7 3

7_2

7_4

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 7 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 3 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X14,8,1,7 X8,14,9,13 X12,6,13,5 X2,10,3,9 X4,12,5,11
Gauss code	1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3
Dowker-Thistlethwaite code	6 10 12 14 2 4 8
Conway Notation	[43]

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

[edit Notes on presentations of 7 3]

Knot 7_3.

A graph, knot 7_3.

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["7 3"];

In[4]:=

PD[K]

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

Out[4]=

X6271 X10,4,11,3 X14,8,1,7 X8,14,9,13 X12,6,13,5 X2,10,3,9 X4,12,5,11

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 12 14 2 4 8

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[43]

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}(3,\{1,1,1,1,1,2,-1,2\})}

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]=

{ 3, 8, 3 }

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

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 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 \{3,4\}} Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (syntax error): {\displaystyle \text{$\$$Failed}} Hyperbolic Volume 4.59213 A-Polynomial See Data:7 3/A-polynomial

[edit Notes for 7 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,3))$ Rasmussen s-Invariant -4

[edit Notes for 7 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}+2t^{-2}-3t-3t^{-1}+3$
Conway polynomial	$2z^{4}+5z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 4 }
Jones polynomial	$-q^{9}+q^{8}-2q^{7}+3q^{6}-2q^{5}+2q^{4}-q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{-8}-2a^{-8}+z^{4}a^{-6}+3z^{2}a^{-6}+2a^{-6}+z^{4}a^{-4}+3z^{2}a^{-4}+a^{-4}$
Kauffman polynomial (db, data sources)	$z^{3}a^{-11}-2za^{-11}+z^{4}a^{-10}-z^{2}a^{-10}+z^{5}a^{-9}-z^{3}a^{-9}+za^{-9}+z^{6}a^{-8}-3z^{4}a^{-8}+6z^{2}a^{-8}-2a^{-8}+2z^{5}a^{-7}-4z^{3}a^{-7}+3za^{-7}+z^{6}a^{-6}-3z^{4}a^{-6}+4z^{2}a^{-6}-2a^{-6}+z^{5}a^{-5}-2z^{3}a^{-5}+z^{4}a^{-4}-3z^{2}a^{-4}+a^{-4}$
The A2 invariant	$q^{-6}+q^{-10}+q^{-14}+2q^{-16}+q^{-18}+q^{-20}-q^{-22}-q^{-24}-q^{-26}-q^{-28}$
The G2 invariant	$q^{-30}+q^{-34}-q^{-36}+q^{-38}+q^{-40}-q^{-42}+3q^{-44}-q^{-46}+2q^{-48}-q^{-52}+2q^{-54}-2q^{-56}+2q^{-58}-q^{-62}+2q^{-64}+q^{-68}+2q^{-70}-q^{-72}+2q^{-74}+3q^{-80}-2q^{-82}+3q^{-84}+2q^{-88}+q^{-90}-3q^{-92}+3q^{-94}-3q^{-96}+2q^{-98}-q^{-100}-3q^{-102}+q^{-104}-q^{-106}-q^{-108}-3q^{-112}-q^{-114}-q^{-116}-2q^{-118}+2q^{-120}-3q^{-122}+q^{-124}-q^{-128}+q^{-130}-q^{-132}+q^{-134}-q^{-136}+q^{-138}-q^{-142}+q^{-144}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 7_3.

A1 Invariants.

