7 4

7_3

7_5

Visit 7 4's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 4's page at Knotilus! Visit 7 4's page at the original Knot Atlas! Simplest version of Endless knot symbol.

Celtic or pseudo-Celtic knot	Mongolian ornament	Susan Williams' medallion [1], the "Endless knot" of Buddhism [2]	Ornamental "Endless knot"
a knot seen at the Castle of Kornik [3]	A 7-4 knot reduced from TakaraMusubi with 9 crossings [4]	TakaraMusubi knot seen in Japanese symbols, or Kolam in South India [5]	Buddhist Endless Knot
Ornamental Endless Knot	Albrecht Dürer knot, 16th-century	A laser cut by Tom Longtin [6]	Unicursal hexagram of occultism
Logo of the raelian sect	Lissajous curve : x=cos 3t , y=sin 2t, z=sin 7t	French europa stamp 2023

Knot presentations

Planar diagram presentation	X6271 X12,6,13,5 X14,8,1,7 X8,14,9,13 X2,12,3,11 X10,4,11,3 X4,10,5,9
Gauss code	1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3
Dowker-Thistlethwaite code	6 10 12 14 4 2 8
Conway Notation	[313]

Minimum Braid Representative: Length is 9, width is 4. Braid index is 4.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 1 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 5.13794 A-Polynomial See Data:7 4/A-polynomial

[edit Notes for 7 4's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,4))$ Rasmussen s-Invariant -2

[edit Notes for 7 4's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t+4t^{-1}-7$
Conway polynomial	$4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 15, 2 }
Jones polynomial	$-q^{8}+q^{7}-2q^{6}+3q^{5}-2q^{4}+3q^{3}-2q^{2}+q$
HOMFLY-PT polynomial (db, data sources)	$-a^{-8}+z^{2}a^{-6}+2z^{2}a^{-4}+2a^{-4}+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{-6}+z^{6}a^{-8}+2z^{5}a^{-5}+3z^{5}a^{-7}+z^{5}a^{-9}+3z^{4}a^{-4}-3z^{4}a^{-8}+2z^{3}a^{-3}-2z^{3}a^{-5}-8z^{3}a^{-7}-4z^{3}a^{-9}+z^{2}a^{-2}-4z^{2}a^{-4}-3z^{2}a^{-6}+2z^{2}a^{-8}+4za^{-7}+4za^{-9}+2a^{-4}-a^{-8}$
The A2 invariant	$q^{-2}-q^{-4}+q^{-8}+q^{-10}+2q^{-12}+q^{-14}+q^{-16}-q^{-20}-q^{-24}-q^{-26}$
The G2 invariant	$q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-22}+4q^{-24}-2q^{-26}+2q^{-28}-q^{-30}+q^{-34}-2q^{-36}+3q^{-38}-2q^{-40}+q^{-44}+2q^{-48}+q^{-50}-q^{-52}+2q^{-54}-q^{-56}+3q^{-58}+q^{-60}-2q^{-62}+6q^{-64}-3q^{-66}+4q^{-68}+2q^{-70}-3q^{-72}+4q^{-74}-3q^{-76}+3q^{-78}-q^{-80}-q^{-82}+q^{-84}-2q^{-86}+2q^{-88}-3q^{-92}-q^{-94}-2q^{-96}-4q^{-102}+2q^{-104}-3q^{-106}+q^{-108}+q^{-110}-4q^{-112}+3q^{-114}-2q^{-116}+q^{-118}-2q^{-122}+2q^{-124}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 7_4.

A1 Invariants.

