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]=

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

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 $\{3,4\}$ 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]=

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

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 $20$ $88$ $200$ ${\frac {1510}{3}}$ ${\frac {242}{3}}$ $1760$ ${\frac {9520}{3}}$ ${\frac {1696}{3}}$ $440$ ${\frac {4000}{3}}$ $3872$ ${\frac {30200}{3}}$ ${\frac {4840}{3}}$ ${\frac {121855}{6}}$ $382$ ${\frac {73862}{9}}$ ${\frac {2437}{18}}$ ${\frac {6559}{6}}$

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 $t^{r}q^{j}$ 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=$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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

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

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.

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