10 58

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10_57

10_59

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 10 58's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10_58's page at Knotilus!

Visit 10 58's page at the original Knot Atlas!

[edit 10 58 Quick Notes]

[edit 10 58 Further Notes and Views]

[edit] Knot presentations

Planar diagram presentation	X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,14,6,15 X_11,19,12,18 X_15,20,16,1 X_19,16,20,17 X_17,13,18,12 X_13,6,14,7
Gauss code	-1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -10, 5, -7, 8, -9, 6, -8, 7
Dowker-Thistlethwaite code	4 8 14 10 2 18 6 20 12 16
Conway Notation	[22,22,2]

Minimum Braid Representative

A Morse Link Presentation

An Arc Presentation

Length is 11, width is 6,

Braid index is 6

[{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 2}, {8, 6}, {1, 3}, {4, 7}, {6, 12}, {2, 5}, {11, 4}, {5, 1}]

[edit Notes on presentations of 10 58]

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["10 58"];

In[4]:= PD[K]

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

Out[4]= X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_5,14,6,15 X_11,19,12,18 X_15,20,16,1 X_19,16,20,17 X_17,13,18,12 X_13,6,14,7

In[5]:= GaussCode[K]

Out[5]= -1, 4, -3, 1, -5, 10, -2, 3, -4, 2, -6, 9, -10, 5, -7, 8, -9, 6, -8, 7

In[6]:= DTCode[K]

Out[6]= 4 8 14 10 2 18 6 20 12 16

(The path below may be different on your system)

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

In[8]:= ConwayNotation[K]

Out[8]= [22,22,2]

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]= $BR(6,{1,-2,1,3,-2,-4,-3,-3,5,-4,5})$

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]= { 6, 11, 6 }

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[{12, 9}, {10, 8}, {9, 11}, {3, 10}, {7, 2}, {8, 6}, {1, 3}, {4, 7}, {6, 12}, {2, 5}, {11, 4}, {5, 1}]

In[14]:= Draw[ap]

Out[14]= -Graphics-

[edit] Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	2
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[-7][-5]
Hyperbolic Volume	12.7213
A-Polynomial	See Data:10 58/A-polynomial

[edit Notes for 10 58's three dimensional invariants]

[edit] Four dimensional invariants

Smooth 4 genus	$1$
Topological 4 genus	$1$
Concordance genus	$2$
Rasmussen s-Invariant	0

[edit Notes for 10 58's four dimensional invariants]

[edit] Polynomial invariants

Alexander polynomial	$3 t 2 -16 t + 27-16 t -1 + 3 t -2$
Conway polynomial	$3 z 4 -4 z 2 + 1$
2nd Alexander ideal (db, data sources)	${1}$
Determinant and Signature	{ 65, 0 }
Jones polynomial	$q 4 -2 q 3 + 5 q 2 -8 q + 10-11 q -1 + 10 q -2 -8 q -3 + 6 q -4 -3 q -5 + q -6$
HOMFLY-PT polynomial (db, data sources)	$a 6 -3 z 2 a 4 -2 a 4 + 2 z 4 a 2 + 3 z 2 a 2 + 3 a 2 + z 4 -2 z 2 -2-2 z 2 a -2 + a -4$
Kauffman polynomial (db, data sources)	$a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 3 a 5 z 7 + 6 a 3 z 7 + 7 a z 7 + 4 z 7 a -1 + a 6 z 6 -5 a 4 z 6 -10 a 2 z 6 + 3 z 6 a -2 - z 6 -9 a 5 z 5 -23 a 3 z 5 -22 a z 5 -6 z 5 a -1 + 2 z 5 a -3 -3 a 6 z 4 -5 a 4 z 4 -4 a 2 z 4 -2 z 4 a -2 + z 4 a -4 -5 z 4 + 7 a 5 z 3 + 18 a 3 z 3 + 21 a z 3 + 8 z 3 a -1 -2 z 3 a -3 + 3 a 6 z 2 + 7 a 4 z 2 + 10 a 2 z 2 -2 z 2 a -4 + 8 z 2 -2 a 5 z -4 a 3 z -6 a z -4 z a -1 - a 6 -2 a 4 -3 a 2 + a -4 -2$
The A2 invariant	$q 20 + q 18 -2 q 16 + q 14 -2 q 10 + 3 q 8 + q 4 -2 + q -2 -3 q -4 + q -6 + 2 q -8 - q -10 + q -12 + q -14$
The G2 invariant	Data:10 58/QuantumInvariant/G2/1,0