Weight	Invariant
1	$q^{-1}-q^{-3}+q^{-5}+q^{-7}+q^{-9}+q^{-11}-q^{-13}-q^{-17}$
2	$q^{-2}-q^{-4}+3q^{-8}-q^{-10}+2q^{-14}-q^{-16}+q^{-20}+2q^{-22}+q^{-28}-q^{-30}-3q^{-32}-2q^{-38}+q^{-40}+q^{-42}-q^{-44}+q^{-48}$
3	$q^{-3}-q^{-5}+2q^{-9}+q^{-11}-q^{-13}-q^{-15}+3q^{-17}+q^{-19}-3q^{-21}+q^{-23}+4q^{-25}+2q^{-27}-3q^{-29}-q^{-31}+q^{-33}+2q^{-35}-q^{-39}-q^{-41}+q^{-43}+3q^{-45}-2q^{-47}-3q^{-49}+2q^{-53}-2q^{-55}-3q^{-57}-2q^{-59}+2q^{-61}-2q^{-65}-q^{-67}+3q^{-69}+3q^{-71}-q^{-73}-2q^{-75}+q^{-77}+3q^{-79}-2q^{-83}-q^{-85}+q^{-87}+q^{-89}-q^{-93}$
4	$q^{-4}-q^{-6}+2q^{-10}+q^{-14}-2q^{-16}+q^{-18}+3q^{-20}-2q^{-22}+q^{-24}-2q^{-26}+4q^{-28}+6q^{-30}-3q^{-32}-4q^{-34}-5q^{-36}+5q^{-38}+9q^{-40}-4q^{-44}-6q^{-46}+6q^{-50}+3q^{-52}-3q^{-56}-3q^{-58}-q^{-60}+2q^{-62}+4q^{-64}-q^{-66}-4q^{-68}-5q^{-70}+q^{-72}+4q^{-74}+2q^{-76}-4q^{-78}-6q^{-80}+q^{-82}+4q^{-84}+2q^{-86}-5q^{-88}-5q^{-90}+2q^{-92}+3q^{-94}+4q^{-96}-2q^{-98}-5q^{-100}+2q^{-102}+3q^{-104}+7q^{-106}+q^{-108}-5q^{-110}-3q^{-112}-q^{-114}+6q^{-116}+4q^{-118}-2q^{-120}-4q^{-122}-5q^{-124}+2q^{-126}+4q^{-128}+2q^{-130}-q^{-132}-5q^{-134}-q^{-136}+q^{-138}+2q^{-140}+2q^{-142}-q^{-144}-q^{-146}-q^{-148}+q^{-152}$
5	$q^{-5}-q^{-7}+2q^{-11}+q^{-21}+q^{-23}-2q^{-27}+q^{-29}+5q^{-31}+5q^{-33}-2q^{-35}-6q^{-37}-7q^{-39}+11q^{-43}+12q^{-45}+3q^{-47}-11q^{-49}-15q^{-51}-5q^{-53}+9q^{-55}+18q^{-57}+12q^{-59}-7q^{-61}-16q^{-63}-12q^{-65}-q^{-67}+11q^{-69}+13q^{-71}+3q^{-73}-7q^{-75}-8q^{-77}-7q^{-79}-q^{-81}+6q^{-83}+7q^{-85}+4q^{-87}-q^{-89}-7q^{-91}-8q^{-93}-4q^{-95}+5q^{-97}+9q^{-99}+2q^{-101}-6q^{-103}-10q^{-105}-5q^{-107}+5q^{-109}+9q^{-111}+q^{-113}-4q^{-115}-9q^{-117}-3q^{-119}+8q^{-121}+10q^{-123}+2q^{-125}-6q^{-127}-10q^{-129}-3q^{-131}+9q^{-133}+13q^{-135}+3q^{-137}-6q^{-139}-11q^{-141}-7q^{-143}+7q^{-145}+13q^{-147}+9q^{-149}-9q^{-153}-12q^{-155}-4q^{-157}+6q^{-159}+11q^{-161}+6q^{-163}-2q^{-165}-10q^{-167}-11q^{-169}-4q^{-171}+6q^{-173}+10q^{-175}+7q^{-177}-q^{-179}-8q^{-181}-9q^{-183}-3q^{-185}+4q^{-187}+8q^{-189}+6q^{-191}-5q^{-195}-6q^{-197}-3q^{-199}+2q^{-201}+5q^{-203}+3q^{-205}+q^{-207}-q^{-209}-3q^{-211}-2q^{-213}+q^{-217}+q^{-219}+q^{-221}-q^{-225}$