Weight	Invariant
1	$q^{-3}+q^{-7}+q^{-11}+q^{-13}-q^{-15}-q^{-19}$
2	$q^{-6}+2q^{-12}+q^{-14}-q^{-16}+q^{-18}+q^{-20}-q^{-22}+q^{-24}+q^{-26}-q^{-28}+q^{-34}-2q^{-36}-q^{-38}+q^{-40}-2q^{-42}-q^{-44}+q^{-46}+q^{-52}$
3	$q^{-9}+q^{-15}+2q^{-17}+q^{-19}-q^{-21}-q^{-23}+2q^{-25}+2q^{-27}-2q^{-31}+3q^{-35}+q^{-37}-2q^{-39}-q^{-41}+2q^{-43}+q^{-45}-2q^{-47}-q^{-49}+q^{-51}-2q^{-55}-q^{-57}-q^{-59}+2q^{-63}-2q^{-65}-2q^{-67}+3q^{-71}-q^{-73}-3q^{-75}+3q^{-79}+q^{-81}-q^{-83}+q^{-87}+q^{-89}-q^{-99}$
4	$q^{-12}+q^{-18}+q^{-20}+2q^{-22}-q^{-26}+4q^{-32}+2q^{-34}-q^{-36}-2q^{-38}-3q^{-40}+2q^{-42}+3q^{-44}+3q^{-46}-5q^{-50}-q^{-52}+2q^{-54}+5q^{-56}+2q^{-58}-5q^{-60}-4q^{-62}-q^{-64}+4q^{-66}+3q^{-68}-4q^{-70}-3q^{-72}-q^{-74}+3q^{-76}+2q^{-78}-3q^{-80}-2q^{-82}-q^{-84}+q^{-86}-2q^{-90}-q^{-92}-q^{-94}-q^{-96}+q^{-98}+4q^{-100}-q^{-102}-2q^{-104}-2q^{-106}+q^{-108}+7q^{-110}+q^{-112}-2q^{-114}-5q^{-116}-q^{-118}+7q^{-120}+3q^{-122}-q^{-124}-4q^{-126}-3q^{-128}+3q^{-130}+2q^{-132}+2q^{-134}-q^{-136}-2q^{-138}-q^{-142}+q^{-144}-q^{-148}-q^{-152}+q^{-160}$
5	$q^{-15}+q^{-21}+q^{-23}+q^{-25}+q^{-27}-q^{-31}+2q^{-35}+2q^{-37}+3q^{-39}+q^{-41}-2q^{-43}-4q^{-45}-2q^{-47}+q^{-49}+4q^{-51}+5q^{-53}+2q^{-55}-q^{-57}-5q^{-59}-5q^{-61}+4q^{-65}+7q^{-67}+5q^{-69}-2q^{-71}-8q^{-73}-8q^{-75}-q^{-77}+6q^{-79}+9q^{-81}+4q^{-83}-5q^{-85}-11q^{-87}-7q^{-89}+3q^{-91}+9q^{-93}+7q^{-95}-2q^{-97}-9q^{-99}-8q^{-101}+7q^{-105}+5q^{-107}-q^{-109}-6q^{-111}-5q^{-113}+4q^{-117}+3q^{-119}-q^{-121}-3q^{-123}-q^{-125}+2q^{-129}+q^{-131}-q^{-133}-q^{-137}-q^{-139}-q^{-141}+2q^{-143}+6q^{-145}+q^{-147}-q^{-149}-3q^{-151}-4q^{-153}+2q^{-155}+10q^{-157}+7q^{-159}-7q^{-163}-11q^{-165}-3q^{-167}+8q^{-169}+11q^{-171}+4q^{-173}-7q^{-175}-12q^{-177}-7q^{-179}+3q^{-181}+9q^{-183}+7q^{-185}+q^{-187}-6q^{-189}-6q^{-191}-3q^{-193}+2q^{-195}+4q^{-197}+3q^{-199}-2q^{-203}-3q^{-205}-q^{-207}+q^{-211}+2q^{-213}-q^{-217}+q^{-225}+q^{-227}-q^{-235}$
6	$q^{-18}+q^{-24}+q^{-26}+q^{-28}+q^{-32}-q^{-36}+q^{-38}+2q^{-40}+3q^{-42}+q^{-44}+2q^{-46}-q^{-48}-4q^{-50}-3q^{-52}-q^{-54}+3q^{-56}+3q^{-58}+7q^{-60}+4q^{-62}-2q^{-64}-5q^{-66}-6q^{-68}-4q^{-70}-3q^{-72}+6q^{-74}+9q^{-76}+8q^{-78}+3q^{-80}-2q^{-82}-8q^{-84}-14q^{-86}-5q^{-88}+2q^{-90}+10q^{-92}+13q^{-94}+10q^{-96}-q^{-98}-16q^{-100}-16q^{-102}-12q^{-104}+q^{-106}+13q^{-108}+21q^{-110}+12q^{-112}-6q^{-114}-16q^{-116}-21q^{-118}-10q^{-120}+5q^{-122}+21q^{-124}+19q^{-126}+2q^{-128}-11q^{-130}-21q^{-132}-15q^{-134}-q^{