© | Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table:

1027 1029

KnotPlot

This page is passe. Go here instead!    The Alternating Knot 1028    Visit 1028's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 1028's page at Knotilus! Acknowledgement

KnotPlot

PD Presentation:

X1425 X3,10,4,11 X13,19,14,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X11,1,12,20 X19,13,20,12 X9,2,10,3

Gauss Code:

{-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, 8}

DT (Dowker-Thistlethwaite) Code:

4 10 14 16 2 20 18 8 6 12

Minimum Braid Representative: Length is 12, width is 5 Braid index is 5

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index Reversible 2 2 2 / NotAvailable 1

Alexander Polynomial:

4t-2 - 13t-1 + 19 - 13t + 4t2

Conway Polynomial:

1 + 3z2 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{1037, ...}

Determinant and Signature:

{53, 0}

Jones Polynomial:

- q-3 + 3q-2 - 5q-1 + 7 - 8q + 9q2 - 7q3 + 6q4 - 4q5 + 2q6 - q7

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-10 + q-8 + q-6 - 2q-4 + q-2 - 1 + 2q4 + q6 + 3q8 + q14 - 2q16 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + a-4z2 + a-4z4 + 3a-2 + 4a-2z2 + 2a-2z4 - 1 + z4 - a2z2

Kauffman Polynomial:

- 4a-7z + 8a-7z3 - 5a-7z5 + a-7z7 + a-6 - 5a-6z2 + 12a-6z4 - 9a-6z6 + 2a-6z8 - 6a-5z + 18a-5z3 - 12a-5z5 + a-5z9 + 11a-4z4 - 16a-4z6 + 5a-4z8 - 2a-3z + 13a-3z3 - 18a-3z5 + 4a-3z7 + a-3z9 - 3a-2 + 10a-2z2 - 12a-2z4 - a-2z6 + 3a-2z8 + a-1z - 2a-1z3 - 6a-1z5 + 5a-1z7 - 1 + 4z2 - 8z4 + 6z6 + az - 4az3 + 5az5 - a2z2 + 3a2z4 + a3z3

V2 and V3, the type 2 and 3 Vassiliev invariants:

{3, 4}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 1028. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                     1 j = 13                   1   j = 11                 3 1   j = 9               3 1     j = 7             4 3       j = 5           5 3         j = 3         3 4           j = 1       4 5             j = -1     2 4               j = -3   1 3                 j = -5   2                   j = -7 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-9 - 3q-8 + 2q-7 + 5q-6 - 13q-5 + 9q-4 + 10q-3 - 28q-2 + 21q-1 + 15 - 44q + 29q2 + 23q3 - 54q4 + 26q5 + 32q6 - 53q7 + 16q8 + 34q9 - 41q10 + 4q11 + 29q12 - 24q13 - 4q14 + 18q15 - 9q16 - 5q17 + 7q18 - q19 - 2q20 + q21
3	- q-18 + 3q-17 - 2q-16 - 2q-15 + q-14 + 6q-13 - 4q-12 - 9q-11 + 9q-10 + 8q-9 - 13q-8 - 9q-7 + 24q-6 + 3q-5 - 34q-4 + 2q-3 + 50q-2 - 10q-1 - 65 + 11q + 87q2 - 14q3 - 96q4 + 112q6 + 9q7 - 107q8 - 35q9 + 112q10 + 44q11 - 92q12 - 71q13 + 87q14 + 75q15 - 63q16 - 91q17 + 51q18 + 90q19 - 28q20 - 90q21 + 9q22 + 83q23 + 8q24 - 70q25 - 23q26 + 55q27 + 29q28 - 34q29 - 34q30 + 21q31 + 26q32 - 5q33 - 21q34 + 11q36 + 4q37 - 6q38 - 2q39 + q40 + 2q41 - q42
4	q-30 - 3q-29 + 2q-28 + 2q-27 - 4q-26 + 6q-25 - 11q-24 + 9q-23 + 4q-22 - 14q-21 + 23q-20 - 30q-19 + 16q-18 + 4q-17 - 26q-16 + 64q-15 - 56q-14 + 7q-13 - 14q-12 - 29q-11 + 151q-10 - 79q-9 - 42q-8 - 73q-7 - 22q-6 + 302q-5 - 68q-4 - 127q-3 - 195q-2 - 38q-1 + 492 + 4q - 182q2 - 349q3 - 126q4 + 627q5 + 116q6 - 139q7 - 446q8 - 261q9 + 630q10 + 182q11 - 12q12 - 418q13 - 371q14 + 518q15 + 167q16 + 121q17 - 306q18 - 415q19 + 367q20 + 103q21 + 220q22 - 172q23 - 409q24 + 213q25 + 30q26 + 284q27 - 39q28 - 360q29 + 69q30 - 59q31 + 299q32 + 88q33 - 256q34 - 31q35 - 159q36 + 234q37 + 169q38 - 105q39 - 45q40 - 225q41 + 104q42 + 160q43 + 23q44 + 16q45 - 200q46 - 12q47 + 74q48 + 62q49 + 75q50 - 107q51 - 47q52 - 4q53 + 28q54 + 72q55 - 26q56 - 22q57 - 23q58 - 5q59 + 33q60 + q61 - 9q63 - 8q64 + 7q65 + q66 + 2q67 - q68 - 2q69 + q70