6	$q^{-6}-q^{-8}+2q^{-12}-q^{-18}+2q^{-20}-q^{-24}+3q^{-26}-2q^{-28}-q^{-30}+2q^{-32}+7q^{-34}+2q^{-36}-3q^{-38}-3q^{-40}-10q^{-42}-4q^{-44}+8q^{-46}+21q^{-48}+10q^{-50}-5q^{-52}-13q^{-54}-25q^{-56}-11q^{-58}+13q^{-60}+34q^{-62}+25q^{-64}+q^{-66}-19q^{-68}-40q^{-70}-28q^{-72}+4q^{-74}+35q^{-76}+39q^{-78}+19q^{-80}-5q^{-82}-36q^{-84}-38q^{-86}-17q^{-88}+15q^{-90}+32q^{-92}+28q^{-94}+13q^{-96}-14q^{-98}-26q^{-100}-26q^{-102}-8q^{-104}+7q^{-106}+15q^{-108}+18q^{-110}+8q^{-112}-2q^{-114}-12q^{-116}-14q^{-118}-12q^{-120}-2q^{-122}+10q^{-124}+17q^{-126}+12q^{-128}-12q^{-132}-19q^{-134}-9q^{-136}+3q^{-138}+15q^{-140}+13q^{-142}-12q^{-146}-18q^{-148}-3q^{-150}+8q^{-152}+18q^{-154}+12q^{-156}-6q^{-158}-16q^{-160}-14q^{-162}+2q^{-164}+14q^{-166}+24q^{-168}+12q^{-170}-9q^{-172}-23q^{-174}-19q^{-176}-2q^{-178}+14q^{-180}+29q^{-182}+20q^{-184}-4q^{-186}-26q^{-188}-27q^{-190}-12q^{-192}+6q^{-194}+29q^{-196}+28q^{-198}+10q^{-200}-17q^{-202}-27q^{-204}-23q^{-206}-11q^{-208}+15q^{-210}+27q^{-212}+23q^{-214}+q^{-216}-13q^{-218}-23q^{-220}-25q^{-222}-5q^{-224}+12q^{-226}+24q^{-228}+18q^{-230}+9q^{-232}-8q^{-234}-23q^{-236}-18q^{-238}-9q^{-240}+8q^{-242}+15q^{-244}+21q^{-246}+12q^{-248}-5q^{-250}-13q^{-252}-17q^{-254}-10q^{-256}-2q^{-258}+12q^{-260}+15q^{-262}+9q^{-264}+2q^{-266}-6q^{-268}-10q^{-270}-12q^{-272}-2q^{-274}+4q^{-276}+7q^{-278}+7q^{-280}+4q^{-282}-6q^{-286}-4q^{-288}-3q^{-290}-q^{-292}+q^{-294}+3q^{-296}+3q^{-298}-q^{-304}-q^{-306}-q^{-308}+q^{-312}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-2}-q^{-4}+q^{-8}+q^{-10}+2q^{-12}+q^{-14}+q^{-16}-q^{-20}-q^{-24}-q^{-26}$
1,1	$q^{-4}-2q^{-6}+2q^{-8}-2q^{-10}+7q^{-12}-4q^{-14}+4q^{-16}-4q^{-18}+7q^{-20}+4q^{-24}+4q^{-26}+q^{-28}+6q^{-30}-8q^{-32}+8q^{-34}-15q^{-36}+8q^{-38}-12q^{-40}+6q^{-42}-7q^{-44}+4q^{-46}+2q^{-48}-2q^{-50}+3q^{-52}-6q^{-54}+6q^{-56}-8q^{-58}+4q^{-60}-4q^{-62}+4q^{-64}+q^{-68}$
2,0	$q^{-4}-q^{-6}-q^{-8}+2q^{-10}+2q^{-12}-q^{-16}+q^{-18}+2q^{-20}+q^{-24}+2q^{-26}+2q^{-28}+2q^{-30}+3q^{-32}+q^{-34}+q^{-36}-q^{-40}-4q^{-42}-4q^{-44}-2q^{-46}-q^{-48}-2q^{-50}-q^{-52}+q^{-54}+q^{-56}-q^{-60}+q^{-62}+q^{-64}+q^{-66}$