-136}+16q^{-138}+17q^{-140}+5q^{-142}-5q^{-144}-14q^{-146}-11q^{-148}-2q^{-150}+11q^{-152}+11q^{-154}+2q^{-156}-3q^{-158}-8q^{-160}-5q^{-162}-q^{-164}+6q^{-166}+5q^{-168}-2q^{-174}+q^{-178}+3q^{-180}+2q^{-182}-2q^{-190}-2q^{-192}-2q^{-194}+2q^{-196}+8q^{-198}+3q^{-200}+2q^{-202}-3q^{-204}-7q^{-206}-8q^{-208}-q^{-210}+14q^{-212}+12q^{-214}+9q^{-216}-3q^{-218}-16q^{-220}-23q^{-222}-11q^{-224}+12q^{-226}+19q^{-228}+20q^{-230}+4q^{-232}-15q^{-234}-28q^{-236}-20q^{-238}+4q^{-240}+16q^{-242}+24q^{-244}+14q^{-246}-2q^{-248}-18q^{-250}-20q^{-252}-6q^{-254}+3q^{-256}+13q^{-258}+14q^{-260}+8q^{-262}-4q^{-264}-9q^{-266}-7q^{-268}-6q^{-270}+q^{-272}+6q^{-274}+6q^{-276}+q^{-278}-q^{-280}-q^{-282}-4q^{-284}-2q^{-286}+q^{-288}+3q^{-290}+q^{-292}+q^{-294}+q^{-296}-2q^{-298}-q^{-300}+q^{-304}+q^{-310}-q^{-312}-q^{-314}-q^{-316}+q^{-324}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-6}+q^{-10}+q^{-14}+2q^{-16}+q^{-18}+q^{-20}-q^{-22}-q^{-24}-q^{-26}-q^{-28}$
1,1	$q^{-12}+2q^{-16}-2q^{-18}+6q^{-20}+8q^{-24}-2q^{-26}+5q^{-28}+2q^{-34}-4q^{-36}+6q^{-38}-8q^{-40}+4q^{-42}-11q^{-44}+4q^{-46}-8q^{-48}+4q^{-50}-q^{-52}+4q^{-56}-2q^{-58}+3q^{-60}-6q^{-62}+2q^{-64}-2q^{-66}+2q^{-68}-2q^{-70}+2q^{-72}+q^{-76}$
2,0	$q^{-12}+q^{-18}+2q^{-20}+q^{-22}+q^{-24}+2q^{-26}+2q^{-28}+q^{-30}+q^{-32}+q^{-34}+2q^{-40}-q^{-46}-q^{-48}-3q^{-50}-3q^{-52}-2q^{-54}-2q^{-56}-2q^{-58}-q^{-60}+q^{-62}+q^{-64}+2q^{-66}+q^{-68}+q^{-70}$
3,0	$q^{-18}+2q^{-26}+3q^{-28}+2q^{-30}+2q^{-36}+4q^{-38}+2q^{-40}+3q^{-46}+4q^{-48}+2q^{-50}+q^{-54}+2q^{-56}+2q^{-58}-q^{-64}-q^{-66}-4q^{-68}-4q^{-70}-2q^{-72}-3q^{-74}-3q^{-76}-5q^{-78}-2q^{-80}-q^{-82}-3q^{-86}-3q^{-88}-q^{-90}+q^{-92}-2q^{-96}-q^{-98}+2q^{-100}+4q^{-102}+4q^{-104}+3q^{-106}+2q^{-108}+3q^{-110}+2q^{-112}+2q^{-114}-q^{-118}-2q^{-120}-2q^{-122}-q^{-124}-q^{-126}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-12}+q^{-16}+q^{-18}+2q^{-22}+3q^{-24}+q^{-26}+3q^{-28}+3q^{-30}+q^{-32}-2q^{-38}-2q^{-40}-3q^{-42}-q^{-44}-2q^{-46}-2q^{-48}+q^{-50}-q^{-52}-q^{-54}+q^{-56}+q^{-58}+q^{-62}$
1,0,0	$q^{-9}+q^{-13}+q^{-17}+q^{-19}+2q^{-21}+2q^{-23}+q^{-25}+q^{-27}-q^{-29}-q^{-31}-2q^{-33}-q^{-35}-q^{-37}$
1,0,1	$q^{-18}+2q^{-22}+q^{-26}+5q^{-28}+2q^{-30}+9q^{-32}+5q^{-34}+6q^{-36}+8q^{-38}+9q^{-42}-q^{-44}+q^{-48}-8q^{-50}-4q^{-52}-7q^{-54}-8q^{-56}-7q^{-58}-3q^{-60}-6q^{-62}-q^{-66}+7q^{-70}-2q^{-72}+7q^{-74}+2q^{-76}-3q^{-78}+3q^{-80}-4q^{-82}-q^{-84}+q^{-86}-2q^{-88}+q^{-90}-q^{-92}+2q^{-96}+q^{-100}$

