7 1

6_3

7_2

Visit 7 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 1's page at Knotilus! Visit 7 1's page at the original Knot Atlas!

7_1 should perhaps be called "The Septafoil Knot", following the trefoil knot and the cinquefoil knot. See also T(7,2).

Interlaced form of 7/2 star polygon or "septagram"		Decorative interlaced form of 7/2 star polygon or "septagram"		3D depiction
Heptagram of intersecting circles.

Knot presentations

Planar diagram presentation	X1829 X3,10,4,11 X5,12,6,13 X7,14,8,1 X9,2,10,3 X11,4,12,5 X13,6,14,7
Gauss code	-1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4
Dowker-Thistlethwaite code	8 10 12 14 2 4 6
Conway Notation	[7]

Three dimensional invariants

Symmetry type Reversible Unknotting number 3 3-genus 3 Bridge index 2 Super bridge index 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 Not hyperbolic A-Polynomial See Data:7 1/A-polynomial

[edit Notes for 7 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $3$ Topological 4 genus $3$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,1))$ Rasmussen s-Invariant -6

[edit Notes for 7 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}+t^{-3}-t^{2}-t^{-2}+t+t^{-1}-1$
Conway polynomial	$z^{6}+5z^{4}+6z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 7, -6 }
Jones polynomial	$-q^{-10}+q^{-9}-q^{-8}+q^{-7}-q^{-6}+q^{-5}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$a^{8}\left(-z^{4}\right)-4a^{8}z^{2}-3a^{8}+a^{6}z^{6}+6a^{6}z^{4}+10a^{6}z^{2}+4a^{6}$
Kauffman polynomial (db, data sources)	$a^{13}z+a^{12}z^{2}+a^{11}z^{3}-a^{11}z+a^{10}z^{4}-2a^{10}z^{2}+a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+a^{8}z^{6}-5a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+a^{7}z^{5}-4a^{7}z^{3}+3a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}-4a^{6}$
The A2 invariant	$-q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}$
The G2 invariant	$q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2q^{84}+2q^{80}+q^{78}+q^{72}+3q^{70}+2q^{68}+q^{64}+2q^{62}+2q^{60}+q^{58}+q^{54}+q^{52}+q^{50}$

Further Quantum Invariants

Further quantum knot invariants for 7_1.

A1 Invariants.

