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

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)

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

The Coloured Jones Polynomials

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

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

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.