3,0	$q^{-6}-q^{-8}-q^{-10}+q^{-12}+3q^{-14}+2q^{-16}-4q^{-18}-2q^{-20}+3q^{-22}+5q^{-24}+3q^{-26}-4q^{-28}-q^{-30}+2q^{-32}+6q^{-34}+4q^{-36}-q^{-38}-q^{-40}+2q^{-42}+3q^{-44}+2q^{-46}+q^{-48}+2q^{-50}+2q^{-52}+q^{-54}+q^{-56}+2q^{-58}+q^{-60}-4q^{-62}-5q^{-64}-4q^{-66}-q^{-68}-4q^{-70}-7q^{-72}-7q^{-74}-4q^{-76}+q^{-78}-2q^{-82}-2q^{-84}+q^{-86}+6q^{-88}+5q^{-90}+2q^{-92}+4q^{-98}+3q^{-100}+q^{-102}-2q^{-104}-3q^{-106}+q^{-110}+2q^{-112}-q^{-116}-q^{-118}-q^{-120}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-4}-q^{-6}-q^{-8}+2q^{-10}+3q^{-16}+q^{-18}+q^{-20}+2q^{-22}+2q^{-24}+q^{-26}+2q^{-28}+2q^{-30}+2q^{-32}-2q^{-34}-q^{-36}-2q^{-38}-5q^{-40}-3q^{-42}-q^{-44}-q^{-46}+q^{-48}+2q^{-50}+q^{-54}$
1,0,0	$q^{-3}-q^{-5}+q^{-11}+q^{-13}+2q^{-15}+2q^{-17}+q^{-19}+q^{-21}-q^{-27}-q^{-31}-q^{-33}-q^{-35}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-6}-q^{-8}-q^{-10}+q^{-12}+q^{-14}+2q^{-20}+2q^{-22}+q^{-26}+3q^{-28}+3q^{-30}+2q^{-32}+5q^{-34}+4q^{-36}+4q^{-38}+3q^{-40}+3q^{-42}-3q^{-46}-3q^{-48}-5q^{-50}-8q^{-52}-7q^{-54}-3q^{-56}-2q^{-58}-q^{-60}+q^{-62}+3q^{-64}+2q^{-66}+q^{-68}+q^{-70}+q^{-72}$
1,0,0,0	$q^{-4}-q^{-6}+q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+2q^{-22}+q^{-24}+q^{-26}-q^{-34}-q^{-38}-q^{-40}-q^{-42}-q^{-44}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-4}-q^{-6}+q^{-8}-2q^{-10}+2q^{-12}+q^{-16}+q^{-18}+q^{-20}+2q^{-22}-2q^{-24}+3q^{-26}-2q^{-28}+2q^{-30}-2q^{-32}+2q^{-34}-q^{-36}+q^{-40}-q^{-42}+q^{-44}-q^{-46}+q^{-48}-2q^{-50}-q^{-54}$
1,0	$q^{-6}-q^{-10}-q^{-12}+2q^{-16}+q^{-18}-q^{-20}+q^{-24}+2q^{-26}+2q^{-34}+q^{-36}+2q^{-42}+2q^{-44}+q^{-46}+q^{-50}+q^{-52}-2q^{-56}-q^{-58}-q^{-62}-3q^{-64}-3q^{-66}-q^{-68}-q^{-74}-q^{-76}+q^{-78}+2q^{-80}+q^{-88}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-6}-q^{-8}-q^{-12}+2q^{-14}-q^{-16}+q^{-18}+2q^{-22}+2q^{-24}+2q^{-26}+2q^{-28}+q^{-30}+3q^{-32}+3q^{-36}-q^{-38}+3q^{-40}+3q^{-44}-q^{-46}+q^{-48}-2q^{-50}-2q^{-52}-3q^{-54}-4q^{-56}-2q^{-58}-3q^{-60}-q^{-62}-q^{-64}+2q^{-66}+2q^{-70}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-10}-q^{-12}+q^{-14}-q^{-16}-q^{-22}+4q^{-24}-2q^{-26}+2q^{-28}-q^{-30}+q^{-34}-2q^{-36}+3q^{-38}-2q^{-40}+q^{-44}+2q^{-48}+q^{-50}-q^{-52}+2q^{-54}-q^{-56}+3q^{-58}+q^{-60}-2q^{-62}+6q^{-64}-3q^{-66}+4q^{-68}+2q^{-70}-3q^{-72}+4q^{-74}-3q^{-76}+3q^{-78}-q^{-80}-q^{-82}+q^{-84}-2q^{-86}+2q^{-88}-3q^{-92}-q^{-94}-2q^{-96}-4q^{-102}+2q^{-104}-3q^{-106}+q^{-108}+q^{-110}-4q^{-112}+3q^{-114}-2q^{-116}+q^{-118}-2q^{-122}+2q^{-124}+q^{-128}$

