7 2

7_1

7_3

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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 $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number Failed to parse (syntax error): {\displaystyle \text{$\$$Failed}} 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 $1$ Topological 4 genus $1$ Concordance genus ${\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,2))$ Rasmussen s-Invariant -2

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

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 7_2.

A1 Invariants.

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

A2 Invariants.

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

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

.

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]=

$3t+3t^{-1}-5$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{2}+1$

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]=

{ 11, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{8}+a^{6}z^{2}+a^{6}+a^{4}z^{2}+a^{2}z^{2}+a^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

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 $12$ $-48$ $72$ $222$ $34$ $-576$ $-1152$ $-192$ $-176$ $288$ $1152$ $2664$ $408$ ${\frac {60431}{10}}$ $-{\frac {826}{15}}$ ${\frac {38942}{15}}$ ${\frac {497}{6}}$ ${\frac {3471}{10}}$

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 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 χ -1               1 1 -3             1 1 0 -5           1     1 -7         1 1     0 -9       1 1       0 -11       1         1 -13   1 1           0 -15                 0 -17 1               -1

Integral Khovanov Homology (db, data source)

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

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, 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