7 2

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7 1.gif

7_1

7 3.gif

7_3

7 2.gif Visit 7 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 7 2's page at Knotilus!

Visit 7 2's page at the original Knot Atlas!

7 2 Quick Notes


7 2 Further Notes and Views

Knot presentations

Planar diagram presentation X1425 X3,10,4,11 X5,14,6,1 X7,12,8,13 X11,8,12,9 X13,6,14,7 X9,2,10,3
Gauss code -1, 7, -2, 1, -3, 6, -4, 5, -7, 2, -5, 4, -6, 3
Dowker-Thistlethwaite code 4 10 14 12 2 8 6
Conway Notation [52]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 1
Bridge index 2
Super bridge index [math]\displaystyle{ \{3,4\} }[/math]
Nakanishi index 1
Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math]
Hyperbolic Volume 3.33174
A-Polynomial See Data:7 2/A-polynomial

[edit Notes for 7 2's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,2)) }[/math]
Rasmussen s-Invariant -2

[edit Notes for 7 2's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ 3 t+3 t^{-1} -5 }[/math]
Conway polynomial [math]\displaystyle{ 3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{1\} }[/math]
Determinant and Signature { 11, -2 }
Jones polynomial [math]\displaystyle{ - q^{-8} + q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ -a^8+a^6 z^2+a^6+a^4 z^2+a^2 z^2+a^2 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ a^9 z^5-4 a^9 z^3+3 a^9 z+a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^5-6 a^7 z^3+3 a^7 z+a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+a^5 z^5-a^5 z^3+a^4 z^4+a^3 z^3+a^2 z^2-a^2 }[/math]
The A2 invariant [math]\displaystyle{ -q^{26}-q^{24}+q^{18}+q^{16}+q^8+q^6+q^2 }[/math]
The G2 invariant [math]\displaystyle{ q^{128}+q^{124}-q^{122}-q^{116}+2 q^{114}-2 q^{112}-q^{110}-q^{106}-2 q^{102}-2 q^{100}-q^{92}-q^{90}+2 q^{88}+q^{78}+3 q^{74}+q^{70}+q^{68}-q^{66}+2 q^{64}-q^{60}+q^{54}-q^{50}-q^{40}+q^{38}+q^{36}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 7_2.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^{17}+q^{11}+q^5+q }[/math]
2 [math]\displaystyle{ q^{48}-q^{44}-q^{38}+q^{34}-q^{26}-q^{24}+2 q^{14}+q^{12}+q^8+q^2 }[/math]
3 [math]\displaystyle{ -q^{93}+q^{89}+q^{87}-q^{83}+q^{79}-q^{75}-q^{73}+q^{69}+q^{61}+q^{59}-q^{55}-q^{49}-q^{47}-q^{41}-q^{39}+q^{33}-q^{29}+q^{25}+q^{23}+q^{17}+q^{15}+q^{13}+q^{11}+q^3 }[/math]
4 [math]\displaystyle{ q^{152}-q^{148}-q^{146}-q^{144}+q^{142}+q^{140}+q^{138}-2 q^{134}+q^{130}+q^{128}+q^{126}-q^{124}-q^{122}-q^{120}+q^{116}-q^{110}-q^{108}+q^{104}+2 q^{102}-q^{98}+q^{94}+2 q^{92}-2 q^{88}-q^{86}+q^{82}-q^{78}+q^{74}+q^{72}-q^{68}+q^{64}-q^{60}-q^{58}-2 q^{56}-2 q^{54}-q^{52}+q^{50}+q^{48}-q^{46}-q^{44}-2 q^{42}+q^{40}+3 q^{38}+2 q^{36}-2 q^{32}+2 q^{28}+q^{26}+q^{24}-q^{22}+q^{18}+q^{16}+2 q^{14}+q^4 }[/math]