.

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 4"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t+4t^{-1}-7$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{2}+1$

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]=

{ 15, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_2, ...}

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

Vassiliev invariants

V2 and V3:

(4, 8)

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 $16$ $64$ $128$ ${\frac {1016}{3}}$ ${\frac {184}{3}}$ $1024$ ${\frac {5824}{3}}$ ${\frac {1024}{3}}$ $320$ ${\frac {2048}{3}}$ $2048$ ${\frac {16256}{3}}$ ${\frac {2944}{3}}$ ${\frac {168062}{15}}$ $-{\frac {1176}{5}}$ ${\frac {233288}{45}}$ ${\frac {898}{9}}$ 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{11102}{15}}

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 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=} 2 is the signature of 7 4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 χ 17               1 -1 15                 0 13           2 1   -1 11         1       1 9       1 2       1 7     2 1         1 5     1           1 3 1 2             -1 1 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}} $i=1$ $i=3$ 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 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}\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}\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^{23}-q^{22}-q^{21}+3 q^{20}-q^{19}-4 q^{18}+5 q^{17}-q^{16}-7 q^{15}+7 q^{14}+q^{13}-8 q^{12}+7 q^{11}+3 q^{10}-9 q^9+6 q^8+2 q^7-6 q^6+4 q^5+q^4-2 q^3+q^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 -q^{45}+q^{44}+q^{43}-3 q^{41}+3 q^{39}+3 q^{38}-5 q^{37}-3 q^{36}+4 q^{35}+7 q^{34}-5 q^{33}-7 q^{32}+3 q^{31}+9 q^{30}-3 q^{29}-11 q^{28}+2 q^{27}+10 q^{26}+q^{25}-13 q^{24}-q^{23}+11 q^{22}+6 q^{21}-15 q^{20}-3 q^{19}+11 q^{18}+7 q^{17}-13 q^{16}-4 q^{15}+9 q^{14}+5 q^{13}-8 q^{12}-2 q^{11}+6 q^{10}+q^9-4 q^8+2 q^6+q^5-2 q^4+q^3}
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^{74}-q^{73}-q^{72}+4 q^{69}-q^{68}-2 q^{67}-2 q^{66}-4 q^{65}+8 q^{64}+2 q^{63}-4 q^{61}-11 q^{60}+9 q^{59}+4 q^{58}+6 q^{57}-2 q^{56}-18 q^{55}+7 q^{54}+2 q^{53}+12 q^{52}+4 q^{51}-22 q^{50}+6 q^{49}-5 q^{48}+15 q^{47}+10 q^{46}-23 q^{45}+5 q^{44}-12 q^{43}+15 q^{42}+17 q^{41}-21 q^{40}+2 q^{39}-19 q^{38}+17 q^{37}+23 q^{36}-19 q^{35}-q^{34}-25 q^{33}+18 q^{32}+26 q^{31}-14 q^{30}-3 q^{29}-28 q^{28}+16 q^{27}+26 q^{26}-11 q^{25}-25 q^{23}+10 q^{22}+20 q^{21}-9 q^{20}+4 q^{19}-16 q^{18}+6 q^{17}+10 q^{16}-8 q^{15}+5 q^{14}-7 q^{13}+4 q^{12}+4 q^{11}-5 q^{10}+2 q^9-2 q^8+2 q^7+q^6-2 q^5+q^4}