Weight	Invariant
1	$-q^{21}+q^{9}+q^{7}+q^{5}$
2	$q^{56}-q^{44}-q^{42}-q^{40}+q^{18}+q^{16}+q^{14}+q^{12}+q^{10}$
3	$-q^{105}+q^{93}+q^{91}+q^{89}-q^{67}-q^{65}-q^{63}-q^{61}-q^{59}+q^{27}+q^{25}+q^{23}+q^{21}+q^{19}+q^{17}+q^{15}$
4	$q^{168}-q^{156}-q^{154}-q^{152}+q^{130}+q^{128}+q^{126}+q^{124}+q^{122}-q^{90}-q^{88}-q^{86}-q^{84}-q^{82}-q^{80}-q^{78}+q^{36}+q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{20}$
5	$-q^{245}+q^{233}+q^{231}+q^{229}-q^{207}-q^{205}-q^{203}-q^{201}-q^{199}+q^{167}+q^{165}+q^{163}+q^{161}+q^{159}+q^{157}+q^{155}-q^{113}-q^{111}-q^{109}-q^{107}-q^{105}-q^{103}-q^{101}-q^{99}-q^{97}+q^{45}+q^{43}+q^{41}+q^{39}+q^{37}+q^{35}+q^{33}+q^{31}+q^{29}+q^{27}+q^{25}$
6	$q^{336}-q^{324}-q^{322}-q^{320}+q^{298}+q^{296}+q^{294}+q^{292}+q^{290}-q^{258}-q^{256}-q^{254}-q^{252}-q^{250}-q^{248}-q^{246}+q^{204}+q^{202}+q^{200}+q^{198}+q^{196}+q^{194}+q^{192}+q^{190}+q^{188}-q^{136}-q^{134}-q^{132}-q^{130}-q^{128}-q^{126}-q^{124}-q^{122}-q^{120}-q^{118}-q^{116}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}+q^{38}+q^{36}+q^{34}+q^{32}+q^{30}$
8	$q^{560}-q^{548}-q^{546}-q^{544}+q^{522}+q^{520}+q^{518}+q^{516}+q^{514}-q^{482}-q^{480}-q^{478}-q^{476}-q^{474}-q^{472}-q^{470}+q^{428}+q^{426}+q^{424}+q^{422}+q^{420}+q^{418}+q^{416}+q^{414}+q^{412}-q^{360}-q^{358}-q^{356}-q^{354}-q^{352}-q^{350}-q^{348}-q^{346}-q^{344}-q^{342}-q^{340}+q^{278}+q^{276}+q^{274}+q^{272}+q^{270}+q^{268}+q^{266}+q^{264}+q^{262}+q^{260}+q^{258}+q^{256}+q^{254}-q^{182}-q^{180}-q^{178}-q^{176}-q^{174}-q^{172}-q^{170}-q^{168}-q^{166}-q^{164}-q^{162}-q^{160}-q^{158}-q^{156}-q^{154}+q^{72}+q^{70}+q^{68}+q^{66}+q^{64}+q^{62}+q^{60}+q^{58}+q^{56}+q^{54}+q^{52}+q^{50}+q^{48}+q^{46}+q^{44}+q^{42}+q^{40}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{30}-q^{28}-q^{26}+q^{18}+q^{16}+2q^{14}+q^{12}+q^{10}$
1,1	$q^{84}-2q^{48}-2q^{46}-4q^{44}-4q^{42}-4q^{40}-2q^{38}-q^{36}+2q^{34}+4q^{32}+4q^{30}+5q^{28}+4q^{26}+4q^{24}+2q^{22}+q^{20}$
2,0	$q^{74}+q^{72}+2q^{70}+q^{68}+q^{66}-q^{62}-2q^{60}-3q^{58}-3q^{56}-3q^{54}-2q^{52}-q^{50}+q^{36}+q^{34}+2q^{32}+2q^{30}+3q^{28}+2q^{26}+2q^{24}+q^{22}+q^{20}$
3,0	$-q^{132}-q^{130}-2q^{128}-2q^{126}-2q^{124}-q^{122}+2q^{118}+4q^{116}+4q^{114}+5q^{112}+4q^{110}+4q^{108}+2q^{106}+q^{104}-q^{94}-2q^{92}-3q^{90}-4q^{88}-5q^{86}-5q^{84}-5q^{82}-4q^{80}-3q^{78}-2q^{76}-q^{74}+q^{54}+q^{52}+2q^{50}+2q^{48}+3q^{46}+3q^{44}+4q^{42}+3q^{40}+3q^{38}+2q^{36}+2q^{34}+q^{32}+q^{30}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{70}-q^{48}-2q^{46}-3q^{44}-3q^{42}-3q^{40}-2q^{38}+q^{34}+3q^{32}+3q^{30}+4q^{28}+3q^{26}+3q^{24}+q^{22}+q^{20}$
1,0,0	$-q^{39}-q^{37}-2q^{35}-q^{33}-q^{31}+q^{27}+q^{25}+2q^{23}+2q^{21}+2q^{19}+q^{17}+q^{15}$
1,0,1	$q^{112}+q^{78}+q^{76}+3q^{74}+3q^{72}+4q^{70}+3q^{68}+q^{66}-3q^{64}-7q^{62}-10q^{60}-14q^{58}-14q^{56}-13q^{54}-8q^{52}-3q^{50}+3q^{48}+8q^{46}+11q^{44}+12q^{42}+11q^{40}+10q^{38}+7q^{36}+5q^{34}+2q^{32}+q^{30}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{88}+q^{86}+q^{84}+q^{82}+q^{80}-q^{66}-2q^{64}-4q^{62}-5q^{60}-7q^{58}-7q^{56}-6q^{54}-4q^{52}-q^{50}+2q^{48}+5q^{46}+6q^{44}+8q^{42}+6q^{40}+6q^{38}+4q^{36}+3q^{34}+q^{32}+q^{30}$
1,0,0,0	$-q^{48}-q^{46}-2q^{44}-2q^{42}-2q^{40}-q^{38}+q^{34}+2q^{32}+2q^{30}+3q^{28}+2q^{26}+2q^{24}+q^{22}+q^{20}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{70}-q^{48}-q^{44}-q^{42}-q^{40}+q^{34}+q^{32}+q^{30}+2q^{28}+q^{26}+q^{24}+q^{22}+q^{20}$
1,0	$q^{112}-q^{78}-q^{76}-q^{74}-q^{72}-2q^{70}-q^{68}-q^{66}-q^{64}-q^{62}+q^{54}+q^{50}+q^{48}+2q^{46}+q^{44}+2q^{42}+q^{40}+2q^{38}+q^{36}+q^{34}+q^{30}$

D4 Invariants.

Weight	Invariant
0,1,0,0	$q^{168}-q^{136}-q^{134}-3q^{132}-3q^{130}-4q^{128}-4q^{126}-q^{124}+3q^{120}+8q^{118}+11q^{116}+12q^{114}+15q^{112}+12q^{110}+12q^{108}+9q^{106}+6q^{104}+2q^{102}-6q^{98}-10q^{96}-16q^{94}-22q^{92}-27q^{90}-32q^{88}-33q^{86}-32q^{84}-27q^{82}-19q^{80}-8q^{78}+12q^{74}+19q^{72}+24q^{70}+26q^{68}+27q^{66}+22q^{64}+20q^{62}+14q^{60}+10q^{58}+6q^{56}+4q^{54}+q^{52}+q^{50}$
1,0,0,0	$q^{98}-q^{66}-q^{64}-3q^{62}-3q^{60}-4q^{58}-4q^{56}-3q^{54}-2q^{52}-q^{50}+2q^{48}+3q^{46}+4q^{44}+5q^{42}+4q^{40}+4q^{38}+3q^{36}+2q^{34}+q^{32}+q^{30}$

G2 Invariants.