A4 Invariants.

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

B2 Invariants.

Weight	Invariant
0,1	$q^{-12}+q^{-16}-q^{-18}+2q^{-20}+q^{-24}+q^{-26}+q^{-28}+q^{-30}-q^{-32}+2q^{-34}-2q^{-36}+2q^{-38}-2q^{-40}+q^{-42}-q^{-44}-q^{-50}+q^{-52}-q^{-54}+q^{-56}-q^{-58}-q^{-62}$
1,0	$q^{-18}+q^{-26}+q^{-28}-q^{-32}+q^{-34}+2q^{-36}+2q^{-38}+q^{-42}+q^{-44}+2q^{-46}+q^{-48}+q^{-54}-q^{-58}-q^{-60}-q^{-66}-2q^{-68}-q^{-70}-q^{-74}-2q^{-76}-q^{-78}+q^{-80}-q^{-84}-q^{-86}+q^{-90}+q^{-92}+q^{-100}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-18}+q^{-22}+2q^{-26}+2q^{-30}+q^{-32}+2q^{-34}+2q^{-36}+2q^{-38}+3q^{-40}+3q^{-42}+3q^{-44}+2q^{-48}-2q^{-50}-4q^{-54}-2q^{-56}-4q^{-58}-q^{-60}-2q^{-62}-q^{-64}-q^{-66}-q^{-68}+q^{-70}-q^{-72}-q^{-76}+q^{-78}+q^{-82}+q^{-86}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-30}+q^{-34}-q^{-36}+q^{-38}+q^{-40}-q^{-42}+3q^{-44}-q^{-46}+2q^{-48}-q^{-52}+2q^{-54}-2q^{-56}+2q^{-58}-q^{-62}+2q^{-64}+q^{-68}+2q^{-70}-q^{-72}+2q^{-74}+3q^{-80}-2q^{-82}+3q^{-84}+2q^{-88}+q^{-90}-3q^{-92}+3q^{-94}-3q^{-96}+2q^{-98}-q^{-100}-3q^{-102}+q^{-104}-q^{-106}-q^{-108}-3q^{-112}-q^{-114}-q^{-116}-2q^{-118}+2q^{-120}-3q^{-122}+q^{-124}-q^{-128}+q^{-130}-q^{-132}+q^{-134}-q^{-136}+q^{-138}-q^{-142}+q^{-144}+q^{-148}$

.

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["7 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

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

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}+5z^{2}+1$

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]=

{ 13, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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["7 3"];

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]=

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

(5, 11)

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 20} 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 88} 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 200} 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{1510}{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{242}{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 1760} 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{9520}{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{1696}{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 440} 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{4000}{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 3872} 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{30200}{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{4840}{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{121855}{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 382} 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{73862}{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 \frac{2437}{18}} 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{6559}{6}}

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 7 3. 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               1 -1 17                 0 15           2 1   -1 13         1       1 11       1 2       1 9     1 1         0 7     1           1 5 1 1             0 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}} 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}\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=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}\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=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}^{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=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}_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=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}} 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} 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}}