Further Quantum Invariants

Further quantum knot invariants for 10_58.

A1 Invariants.

Weight	Invariant
1	$q 13 -2 q 11 + 3 q 9 -2 q 7 + 2 q 5 - q 3 - q + 2 q -1 -3 q -3 + 3 q -5 - q -7 + q -9$
2	$q 38 -2 q 36 -2 q 34 + 8 q 32 -2 q 30 -12 q 28 + 13 q 26 + 6 q 24 -20 q 22 + 7 q 20 + 15 q 18 -17 q 16 -2 q 14 + 16 q 12 -7 q 10 -9 q 8 + 8 q 6 + 8 q 4 -11 q 2 -5 + 20 q -2 -7 q -4 -16 q -6 + 18 q -8 -13 q -12 + 9 q -14 + q -16 -5 q -18 + 3 q -20 - q -24 + q -26$
3	$q 75 -2 q 73 -2 q 71 + 3 q 69 + 8 q 67 -2 q 65 -19 q 63 -4 q 61 + 28 q 59 + 21 q 57 -31 q 55 -46 q 53 + 25 q 51 + 67 q 49 -82 q 45 -34 q 43 + 82 q 41 + 67 q 39 -66 q 37 -93 q 35 + 38 q 33 + 108 q 31 -10 q 29 -109 q 27 -12 q 25 + 100 q 23 + 36 q 21 -87 q 19 -50 q 17 + 65 q 15 + 64 q 13 -42 q 11 -75 q 9 + 8 q 7 + 81 q 5 + 32 q 3 -79 q -66 q -1 + 64 q -3 + 96 q -5 -38 q -7 -109 q -9 + 9 q -11 + 104 q -13 + 12 q -15 -79 q -17 -28 q -19 + 55 q -21 + 27 q -23 -31 q -25 -18 q -27 + 14 q -29 + 11 q -31 -7 q -33 -4 q -35 + 3 q -37 + q -39 -2 q -41 + q -43 - q -49 + q -51$

A2 Invariants.

Weight	Invariant
1,0	$q 20 + q 18 -2 q 16 + q 14 -2 q 10 + 3 q 8 + q 4 -2 + q -2 -3 q -4 + q -6 + 2 q -8 - q -10 + q -12 + q -14$

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["10 58"];

In[4]:= Alexander[K][t]

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

Out[4]= $3 t 2 -16 t + 27-16 t -1 + 3 t -2$

In[5]:= Conway[K][z]

Out[5]= $3 z 4 -4 z 2 + 1$

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]= { 65, 0 }

In[8]:= Jones[K][q]

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

Out[8]= $q 4 -2 q 3 + 5 q 2 -8 q + 10-11 q -1 + 10 q -2 -8 q -3 + 6 q -4 -3 q -5 + q -6$

In[9]:= HOMFLYPT[K][a, z]

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

Out[9]= $a 6 -3 z 2 a 4 -2 a 4 + 2 z 4 a 2 + 3 z 2 a 2 + 3 a 2 + z 4 -2 z 2 -2-2 z 2 a -2 + a -4$