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^{110}+q^{109}+q^{108}-q^{105}-3 q^{104}+3 q^{102}+2 q^{101}+2 q^{100}+q^{99}-6 q^{98}-5 q^{97}+3 q^{95}+7 q^{94}+7 q^{93}-4 q^{92}-9 q^{91}-7 q^{90}-3 q^{89}+8 q^{88}+14 q^{87}+4 q^{86}-6 q^{85}-11 q^{84}-13 q^{83}+q^{82}+15 q^{81}+12 q^{80}+2 q^{79}-6 q^{78}-18 q^{77}-9 q^{76}+7 q^{75}+15 q^{74}+11 q^{73}+3 q^{72}-14 q^{71}-15 q^{70}-7 q^{69}+11 q^{68}+16 q^{67}+12 q^{66}-4 q^{65}-19 q^{64}-19 q^{63}+4 q^{62}+20 q^{61}+20 q^{60}+4 q^{59}-21 q^{58}-30 q^{57}-2 q^{56}+25 q^{55}+25 q^{54}+12 q^{53}-25 q^{52}-40 q^{51}-7 q^{50}+29 q^{49}+33 q^{48}+19 q^{47}-29 q^{46}-49 q^{45}-11 q^{44}+30 q^{43}+39 q^{42}+24 q^{41}-26 q^{40}-50 q^{39}-18 q^{38}+24 q^{37}+38 q^{36}+25 q^{35}-16 q^{34}-40 q^{33}-20 q^{32}+12 q^{31}+27 q^{30}+21 q^{29}-7 q^{28}-21 q^{27}-14 q^{26}+3 q^{25}+13 q^{24}+11 q^{23}-3 q^{22}-7 q^{21}-5 q^{20}+2 q^{19}+2 q^{18}+4 q^{17}-2 q^{16}-3 q^{15}+2 q^{14}+2 q^{13}-2 q^{12}+q^{11}-2 q^9+2 q^8+q^7-2 q^6+q^5}
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^{153}-q^{152}-q^{151}+q^{148}+4 q^{146}-q^{145}-3 q^{144}-2 q^{143}-2 q^{142}-2 q^{140}+10 q^{139}+3 q^{138}-2 q^{136}-5 q^{135}-6 q^{134}-12 q^{133}+12 q^{132}+7 q^{131}+8 q^{130}+5 q^{129}+q^{128}-9 q^{127}-26 q^{126}+4 q^{125}+12 q^{123}+13 q^{122}+18 q^{121}-32 q^{119}-3 q^{118}-17 q^{117}+3 q^{116}+8 q^{115}+33 q^{114}+17 q^{113}-23 q^{112}+3 q^{111}-29 q^{110}-14 q^{109}-12 q^{108}+35 q^{107}+27 q^{106}-9 q^{105}+25 q^{104}-25 q^{103}-26 q^{102}-38 q^{101}+23 q^{100}+23 q^{99}+q^{98}+52 q^{97}-7 q^{96}-25 q^{95}-61 q^{94}+5 q^{93}+8 q^{92}+2 q^{91}+74 q^{90}+17 q^{89}-16 q^{88}-76 q^{87}-11 q^{86}-9 q^{85}-2 q^{84}+88 q^{83}+38 q^{82}-4 q^{81}-86 q^{80}-23 q^{79}-25 q^{78}-4 q^{77}+98 q^{76}+56 q^{75}+2 q^{74}-96 q^{73}-34 q^{72}-40 q^{71}+2 q^{70}+110 q^{69}+69 q^{68}+4 q^{67}-108 q^{66}-46 q^{65}-50 q^{64}+9 q^{63}+122 q^{62}+81 q^{61}+9 q^{60}-115 q^{59}-58 q^{58}-60 q^{57}+7 q^{56}+124 q^{55}+91 q^{54}+19 q^{53}-105 q^{52}-61 q^{51}-68 q^{50}-8 q^{49}+106 q^{48}+91 q^{47}+31 q^{46}-78 q^{45}-46 q^{44}-64 q^{43}-25 q^{42}+74 q^{41}+70 q^{40}+33 q^{39}-47 q^{38}-22 q^{37}-44 q^{36}-29 q^{35}+43 q^{34}+38 q^{33}+21 q^{32}-26 q^{31}-2 q^{30}-20 q^{29}-20 q^{28}+22 q^{27}+14 q^{26}+7 q^{25}-14 q^{24}+6 q^{23}-5 q^{22}-9 q^{21}+9 q^{20}+2 q^{19}+q^{18}-7 q^{17}+6 q^{16}-4 q^{14}+4 q^{13}-q^{12}-2 q^{10}+2 q^9+q^8-2 q^7+q^6}
