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]

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]=

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

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

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 $\chi$ (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=$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)

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[7, 4]]
Out[2]=	7
In[3]:=	PD[Knot[7, 4]]
Out[3]=	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[4]:=	GaussCode[Knot[7, 4]]
Out[4]=	GaussCode[1, -5, 6, -7, 2, -1, 3, -4, 7, -6, 5, -2, 4, -3]
In[5]:=	BR[Knot[7, 4]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, 2, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[7, 4]][t]
Out[6]=	4 -7 + - + 4 t t
In[7]:=	Conway[Knot[7, 4]][z]
Out[7]=	2 1 + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 4], Knot[9, 2]}
In[9]:=	{KnotDet[Knot[7, 4]], KnotSignature[Knot[7, 4]]}
Out[9]=	{15, 2}
In[10]:=	J=Jones[Knot[7, 4]][q]
Out[10]=	2 3 4 5 6 7 8 q - 2 q + 3 q - 2 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 4]}
In[12]:=	A2Invariant[Knot[7, 4]][q]
Out[12]=	2 4 8 10 12 14 16 20 24 26 q - q + q + q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[Knot[7, 4]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[7, 4]], Vassiliev[3][Knot[7, 4]]}
Out[14]=	{0, 8}
In[15]:=	Kh[Knot[7, 4]][q, t]
Out[15]=	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