9 49

9_48

10_1

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 49's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 49 at Knotilus!

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]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 11, width is 4, Braid index is 4		[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

[edit Notes on presentations of 9 49]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 49"];

In[4]:=

PD[K]

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

Out[4]=

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

In[5]:=

GaussCode[K]

Out[5]=

1, 5, -4, 6, -3, -1, 2, 7, -8, 4, -5, -2, 9, 3, -6, 8, -7, -9

In[6]:=

DTCode[K]

Out[6]=

6 -10 -14 12 -16 -2 18 -4 -8

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[-20:-20:-20]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

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 \textrm{BR}(4,\{1,1,2,1,1,-3,2,-1,2,3,3\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 4, 11, 4 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{2, 7}, {1, 5}, {8, 3}, {7, 9}, {6, 2}, {4, 1}, {5, 8}, {3, 6}, {9, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

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 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} Topological 4 genus $2$ Concordance genus $2$ Rasmussen s-Invariant -4

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

Polynomial invariants

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

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	$q^{-12}-2q^{-14}+4q^{-16}-8q^{-18}+19q^{-20}-22q^{-22}+36q^{-24}-42q^{-26}+48q^{-28}-40q^{-30}+34q^{-32}-10q^{-34}-6q^{-36}+38q^{-38}-56q^{-40}+70q^{-42}-87q^{-44}+78q^{-46}-84q^{-48}+58q^{-50}-45q^{-52}+18q^{-54}+10q^{-56}-22q^{-58}+45q^{-60}-48q^{-62}+48q^{-64}-42q^{-66}+25q^{-68}-22q^{-70}+10q^{-72}-2q^{-74}+2q^{-76}+2q^{-78}$
2,0	$q^{-12}-q^{-14}-q^{-16}+3q^{-18}+3q^{-20}-q^{-22}-q^{-24}+2q^{-26}+2q^{-28}-2q^{-30}+q^{-32}+4q^{-34}+2q^{-36}+3q^{-38}+5q^{-40}+2q^{-42}-q^{-44}-q^{-46}-2q^{-48}-6q^{-50}-4q^{-52}-q^{-54}-q^{-56}-5q^{-58}-q^{-60}+q^{-62}+3q^{-70}+q^{-72}+q^{-74}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+2q^{-22}+6q^{-24}-2q^{-26}+5q^{-28}+6q^{-30}+q^{-34}+2q^{-36}-q^{-38}-2q^{-40}-3q^{-42}-3q^{-46}-6q^{-48}+2q^{-50}-2q^{-52}-5q^{-54}+4q^{-56}+q^{-58}-q^{-60}+3q^{-62}$
1,0,0	$q^{-9}-q^{-11}+q^{-13}-q^{-15}+q^{-17}+q^{-19}+3q^{-21}+3q^{-23}+2q^{-25}+2q^{-27}-q^{-29}-q^{-31}-3q^{-33}-q^{-35}-2q^{-37}$

A4 Invariants.

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

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

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}-q^{-20}+q^{-22}-2q^{-24}+3q^{-26}-4q^{-28}+4q^{-30}-2q^{-32}+5q^{-34}+q^{-36}+4q^{-38}+4q^{-40}+5q^{-42}+6q^{-44}-2q^{-46}+6q^{-48}-5q^{-50}+5q^{-52}-8q^{-54}+2q^{-56}-8q^{-58}+3q^{-60}-5q^{-62}-4q^{-66}-2q^{-68}+q^{-70}-4q^{-72}+q^{-74}-4q^{-76}+4q^{-78}-2q^{-80}+4q^{-82}-2q^{-84}+3q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

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

.

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]=

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

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]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 49"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]=

{ $3t^{2}-6t+7-6t^{-1}+3t^{-2}$, $-2q^{9}+3q^{8}-4q^{7}+5q^{6}-4q^{5}+4q^{4}-2q^{3}+q^{2}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

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 $24$ $112$ $288$ $700$ $116$ $2688$ ${\frac {14368}{3}}$ 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}} 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{1876}{5}} ${\frac {207244}{15}}$ 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

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=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 i=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 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}} 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}} 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^{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=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^{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=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}^{3}\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^{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=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}\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=7} 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^{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}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the 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+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^{26}+2 q^{25}-6 q^{24}+q^{23}+10 q^{22}-13 q^{21}-3 q^{20}+21 q^{19}-17 q^{18}-9 q^{17}+27 q^{16}-17 q^{15}-11 q^{14}+25 q^{13}-10 q^{12}-11 q^{11}+16 q^{10}-3 q^9-8 q^8+6 q^7+q^6-2 q^5+q^4}
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 -2 q^{50}+q^{48}+9 q^{47}-5 q^{46}-12 q^{45}-2 q^{44}+24 q^{43}+8 q^{42}-31 q^{41}-22 q^{40}+39 q^{39}+35 q^{38}-40 q^{37}-51 q^{36}+40 q^{35}+65 q^{34}-40 q^{33}-72 q^{32}+33 q^{31}+80 q^{30}-33 q^{29}-78 q^{28}+22 q^{27}+80 q^{26}-18 q^{25}-70 q^{24}+5 q^{23}+64 q^{22}+2 q^{21}-48 q^{20}-14 q^{19}+39 q^{18}+14 q^{17}-21 q^{16}-18 q^{15}+12 q^{14}+13 q^{13}-3 q^{12}-8 q^{11}+q^{10}+3 q^9+q^8-2 q^7+q^6}
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^{82}+2 q^{81}-6 q^{79}-4 q^{78}-5 q^{77}+14 q^{76}+21 q^{75}-7 q^{74}-18 q^{73}-45 q^{72}+12 q^{71}+65 q^{70}+31 q^{69}-5 q^{68}-120 q^{67}-46 q^{66}+90 q^{65}+105 q^{64}+70 q^{63}-180 q^{62}-142 q^{61}+59 q^{60}+168 q^{59}+182 q^{58}-196 q^{57}-226 q^{56}-q^{55}+192 q^{54}+274 q^{53}-183 q^{52}-269 q^{51}-52 q^{50}+187 q^{49}+324 q^{48}-158 q^{47}-276 q^{46}-86 q^{45}+162 q^{44}+333 q^{43}-116 q^{42}-248 q^{41}-117 q^{40}+110 q^{39}+309 q^{38}-50 q^{37}-181 q^{36}-137 q^{35}+31 q^{34}+240 q^{33}+19 q^{32}-84 q^{31}-122 q^{30}-43 q^{29}+137 q^{28}+46 q^{27}+q^{26}-65 q^{25}-63 q^{24}+44 q^{23}+25 q^{22}+30 q^{21}-13 q^{20}-35 q^{19}+5 q^{18}+15 q^{16}+3 q^{15}-9 q^{14}+q^{13}-2 q^{12}+3 q^{11}+q^{10}-2 q^9+q^8}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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