5 [math]\displaystyle{ -q^{225}+q^{221}+q^{219}+q^{217}-q^{213}-2 q^{211}-q^{209}+q^{205}+2 q^{203}+q^{201}-q^{199}-2 q^{197}-q^{195}+q^{191}+2 q^{189}+q^{187}-q^{183}-q^{181}-q^{179}+q^{175}+q^{173}+q^{171}-q^{167}-2 q^{165}-2 q^{163}+2 q^{159}+2 q^{157}-2 q^{153}-2 q^{151}-q^{149}+2 q^{147}+4 q^{145}+2 q^{143}-q^{141}-2 q^{139}-2 q^{137}+2 q^{133}+q^{131}-q^{129}-3 q^{127}-2 q^{125}+2 q^{121}+2 q^{119}+q^{117}-q^{115}-q^{113}+q^{111}+2 q^{109}+q^{107}-q^{103}+2 q^{99}+q^{97}-q^{93}-q^{91}-q^{89}-3 q^{77}-3 q^{75}-2 q^{73}+q^{71}+2 q^{69}+3 q^{67}+q^{65}-3 q^{63}-5 q^{61}-3 q^{59}+2 q^{57}+4 q^{55}+3 q^{53}-q^{51}-4 q^{49}-3 q^{47}+q^{45}+4 q^{43}+3 q^{41}-q^{37}-q^{35}+q^{33}+2 q^{31}+2 q^{29}-q^{25}+q^{19}+2 q^{17}+q^{15}+q^5 }[/math]
6 [math]\displaystyle{ q^{312}-q^{308}-q^{306}-q^{304}+2 q^{298}+2 q^{296}+q^{294}-q^{290}-2 q^{288}-3 q^{286}+q^{282}+2 q^{280}+2 q^{278}+q^{276}-3 q^{272}-2 q^{270}-q^{268}+q^{264}+2 q^{262}+2 q^{260}-q^{254}-q^{252}-2 q^{250}-q^{248}+q^{246}+q^{244}+2 q^{242}+2 q^{240}+2 q^{238}-q^{236}-3 q^{234}-3 q^{232}-2 q^{230}+q^{228}+3 q^{226}+4 q^{224}+q^{222}-2 q^{220}-4 q^{218}-5 q^{216}-2 q^{214}+2 q^{212}+5 q^{210}+4 q^{208}+2 q^{206}-q^{204}-4 q^{202}-3 q^{200}+3 q^{196}+4 q^{194}+3 q^{192}-4 q^{188}-5 q^{186}-3 q^{184}+2 q^{180}+3 q^{178}+2 q^{176}-2 q^{174}-4 q^{172}-2 q^{170}+q^{168}+3 q^{166}+3 q^{164}+2 q^{162}-q^{160}-4 q^{158}-2 q^{156}+q^{154}+2 q^{152}+2 q^{150}+q^{148}-q^{146}-2 q^{144}+2 q^{140}+q^{138}-q^{134}-q^{132}-q^{130}+q^{128}+2 q^{126}+2 q^{124}+q^{122}-q^{114}-q^{112}+q^{110}+2 q^{108}+2 q^{106}+q^{104}-q^{102}-4 q^{100}-6 q^{98}-4 q^{96}-q^{94}+2 q^{92}+6 q^{90}+4 q^{88}-q^{86}-6 q^{84}-7 q^{82}-5 q^{80}+6 q^{76}+7 q^{74}+3 q^{72}-3 q^{70}-5 q^{68}-4 q^{66}-q^{64}+4 q^{62}+4 q^{60}+q^{58}-2 q^{56}-2 q^{54}-q^{52}+3 q^{48}+2 q^{46}-q^{42}+q^{38}+q^{36}+3 q^{34}+q^{32}-q^{28}+2 q^{20}+q^{18}+q^{16}+q^6 }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^{26}-q^{24}+q^{18}+q^{16}+q^8+q^6+q^2 }[/math]
1,1 [math]\displaystyle{ q^{68}+2 q^{64}-2 q^{62}+2 q^{60}-4 q^{58}-2 q^{54}+2 q^{50}+4 q^{46}-3 q^{44}+2 q^{42}-4 q^{40}+2 q^{38}-4 q^{36}-2 q^{32}-2 q^{30}+2 q^{24}+2 q^{22}+3 q^{20}+2 q^{18}+2 q^{16}+2 q^{12}+2 q^8+q^4 }[/math]
2,0 [math]\displaystyle{ 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^{38}-2 q^{36}-2 q^{34}-q^{32}+q^{26}+q^{24}+q^{22}+3 q^{20}+2 q^{18}+q^{16}+q^{12}+q^{10}+q^4 }[/math]
3,0 [math]\displaystyle{ -q^{120}-q^{118}-q^{116}+2 q^{112}+2 q^{110}+2 q^{108}-2 q^{96}-2 q^{94}-2 q^{92}+q^{88}+q^{86}+q^{84}+q^{82}+2 q^{80}+3 q^{78}+2 q^{76}-q^{72}-2 q^{70}-q^{68}-3 q^{66}-3 q^{64}-3 q^{62}-q^{60}-q^{56}-q^{54}-q^{52}-q^{48}-q^{40}-q^{38}+2 q^{36}+3 q^{34}+3 q^{32}+2 q^{30}+2 q^{26}+3 q^{24}+3 q^{22}+q^{20}+q^{16}+q^{14}+q^6 }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{54}+q^{50}-q^{46}-q^{44}-2 q^{42}-2 q^{40}-q^{38}-q^{34}+q^{32}+q^{30}+q^{28}+q^{26}+q^{24}+q^{22}+q^{16}+q^{12}+2 q^{10}+q^8+q^4 }[/math]
1,0,0 [math]\displaystyle{ -q^{35}-q^{33}-q^{31}+q^{25}+q^{23}+q^{21}+q^{11}+q^9+q^7+q^3 }[/math]

