7 1

From Knot Atlas

Jump to: navigation, search

6_3

7_2

1 Knot presentations
2 Three dimensional invariants
3 Four dimensional invariants
4 Polynomial invariants
5 "Similar" Knots (within the Atlas)
6 Vassiliev invariants
7 Khovanov Homology
8 The Coloured Jones Polynomials
9 Computer Talk
10 Modifying This Page

(KnotPlot image)

See the full Rolfsen Knot Table.

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

Visit [http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,5,-2,6,-3,7,-4,1,-5,2,-6,3,-7,4/goTop.html 7_1's page] at Knotilus!

Visit 7 1's page at the original Knot Atlas!

[edit 7 1 Quick Notes]

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

[edit 7 1 Further Notes and Views]

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.

[edit] Knot presentations

Planar diagram presentation	X₁₈₂₉ 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
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]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 7, width is 2,

Braid index is 2

[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

[edit Notes on presentations of 7 1]

Knot 7_1.

A graph, knot 7_1.

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091.

Read more at http://katlas.org/wiki/KnotTheory.

In[3]:= K = Knot["7 1"];

In[4]:= PD[K]

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

Out[4]= X₁₈₂₉ 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[5]:= GaussCode[K]

Out[5]= -1, 5, -2, 6, -3, 7, -4, 1, -5, 2, -6, 3, -7, 4

In[6]:= DTCode[K]

Out[6]= 8 10 12 14 2 4 6

(The path below may be different on your system)

In[7]:= AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:= ConwayNotation[K]

Out[8]= [7]

In[9]:= br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]= $BR(2,{-1,-1,-1,-1,-1,-1,-1})$

In[10]:= {First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]= { 2, 7, 2 }

In[11]:= Show[BraidPlot[br]]

Out[11]= -Graphics-

In[12]:= Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]= -Graphics-

In[13]:= ap = ArcPresentation[K]

Out[13]= ArcPresentation[{9, 2}, {1, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

[edit] 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 (unknown error\text): \text{$\$$Failed}
Hyperbolic Volume	Not hyperbolic
A-Polynomial	See Data:7 1/A-polynomial

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

[edit] Four dimensional invariants

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

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

[edit] Polynomial invariants

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

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

A3 Invariants.

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

A4 Invariants.

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

B2 Invariants.

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

D4 Invariants.

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

G2 Invariants.

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

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.

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]= $z 6 + 5 z 4 + 6 z 2 + 1$

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

In[10]:= Kauffman[K][a, z]

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

Out[10]= $a 13 z + a 12 z 2 + a 11 z 3 - a 11 z + a 10 z 4 -2 a 10 z 2 + a 9 z 5 -3 a 9 z 3 + a 9 z + a 8 z 6 -5 a 8 z 4 + 7 a 8 z 2 -3 a 8 + a 7 z 5 -4 a 7 z 3 + 3 a 7 z + a 6 z 6 -6 a 6 z 4 + 10 a 6 z 2 -4 a 6$

[edit] "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {}

Computer Talk

(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 May 31, 2006, 14:15:20.091.

In[3]:= K = Knot["7 1"];

In[4]:= {A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[4]= { $t 3 + t -3 - t 2 - t -2 + t + t -1 -1$ , $- q -10 + q -9 - q -8 + q -7 - q -6 + q -5 + q -3$ }

In[5]:= DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]= {}

[edit] Vassiliev invariants

V₂ and V₃:

(6, -14)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

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.

[edit] Khovanov Homology

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

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

Integral Khovanov Homology

(db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -7$	$i = -5$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}$
$r = -2$	${\mathbb Z}_2$	${\mathbb Z}$
$r = -1$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

$n$	$J n$
2	$q -6 + q -9 - q -11 + q -12 - q -14 + q -15 - q -17 + q -18 - q -20 - q -23 + q -24 - q -26 + q -27$
3	$q -9 + q -13 - q -16 + q -17 - q -20 + q -21 - q -24 + q -25 - q -28 + q -29 - q -31 - q -32 + q -33 - q -35 + q -37 - q -39 + q -41 - q -43 + q -45 + q -46 - q -47 + q -50 - q -51$
4	$q -12 + q -17 - q -21 + q -22 - q -26 + q -27 - q -31 + q -32 - q -36 + q -37 -2 q -41 + q -42 -2 q -46 + q -47 + q -48 -2 q -51 + q -52 + q -53 -2 q -56 + q -57 + q -58 -2 q -61 + q -62 + 2 q -63 -2 q -66 + q -67 + q -68 -2 q -71 + q -72 + q -73 -2 q -76 + q -77 - q -81 + q -82$
5	$q -15 + q -21 - q -26 + q -27 - q -32 + q -33 - q -38 + q -39 - q -44 + q -45 - q -50 - q -56 + q -60 - q -62 + q -66 - q -68 + q -72 - q -74 + q -78 + q -84 - q -87 + q -90 - q -93 + q -96 - q -99 - q -105 + q -107 - q -111 + q -113 + q -119 - q -120$
6	$q -18 + q -25 - q -31 + q -32 - q -38 + q -39 - q -45 + q -46 - q -52 + q -53 - q -59 + q -60 - q -61 - q -66 + q -67 - q -68 + q -72 - q -73 + q -74 - q -75 + q -79 - q -80 + q -81 - q -82 + q -86 - q -87 + q -88 - q -89 + q -93 - q -94 + q -95 - q -96 + q -97 + q -100 - q -101 + q -102 - q -103 + q -104 - q -106 + q -107 - q -108 + q -109 - q -110 + q -111 - q -113 + q -114 - q -115 + q -116 - q -117 + q -118 - q -120 + q -121 - q -122 + q -123 - q -124 + q -125 - q -126 - q -127 + q -128 - q -129 + q -130 - q -131 + q -132 - q -134 + q -135 - q -136 + q -137 - q -138 + q -139 - q -141 + q -142 - q -143 + q -144 - q -145 + q -146 + q -149 - q -150 + q -151 - q -152 + q -156 - q -157 + q -158 - q -159 - q -164 + q -165$
7	$q -21 + q -29 - q -36 + q -37 - q -44 + q -45 - q -52 + q -53 - q -60 + q -61 - q -68 + q -69 - q -71 - q -76 + q -77 - q -79 + q -85 - q -87 + q -93 - q -95 + q -101 - q -103 + q -109 - q -111 + q -114 + q -117 - q -119 + q -122 - q -127 + q -130 - q -135 + q -138 - q -143 + q -146 - q -150 - q -151 + q -154 - q -158 + q -162 - q -166 + q -170 - q -174 + q -178 + q -179 - q -182 + q -187 - q -190 + q -195 - q -198 - q -201 + q -203 - q -209 + q -211 + q -216 - q -217$

[edit] Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.