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

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

Out[10]= $a 3 z 9 + a z 9 + 3 a 4 z 8 + 6 a 2 z 8 + 3 z 8 + 3 a 5 z 7 + 6 a 3 z 7 + 7 a z 7 + 4 z 7 a -1 + a 6 z 6 -5 a 4 z 6 -10 a 2 z 6 + 3 z 6 a -2 - z 6 -9 a 5 z 5 -23 a 3 z 5 -22 a z 5 -6 z 5 a -1 + 2 z 5 a -3 -3 a 6 z 4 -5 a 4 z 4 -4 a 2 z 4 -2 z 4 a -2 + z 4 a -4 -5 z 4 + 7 a 5 z 3 + 18 a 3 z 3 + 21 a z 3 + 8 z 3 a -1 -2 z 3 a -3 + 3 a 6 z 2 + 7 a 4 z 2 + 10 a 2 z 2 -2 z 2 a -4 + 8 z 2 -2 a 5 z -4 a 3 z -6 a z -4 z a -1 - a 6 -2 a 4 -3 a 2 + a -4 -2$

[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["10 58"];

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]= { $3 t 2 -16 t + 27-16 t -1 + 3 t -2$ , $q 4 -2 q 3 + 5 q 2 -8 q + 10-11 q -1 + 10 q -2 -8 q -3 + 6 q -4 -3 q -5 + q -6$ }

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:

(-4, 1)

[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 =

0 is the signature of 10 58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-6

-5

-4

-3

-2

-1

-3

-1

-3

-1

-5

-7

-2

-9

-11

-2

-13

Integral Khovanov Homology

(db, data source)

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

[edit] The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the

n + 1

-dimensional representation of

s l (2)

)