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^{203}+q^{202}+q^{201}-q^{198}-q^{196}-3 q^{195}+q^{194}+3 q^{193}+2 q^{192}+3 q^{191}-q^{190}-9 q^{187}-4 q^{186}+2 q^{184}+8 q^{183}+3 q^{182}+7 q^{181}+8 q^{180}-10 q^{179}-10 q^{178}-10 q^{177}-10 q^{176}+5 q^{175}+2 q^{174}+13 q^{173}+24 q^{172}+3 q^{171}-q^{170}-12 q^{169}-25 q^{168}-10 q^{167}-15 q^{166}+3 q^{165}+31 q^{164}+17 q^{163}+22 q^{162}+9 q^{161}-22 q^{160}-15 q^{159}-36 q^{158}-24 q^{157}+14 q^{156}+9 q^{155}+34 q^{154}+37 q^{153}+5 q^{152}+7 q^{151}-36 q^{150}-45 q^{149}-12 q^{148}-23 q^{147}+14 q^{146}+44 q^{145}+29 q^{144}+47 q^{143}-6 q^{142}-37 q^{141}-22 q^{140}-61 q^{139}-26 q^{138}+18 q^{137}+25 q^{136}+78 q^{135}+40 q^{134}-3 q^{133}-5 q^{132}-80 q^{131}-65 q^{130}-25 q^{129}-10 q^{128}+86 q^{127}+79 q^{126}+41 q^{125}+34 q^{124}-77 q^{123}-88 q^{122}-69 q^{121}-58 q^{120}+70 q^{119}+99 q^{118}+81 q^{117}+79 q^{116}-55 q^{115}-97 q^{114}-101 q^{113}-104 q^{112}+41 q^{111}+105 q^{110}+112 q^{109}+120 q^{108}-29 q^{107}-96 q^{106}-124 q^{105}-144 q^{104}+15 q^{103}+108 q^{102}+136 q^{101}+150 q^{100}-10 q^{99}-99 q^{98}-143 q^{97}-174 q^{96}+q^{95}+118 q^{94}+157 q^{93}+169 q^{92}-2 q^{91}-110 q^{90}-165 q^{89}-194 q^{88}-3 q^{87}+137 q^{86}+179 q^{85}+185 q^{84}+3 q^{83}-129 q^{82}-188 q^{81}-212 q^{80}-7 q^{79}+151 q^{78}+199 q^{77}+204 q^{76}+14 q^{75}-133 q^{74}-202 q^{73}-229 q^{72}-30 q^{71}+138 q^{70}+204 q^{69}+220 q^{68}+40 q^{67}-107 q^{66}-187 q^{65}-233 q^{64}-60 q^{63}+93 q^{62}+173 q^{61}+213 q^{60}+67 q^{59}-59 q^{58}-142 q^{57}-200 q^{56}-72 q^{55}+39 q^{54}+113 q^{53}+167 q^{52}+69 q^{51}-21 q^{50}-83 q^{49}-138 q^{48}-52 q^{47}+10 q^{46}+53 q^{45}+103 q^{44}+43 q^{43}-5 q^{42}-37 q^{41}-75 q^{40}-21 q^{39}+4 q^{38}+18 q^{37}+48 q^{36}+13 q^{35}-q^{34}-10 q^{33}-34 q^{32}-3 q^{31}+5 q^{30}+4 q^{29}+15 q^{28}-q^{27}+q^{26}+2 q^{25}-14 q^{24}+2 q^{23}+4 q^{22}+2 q^{20}-4 q^{19}+2 q^{18}+4 q^{17}-6 q^{16}+2 q^{15}+2 q^{14}-q^{13}-2 q^{11}+2 q^{10}+q^9-2 q^8+q^7}