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^{25}-q^{24}+2 q^{22}-3 q^{21}-q^{20}+5 q^{19}-5 q^{18}-2 q^{17}+8 q^{16}-6 q^{15}-2 q^{14}+7 q^{13}-4 q^{12}-2 q^{11}+5 q^{10}-2 q^9-2 q^8+3 q^7-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 -q^{48}+q^{47}-q^{44}+2 q^{43}-q^{41}-2 q^{40}+4 q^{39}+2 q^{38}-4 q^{37}-5 q^{36}+6 q^{35}+6 q^{34}-7 q^{33}-7 q^{32}+6 q^{31}+10 q^{30}-9 q^{29}-8 q^{28}+6 q^{27}+9 q^{26}-7 q^{25}-7 q^{24}+4 q^{23}+8 q^{22}-4 q^{21}-6 q^{20}+q^{19}+7 q^{18}-q^{17}-4 q^{16}-2 q^{15}+5 q^{14}+q^{13}-2 q^{12}-2 q^{11}+2 q^{10}+q^9-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^{78}-q^{77}-q^{74}+2 q^{73}-2 q^{72}+q^{71}+q^{70}-3 q^{69}+3 q^{68}-4 q^{67}+2 q^{66}+4 q^{65}-3 q^{64}+4 q^{63}-10 q^{62}+q^{61}+7 q^{60}+q^{59}+8 q^{58}-18 q^{57}-3 q^{56}+10 q^{55}+4 q^{54}+14 q^{53}-24 q^{52}-6 q^{51}+10 q^{50}+5 q^{49}+19 q^{48}-27 q^{47}-8 q^{46}+10 q^{45}+5 q^{44}+18 q^{43}-25 q^{42}-7 q^{41}+8 q^{40}+4 q^{39}+17 q^{38}-20 q^{37}-6 q^{36}+4 q^{35}+2 q^{34}+16 q^{33}-13 q^{32}-5 q^{31}-q^{30}-q^{29}+15 q^{28}-6 q^{27}-2 q^{26}-4 q^{25}-4 q^{24}+11 q^{23}-q^{22}+q^{21}-4 q^{20}-5 q^{19}+6 q^{18}+2 q^{16}-q^{15}-3 q^{14}+2 q^{13}+q^{11}-q^9+q^8}
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^{115}+q^{114}+q^{111}-2 q^{109}+q^{108}-q^{106}+2 q^{105}+2 q^{104}-3 q^{103}-q^{101}-3 q^{100}+3 q^{99}+4 q^{98}+q^{96}-3 q^{95}-8 q^{94}+4 q^{92}+7 q^{91}+7 q^{90}-q^{89}-14 q^{88}-10 q^{87}-q^{86}+12 q^{85}+18 q^{84}+6 q^{83}-17 q^{82}-21 q^{81}-9 q^{80}+16 q^{79}+25 q^{78}+13 q^{77}-14 q^{76}-29 q^{75}-15 q^{74}+17 q^{73}+27 q^{72}+15 q^{71}-9 q^{70}-33 q^{69}-18 q^{68}+17 q^{67}+27 q^{66}+16 q^{65}-10 q^{64}-31 q^{63}-17 q^{62}+15 q^{61}+26 q^{60}+14 q^{59}-8 q^{58}-27 q^{57}-16 q^{56}+11 q^{55}+21 q^{54}+13 q^{53}-3 q^{52}-21 q^{51}-14 q^{50}+4 q^{49}+13 q^{48}+12 q^{47}+4 q^{46}-12 q^{45}-12 q^{44}-2 q^{43}+3 q^{42}+8 q^{41}+10 q^{40}-3 q^{39}-7 q^{38}-5 q^{37}-4 q^{36}+q^{35}+10 q^{34}+3 q^{33}-3 q^{31}-7 q^{30}-3 q^{29}+5 q^{28}+3 q^{27}+4 q^{26}-4 q^{24}-4 q^{23}+2 q^{22}+2 q^{20}+2 q^{19}-q^{18}-2 q^{17}+q^{16}+q^{13}-q^{11}+q^{10}}
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^{159}-q^{158}-q^{155}+3 q^{152}-2 q^{151}+q^{149}-2 q^{148}-q^{147}-q^{146}+6 q^{145}-2 q^{144}+3 q^{142}-4 q^{141}-4 q^{140}-3 q^{139}+9 q^{138}-2 q^{137}+2 q^{136}+8 q^{135}-4 q^{134}-9 q^{133}-10 q^{132}+8 q^{131}-4 q^{130}+7 q^{129}+20 q^{128}+2 q^{127}-10 q^{126}-20 q^{125}-q^{124}-18 q^{123}+9 q^{122}+36 q^{121}+18 q^{120}-28 q^{118}-13 q^{117}-42 q^{116}+q^{115}+49 q^{114}+37 q^{113}+16 q^{112}-29 q^{111}-20 q^{110}-65 q^{109}-11 q^{108}+56 q^{107}+50 q^{106}+28 q^{105}-26 q^{104}-18 q^{103}-80 q^{102}-18 q^{101}+57 q^{100}+54 q^{99}+33 q^{98}-25 