Weight	Invariant
1,0	$q^{168}-q^{136}-q^{134}-q^{128}-q^{126}-q^{124}-q^{118}-q^{116}-q^{102}-q^{96}-q^{94}-q^{92}+q^{88}-2q^{84}+2q^{80}+q^{78}+q^{72}+3q^{70}+2q^{68}+q^{64}+2q^{62}+2q^{60}+q^{58}+q^{54}+q^{52}+q^{50}$

.

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 1"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}+t^{-3}-t^{2}-t^{-2}+t+t^{-1}-1$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+5z^{4}+6z^{2}+1$

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]=

{ 7, -6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{-10}+q^{-9}-q^{-8}+q^{-7}-q^{-6}+q^{-5}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{8}\left(-z^{4}\right)-4a^{8}z^{2}-3a^{8}+a^{6}z^{6}+6a^{6}z^{4}+10a^{6}z^{2}+4a^{6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{13}z+a^{12}z^{2}+a^{11}z^{3}-a^{11}z+a^{10}z^{4}-2a^{10}z^{2}+a^{9}z^{5}-3a^{9}z^{3}+a^{9}z+a^{8}z^{6}-5a^{8}z^{4}+7a^{8}z^{2}-3a^{8}+a^{7}z^{5}-4a^{7}z^{3}+3a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+10a^{6}z^{2}-4a^{6}$

Vassiliev invariants

V2 and V3:

(6, -14)

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 $24$ $-112$ $288$ $684$ $100$ $-2688$ $-{\frac {13888}{3}}$ $-{\frac {2464}{3}}$ $-560$ $2304$ $6272$ $16416$ $2400$ ${\frac {160231}{5}}$ ${\frac {21548}{15}}$ ${\frac {58148}{5}}$ $163$ ${\frac {7351}{5}}$

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 $s=$-6 is the signature of 7 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-7	-6	-5	-4	-3	-2	-1	0	χ
-5								1	1
-7								1	1
-9						1			1
-11									0
-13				1	1				0
-15									0
-17		1	1						0
-19									0
-21	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, 1]]
Out[2]=	7
In[3]:=	PD[Knot[7, 1]]
Out[3]=	PD[X[1, 8, 2, 9], X[3, 10, 4, 11], X[5, 12, 6, 13], X[7, 14, 8, 1], X[9, 2, 10, 3], X[11, 4, 12, 5], X[13, 6, 14, 7]]
In[4]:=	GaussCode[Knot[7, 1]]
Out[4]=	GaussCode[-1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4]
In[5]:=	BR[Knot[7, 1]]
Out[5]=	BR[2, {-1, -1, -1, -1, -1, -1, -1}]
In[6]:=	alex = Alexander[Knot[7, 1]][t]
Out[6]=	-3 -2 1 2 3 -1 + t - t + - + t - t + t t
In[7]:=	Conway[Knot[7, 1]][z]
Out[7]=	2 4 6 1 + 6 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 1]}
In[9]:=	{KnotDet[Knot[7, 1]], KnotSignature[Knot[7, 1]]}
Out[9]=	{7, -6}
In[10]:=	J=Jones[Knot[7, 1]][q]
Out[10]=	-10 -9 -8 -7 -6 -5 -3 -q + q - q + q - q + q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 1]}
In[12]:=	A2Invariant[Knot[7, 1]][q]
Out[12]=	-30 -28 -26 -18 -16 2 -12 -10 -q - q - q + q + q + --- + q + q 14 q
In[13]:=	Kauffman[Knot[7, 1]][a, z]
Out[13]=	6 8 7 9 11 13 6 2 8 2 -4 a - 3 a + 3 a z + a z - a z + a z + 10 a z + 7 a z - 10 2 12 2 7 3 9 3 11 3 6 4 8 4 2 a z + a z - 4 a z - 3 a z + a z - 6 a z - 5 a z + 10 4 7 5 9 5 6 6 8 6 a z + a z + a z + a z + a z
In[14]:=	{Vassiliev[2][Knot[7, 1]], Vassiliev[3][Knot[7, 1]]}
Out[14]=	{0, -14}
In[15]:=	Kh[Knot[7, 1]][q, t]
Out[15]=	-7 -5 1 1 1 1 1 1 q + q + ------ + ------ + ------ + ------ + ------ + ----- 21 7 17 6 17 5 13 4 13 3 9 2 q t q t q t q t q t q t