A4 Invariants.

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

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{54}-q^{50}-q^{46}+q^{44}+q^{38}+q^{34}-q^{32}+q^{30}-q^{28}+q^{26}-q^{24}+q^{22}+q^{16}+q^{12}+q^8+q^4 }[/math]
1,0 [math]\displaystyle{ q^{88}+q^{80}-q^{76}-q^{74}-q^{68}-q^{66}-q^{64}-q^{56}+q^{44}+q^{42}+q^{36}+q^{34}+q^{26}+q^{18}+q^{16}+q^{14}+q^6 }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{74}+q^{70}+q^{66}-q^{64}-q^{62}-2 q^{60}-2 q^{58}-2 q^{56}-2 q^{54}-q^{52}-q^{50}+q^{48}+2 q^{44}+q^{42}+2 q^{40}+2 q^{36}+q^{32}+q^{22}+q^{18}+q^{16}+2 q^{14}+q^{12}+q^{10}+q^6 }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{128}+q^{124}-q^{122}-q^{116}+2 q^{114}-2 q^{112}-q^{110}-q^{106}-2 q^{102}-2 q^{100}-q^{92}-q^{90}+2 q^{88}+q^{78}+3 q^{74}+q^{70}+q^{68}-q^{66}+2 q^{64}-q^{60}+q^{54}-q^{50}-q^{40}+q^{38}+q^{36}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10} }[/math]

.

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 2"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ 3 t+3 t^{-1} -5 }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ 3 z^2+1 }[/math]
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]= [math]\displaystyle{ \{1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 11, -2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ - q^{-8} + q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +2 q^{-3} - q^{-2} + q^{-1} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ -a^8+a^6 z^2+a^6+a^4 z^2+a^2 z^2+a^2 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ a^9 z^5-4 a^9 z^3+3 a^9 z+a^8 z^6-4 a^8 z^4+4 a^8 z^2-a^8+2 a^7 z^5-6 a^7 z^3+3 a^7 z+a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+a^5 z^5-a^5 z^3+a^4 z^4+a^3 z^3+a^2 z^2-a^2 }[/math]