q^{97}-13 q^{96}-86 q^{95}-22 q^{94}+57 q^{93}+54 q^{92}+35 q^{91}-25 q^{90}-13 q^{89}-84 q^{88}-21 q^{87}+55 q^{86}+53 q^{85}+33 q^{84}-23 q^{83}-13 q^{82}-79 q^{81}-20 q^{80}+48 q^{79}+49 q^{78}+30 q^{77}-18 q^{76}-8 q^{75}-70 q^{74}-20 q^{73}+35 q^{72}+40 q^{71}+27 q^{70}-9 q^{69}+2 q^{68}-58 q^{67}-21 q^{66}+18 q^{65}+26 q^{64}+21 q^{63}+q^{62}+15 q^{61}-41 q^{60}-19 q^{59}+2 q^{58}+11 q^{57}+10 q^{56}+6 q^{55}+25 q^{54}-23 q^{53}-10 q^{52}-6 q^{51}-q^{50}-3 q^{49}+2 q^{48}+25 q^{47}-8 q^{46}+q^{45}-3 q^{44}-4 q^{43}-11 q^{42}-5 q^{41}+16 q^{40}-2 q^{39}+7 q^{38}+2 q^{37}+q^{36}-10 q^{35}-8 q^{34}+7 q^{33}-3 q^{32}+5 q^{31}+3 q^{30}+4 q^{29}-4 q^{28}-5 q^{27}+3 q^{26}-3 q^{25}+q^{24}+q^{23}+3 q^{22}-q^{21}-2 q^{20}+2 q^{19}-q^{18}+q^{15}-q^{13}+q^{12}}
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 -q^{210}+q^{209}+q^{206}-q^{203}-2 q^{202}+2 q^{201}-q^{199}+2 q^{198}+q^{196}-q^{195}-5 q^{194}+3 q^{193}+q^{192}-2 q^{191}+3 q^{190}+4 q^{188}-2 q^{187}-9 q^{186}+3 q^{185}+q^{184}-2 q^{183}+5 q^{182}+q^{181}+9 q^{180}-q^{179}-13 q^{178}-q^{177}-6 q^{176}-6 q^{175}+8 q^{174}+6 q^{173}+19 q^{172}+9 q^{171}-11 q^{170}-5 q^{169}-22 q^{168}-23 q^{167}+q^{166}+5 q^{165}+35 q^{164}+34 q^{163}+8 q^{162}+3 q^{161}-36 q^{160}-53 q^{159}-27 q^{158}-12 q^{157}+43 q^{156}+66 q^{155}+40 q^{154}+32 q^{153}-38 q^{152}-82 q^{151}-63 q^{150}-46 q^{149}+39 q^{148}+88 q^{147}+74 q^{146}+65 q^{145}-30 q^{144}-96 q^{143}-88 q^{142}-79 q^{141}+25 q^{140}+101 q^{139}+93 q^{138}+88 q^{137}-22 q^{136}-98 q^{135}-95 q^{134}-98 q^{133}+14 q^{132}+104 q^{131}+100 q^{130}+97 q^{129}-19 q^{128}-97 q^{127}-92 q^{126}-106 q^{125}+9 q^{124}+104 q^{123}+100 q^{122}+100 q^{121}-19 q^{120}-95 q^{119}-93 q^{118}-104 q^{117}+10 q^{116}+102 q^{115}+99 q^{114}+98 q^{113}-17 q^{112}-95 q^{111}-91 q^{110}-99 q^{109}+8 q^{108}+96 q^{107}+94 q^{106}+95 q^{105}-13 q^{104}-88 q^{103}-83 q^{102}-93 q^{101}+2 q^{100}+82 q^{99}+83 q^{98}+88 q^{97}-2 q^{96}-69 q^{95}-69 q^{94}-87 q^{93}-11 q^{92}+60 q^{91}+63 q^{90}+79 q^{89}+14 q^{88}-41 q^{87}-46 q^{86}-78 q^{85}-25 q^{84}+30 q^{83}+37 q^{82}+64 q^{81}+26 q^{80}-12 q^{79}-16 q^{78}-58 q^{77}-34 q^{76}+4 q^{75}+8 q^{74}+41 q^{73}+25 q^{72}+7 q^{71}+12 q^{70}-31 q^{69}-26 q^{68}-8 q^{67}-13 q^{66}+14 q^{65}+12 q^{64}+8 q^{63}+24 q^{62}-7 q^{61}-8 q^{60}-q^{59}-18 q^{58}-3 q^{57}-4 q^{56}-2 q^{55}+18 q^{54}+q^{53}+5 q^{52}+11 q^{51}-8 q^{50}-6 q^{49}-8 q^{48}-10 q^{47}+7 q^{46}-3 q^{45}+5 q^{44}+13 q^{43}+q^{42}+q^{41}-5 q^{40}-8 q^{39}+2 q^{38}-5 q^{37}-q^{36}+6 q^{35}+3 q^{34}+3 q^{33}-q^{32}-4 q^{31}+3 q^{30}-3 q^{29}-2 q^{28}+q^{27}+q^{26}+2 q^{25}-2 q^{23}+2 q^{22}-q^{20}+q^{17}-q^{15}+q^{14}}

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.

7_2 7_4