2						2
5	1	2							-1
3	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, 49]]
Out[2]=	9
In[3]:=	PD[Knot[9, 49]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 8, 13, 7], X[5, 15, 6, 14], X[3, 11, 4, 10], X[11, 3, 12, 2], X[15, 5, 16, 4], X[17, 9, 18, 8], X[9, 17, 10, 16], X[18, 14, 1, 13]]
In[4]:=	GaussCode[Knot[9, 49]]
Out[4]=	GaussCode[1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9]
In[5]:=	BR[Knot[9, 49]]
Out[5]=	BR[4, {1, 1, 2, 1, 1, -3, 2, -1, 2, 3, 3}]
In[6]:=	alex = Alexander[Knot[9, 49]][t]
Out[6]=	3 6 2 7 + -- - - - 6 t + 3 t 2 t t
In[7]:=	Conway[Knot[9, 49]][z]
Out[7]=	2 4 1 + 6 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 49]}
In[9]:=	{KnotDet[Knot[9, 49]], KnotSignature[Knot[9, 49]]}
Out[9]=	{25, 4}
In[10]:=	J=Jones[Knot[9, 49]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - 2 q + 4 q - 4 q + 5 q - 4 q + 3 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 49]}
In[12]:=	A2Invariant[Knot[9, 49]][q]
Out[12]=	6 8 10 14 16 18 20 22 24 26 28 q - q + q + q + 3 q + q + 2 q - q - q - q - 2 q
In[13]:=	Kauffman[Knot[9, 49]][a, z]
Out[13]=	2 2 2 2 3 3 -3 4 4 z 2 z 2 z z 10 z 9 z 2 z 3 z 3 z -- - -- - --- - --- + --- - --- + ----- + ---- - ---- + ---- + ---- - 8 6 11 9 7 10 8 6 4 11 9 a a a a a a a a a a a 3 3 4 4 4 5 5 5 6 6 6 3 z 3 z 9 z 8 z z z z 2 z z 4 z 3 z ---- - ---- - ---- - ---- + -- - -- + -- + ---- + --- + ---- + ---- + 7 5 8 6 4 9 7 5 10 8 6 a a a a a a a a a a a 7 7 z z -- + -- 9 7 a a
In[14]:=	{Vassiliev[2][Knot[9, 49]], Vassiliev[3][Knot[9, 49]]}
Out[14]=	{0, 14}
In[15]:=	Kh[Knot[9, 49]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 3 q t + 13 4 13 5 15 5 15 6 17 6 19 7 2 q t + q t + 3 q t + 2 q t + q t + 2 q t