Vassiliev invariants

V2 and V3: (3, -6)
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
[math]\displaystyle{ 12 }[/math] [math]\displaystyle{ -48 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 222 }[/math] [math]\displaystyle{ 34 }[/math] [math]\displaystyle{ -576 }[/math] [math]\displaystyle{ -1152 }[/math] [math]\displaystyle{ -192 }[/math] [math]\displaystyle{ -176 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 1152 }[/math] [math]\displaystyle{ 2664 }[/math] [math]\displaystyle{ 408 }[/math] [math]\displaystyle{ \frac{60431}{10} }[/math] [math]\displaystyle{ -\frac{826}{15} }[/math] [math]\displaystyle{ \frac{38942}{15} }[/math] [math]\displaystyle{ \frac{497}{6} }[/math] [math]\displaystyle{ \frac{3471}{10} }[/math]

V2,1 through V6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]-2 is the signature of 7 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
 -7-6-5-4-3-2-10χ
-1       11
-3      110
-5     1  1
-7    11  0
-9   11   0
-11   1    1
-13 11     0
-15        0
-171       -1
Integral Khovanov Homology

(db, data source)

  
[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math]
[math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]
[math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[7, 2]]
Out[2]=  
7
In[3]:=
PD[Knot[7, 2]]
Out[3]=  
PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[5, 14, 6, 1], X[7, 12, 8, 13], 
  X[11, 8, 12, 9], X[13, 6, 14, 7], X[9, 2, 10, 3]]
In[4]:=
GaussCode[Knot[7, 2]]
Out[4]=  
GaussCode[-1, 7, -2, 1, -3, 6, -4, 5, -7, 2, -5, 4, -6, 3]
In[5]:=
BR[Knot[7, 2]]
Out[5]=  
BR[4, {-1, -1, -1, -2, 1, -2, -3, 2, -3}]
In[6]:=
alex = Alexander[Knot[7, 2]][t]
Out[6]=  
     3

-5 + - + 3 t


     t
In[7]:=
Conway[Knot[7, 2]][z]
Out[7]=  
       2
1 + 3 z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[7, 2]}
In[9]:=
{KnotDet[Knot[7, 2]], KnotSignature[Knot[7, 2]]}
Out[9]=  
{11, -2}
In[10]:=
J=Jones[Knot[7, 2]][q]
Out[10]=  
  -8    -7    -6   2    2    2     -2   1

-q   + q   - q   + -- - -- + -- - q   + -



                   5    4    3         q


                   q    q    q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[7, 2], Knot[11, NonAlternating, 88]}
In[12]:=
A2Invariant[Knot[7, 2]][q]
Out[12]=  
  -26    -24    -18    -16    -8    -6    -2
-q    - q    + q    + q    + q   + q   + q
In[13]:=
Kauffman[Knot[7, 2]][a, z]
Out[13]=  
  2    6    8      7        9      2  2      6  2      8  2    3  3

-a  - a  - a  + 3 a  z + 3 a  z + a  z  + 3 a  z  + 4 a  z  + a  z  - 



  5  3      7  3      9  3    4  4      6  4      8  4    5  5
 a  z  - 6 a  z  - 4 a  z  + a  z  - 3 a  z  - 4 a  z  + a  z  + 

    7  5    9  5    6  6    8  6


  2 a  z  + a  z  + a  z  + a  z
In[14]:=
{Vassiliev[2][Knot[7, 2]], Vassiliev[3][Knot[7, 2]]}
Out[14]=  
{0, -6}
In[15]:=
Kh[Knot[7, 2]][q, t]
Out[15]=  
 -3   1     1        1        1        1        1       1       1

q   + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + 



     q    17  7    13  6    13  5    11  4    9  4    9  3    7  3
         q   t    q   t    q   t    q   t    q  t    q  t    q  t

   1       1      1
 ----- + ----- + ----
  7  2    5  2    3


  q  t    q  t    q  t
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