7 3

7_2

7_4

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7 3 Quick Notes

7 3 Further Notes and Views

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]

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 (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 \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]=

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

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

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, 3]]
Out[2]=	7
In[3]:=	PD[Knot[7, 3]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 1, 7], X[8, 14, 9, 13], X[12, 6, 13, 5], X[2, 10, 3, 9], X[4, 12, 5, 11]]
In[4]:=	GaussCode[Knot[7, 3]]
Out[4]=	GaussCode[1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3]
In[5]:=	BR[Knot[7, 3]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[7, 3]][t]
Out[6]=	2 3 2 3 + -- - - - 3 t + 2 t 2 t t
In[7]:=	Conway[Knot[7, 3]][z]
Out[7]=	2 4 1 + 5 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 3]}
In[9]:=	{KnotDet[Knot[7, 3]], KnotSignature[Knot[7, 3]]}
Out[9]=	{13, 4}
In[10]:=	J=Jones[Knot[7, 3]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - q + 2 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, 3]}
In[12]:=	A2Invariant[Knot[7, 3]][q]
Out[12]=	6 10 14 16 18 20 22 24 26 28 q + q + q + 2 q + q + q - q - q - q - q
In[13]:=	Kauffman[Knot[7, 3]][a, z]
Out[13]=	2 2 2 2 3 3 -2 2 -4 2 z z 3 z z 6 z 4 z 3 z z z -- - -- + a - --- + -- + --- - --- + ---- + ---- - ---- + --- - -- - 8 6 11 9 7 10 8 6 4 11 9 a a a a a a a a a a a 3 3 4 4 4 4 5 5 5 6 6 4 z 2 z z 3 z 3 z z z 2 z z z z ---- - ---- + --- - ---- - ---- + -- + -- + ---- + -- + -- + -- 7 5 10 8 6 4 9 7 5 8 6 a a a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[7, 3]], Vassiliev[3][Knot[7, 3]]}
Out[14]=	{0, 11}
In[15]:=	Kh[Knot[7, 3]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 13 4 q + q + q t + q t + q t + q t + q t + 2 q t + q t + 15 5 15 6 19 7 2 q t + q t + q t