Computer Talk

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

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[7, 4]]
Out[2]=	PD[X[6, 2, 7, 1], X[12, 6, 13, 5], X[14, 8, 1, 7], X[8, 14, 9, 13], X[2, 12, 3, 11], X[10, 4, 11, 3], X[4, 10, 5, 9]]
In[3]:=	GaussCode[Knot[7, 4]]
Out[3]=	GaussCode[1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3]
In[4]:=	DTCode[Knot[7, 4]]
Out[4]=	DTCode[6, 10, 12, 14, 4, 2, 8]
In[5]:=	br = BR[Knot[7, 4]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 9}
In[7]:=	BraidIndex[Knot[7, 4]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[7, 4]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[7, 4]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 1, 2, {3, 4}, 1}
In[10]:=	alex = Alexander[Knot[7, 4]][t]
Out[10]=	4 -7 + - + 4 t t
In[11]:=	Conway[Knot[7, 4]][z]
Out[11]=	2 1 + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[7, 4], Knot[9, 2]}
In[13]:=	{KnotDet[Knot[7, 4]], KnotSignature[Knot[7, 4]]}
Out[13]=	{15, 2}
In[14]:=	Jones[Knot[7, 4]][q]
Out[14]=	2 3 4 5 6 7 8 q - 2 q + 3 q - 2 q + 3 q - 2 q + q - q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[7, 4]}
In[16]:=	A2Invariant[Knot[7, 4]][q]
Out[16]=	2 4 8 10 12 14 16 20 24 26 q - q + q + q + 2 q + q + q - q - q - q
In[17]:=	HOMFLYPT[Knot[7, 4]][a, z]
Out[17]=	2 2 2 -8 2 z 2 z z -a + -- + -- + ---- + -- 4 6 4 2 a a a a
In[18]:=	Kauffman[Knot[7, 4]][a, z]
Out[18]=	2 2 2 2 3 3 3 -8 2 4 z 4 z 2 z 3 z 4 z z 4 z 8 z 2 z -a + -- + --- + --- + ---- - ---- - ---- + -- - ---- - ---- - ---- + 4 9 7 8 6 4 2 9 7 5 a a a a a a a a a a 3 4 4 5 5 5 6 6 2 z 3 z 3 z z 3 z 2 z z z ---- - ---- + ---- + -- + ---- + ---- + -- + -- 3 8 4 9 7 5 8 6 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[7, 4]], Vassiliev[3][Knot[7, 4]]}
Out[19]=	{4, 8}
In[20]:=	Kh[Knot[7, 4]][q, t]
Out[20]=	3 3 5 2 7 2 7 3 9 3 9 4 11 4 q + q + 2 q t + q t + 2 q t + q t + q t + 2 q t + q t + 13 5 13 6 17 7 2 q t + q t + q t
In[21]:=	ColouredJones[Knot[7, 4], 2][q]
Out[21]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + q + 4 q - 6 q + 2 q + 6 q - 9 q + 3 q + 7 q - 12 13 14 15 16 17 18 19 20 8 q + q + 7 q - 7 q - q + 5 q - 4 q - q + 3 q - 21 22 23 q - q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

7_3 7_5