$n$	$J n$
2	$q 12 -2 q 11 + q 10 + 4 q 9 -10 q 8 + 7 q 7 + 12 q 6 -32 q 5 + 20 q 4 + 30 q 3 -66 q 2 + 29 q + 57-91 q -1 + 23 q -2 + 76 q -3 -91 q -4 + 6 q -5 + 78 q -6 -68 q -7 -12 q -8 + 63 q -9 -36 q -10 -20 q -11 + 36 q -12 -10 q -13 -13 q -14 + 11 q -15 -3 q -17 + q -18$
3	$q 24 -2 q 23 + q 22 + 2 q 20 -5 q 19 + 4 q 18 + 2 q 17 -5 q 16 -8 q 15 + 22 q 14 + 5 q 13 -37 q 12 -21 q 11 + 80 q 10 + 33 q 9 -120 q 8 -72 q 7 + 171 q 6 + 125 q 5 -215 q 4 -190 q 3 + 242 q 2 + 259 q -247-320 q -1 + 229 q -2 + 370 q -3 -198 q -4 -393 q -5 + 146 q -6 + 403 q -7 -92 q -8 -392 q -9 + 31 q -10 + 366 q -11 + 31 q -12 -328 q -13 -81 q -14 + 269 q -15 + 130 q -16 -210 q -17 -151 q -18 + 138 q -19 + 157 q -20 -77 q -21 -136 q -22 + 22 q -23 + 109 q -24 + 5 q -25 -69 q -26 -20 q -27 + 38 q -28 + 20 q -29 -17 q -30 -13 q -31 + 6 q -32 + 5 q -33 -3 q -35 + q -36$
4	$q 40 -2 q 39 + q 38 -2 q 36 + 7 q 35 -8 q 34 + 4 q 33 - q 32 -11 q 31 + 25 q 30 -15 q 29 + 11 q 28 -15 q 27 -45 q 26 + 73 q 25 + 10 q 24 + 36 q 23 -85 q 22 -173 q 21 + 155 q 20 + 152 q 19 + 179 q 18 -229 q 17 -550 q 16 + 148 q 15 + 464 q 14 + 639 q 13 -300 q 12 -1234 q 11 -182 q 10 + 755 q 9 + 1456 q 8 -29 q 7 -1959 q 6 -867 q 5 + 712 q 4 + 2296 q 3 + 600 q 2 -2318 q -1568 + 273 q -1 + 2743 q -2 + 1280 q -3 -2192 q -4 -1939 q -5 -311 q -6 + 2691 q -7 + 1722 q -8 -1742 q -9 -1928 q -10 -840 q -11 + 2279 q -12 + 1905 q -13 -1119 q -14 -1659 q -15 -1274 q -16 + 1629 q -17 + 1873 q -18 -392 q -19 -1172 q -20 -1561 q -21 + 812 q -22 + 1571 q -23 + 276 q -24 -490 q -25 -1520 q -26 + 26 q -27 + 963 q -28 + 608 q -29 + 188 q -30 -1071 q -31 -412 q -32 + 277 q -33 + 482 q -34 + 529 q -35 -456 q -36 -383 q -37 -135 q -38 + 148 q -39 + 446 q -40 -56 q -41 -143 q -42 -175 q -43 -53 q -44 + 198 q -45 + 41 q -46 + 5 q -47 -71 q -48 -62 q -49 + 47 q -50 + 15 q -51 + 20 q -52 -10 q -53 -20 q -54 + 6 q -55 + 5 q -57 -3 q -59 + q -60$
5	$q 60 -2 q 59 + q 58 -2 q 56 + 3 q 55 + 4 q 54 -8 q 53 + q 52 + 3 q 51 -8 q 50 + 10 q 49 + 14 q 48 -21 q 47 -9 q 46 + 3 q 45 -4 q 44 + 36 q 43 + 41 q 42 -47 q 41 -82 q 40 -45 q 39 + 34 q 38 + 184 q 37 + 181 q 36 -96 q 35 -365 q 34 -366 q 33 + 36 q 32 + 668 q 31 + 826 q 30 + 63 q 29 -1048 q 28 -1507 q 27 -542 q 26 + 1488 q 25 + 2637 q 24 + 1327 q 23 -1766 q 22 -4006 q 21 -2804 q 20 + 1707 q 19 + 5681 q 18 + 4795 q 17 -1109 q 16 -7200 q 15 -7358 q 14 -202 q 13 + 8431 q 12 + 10157 q 11 + 2142 q 10 -9044 q 9 -12831 q 8 -4599 q 7 + 8938 q 6 + 15092 q 5 + 7226 q 4 -8168 q 3 -16662 q 2 -9688 q + 6860 + 17481 q -1 + 11752 q -2 -5299 q -3 -17577 q -4 -13272 q -5 + 3673 q -6 + 17126 q -7 + 14227 q -8 -2146 q -9 -16230 q -10 -14734 q -11 + 700 q -12 + 15120 q -13 + 14863 q -14 + 628 q -15 -13722 q -16 -14762 q -17 -1986 q -18 + 12146 q -19 + 14447 q -20 + 3341 q -21 -10288 q -22 -13866 q -23 -4733 q -24 + 8135 q -25 + 12989 q -26 + 6033 q -27 -5764 q -28 -11665 q -29 -7050 q -30 + 3187 q -31 + 9899 q -32 + 7679 q -33 -758 q -34 -7696 q -35 -7632 q -36 -1427 q -37 + 5246 q -38 + 6977 q -39 + 2970 q -40 -2803 q -41 -5663 q -42 -3832 q -43 + 685 q -44 + 4017 q -45 + 3853 q -46 + 893 q -47 -2264 q -48 -3300 q -49 -1753 q -50 + 801 q -51 + 2302 q -52 + 1953 q -53 + 287 q -54 -1310 q -55 -1672 q -56 -786 q -57 + 446 q -58 + 1126 q -59 + 915 q -60 + 86 q -61 -609 q -62 -727 q -63 -322 q -64 + 209 q -65 + 458 q -66 + 340 q -67 + 19 q -68 -233 q -69 -253 q -70 -84 q -71 + 80 q -72 + 132 q -73 + 95 q -74 -6 q -75 -71 q -76 -55 q -77 -4 q -78 + 15 q -79 + 24 q -80 + 20 q -81 -10 q -82 -13 q -83 - q -84 + 5 q -87 -3 q -89 + q -90$
6

[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.