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 28]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[13, 19, 14, 18], X[5, 15, 6, 14], > X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[11, 1, 12, 20], > X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[3]:=	GaussCode[Knot[10, 28]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -8, 9, -3, 4, -7, 6, -5, 3, -9, > 8]
In[4]:=	DTCode[Knot[10, 28]]
Out[4]=	DTCode[4, 10, 14, 16, 2, 20, 18, 8, 6, 12]
In[5]:=	br = BR[Knot[10, 28]]
Out[5]=	BR[5, {1, 1, 2, -1, 2, 2, 3, -2, -4, 3, -4, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 28]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 28]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 28]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 2, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 28]][t]
Out[10]=	4 13 2 19 + -- - -- - 13 t + 4 t 2 t t
In[11]:=	Conway[Knot[10, 28]][z]
Out[11]=	2 4 1 + 3 z + 4 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 28], Knot[10, 37]}
In[13]:=	{KnotDet[Knot[10, 28]], KnotSignature[Knot[10, 28]]}
Out[13]=	{53, 0}
In[14]:=	Jones[Knot[10, 28]][q]
Out[14]=	-3 3 5 2 3 4 5 6 7 7 - q + -- - - - 8 q + 9 q - 7 q + 6 q - 4 q + 2 q - q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 28]}
In[16]:=	A2Invariant[Knot[10, 28]][q]
Out[16]=	-10 -8 -6 2 -2 4 6 8 14 16 22 -1 - q + q + q - -- + q + 2 q + q + 3 q + q - 2 q - q 4 q
In[17]:=	HOMFLYPT[Knot[10, 28]][a, z]
Out[17]=	2 2 2 4 4 -6 3 z z 4 z 2 2 4 z 2 z -1 - a + -- - -- + -- + ---- - a z + z + -- + ---- 2 6 4 2 4 2 a a a a a a
In[18]:=	Kauffman[Knot[10, 28]][a, z]
Out[18]=	2 2 -6 3 4 z 6 z 2 z z 2 5 z 10 z 2 2 -1 + a - -- - --- - --- - --- + - + a z + 4 z - ---- + ----- - a z + 2 7 5 3 a 6 2 a a a a a a 3 3 3 3 4 4 8 z 18 z 13 z 2 z 3 3 3 4 12 z 11 z > ---- + ----- + ----- - ---- - 4 a z + a z - 8 z + ----- + ----- - 7 5 3 a 6 4 a a a a a 4 5 5 5 5 6 12 z 2 4 5 z 12 z 18 z 6 z 5 6 9 z > ----- + 3 a z - ---- - ----- - ----- - ---- + 5 a z + 6 z - ---- - 2 7 5 3 a 6 a a a a a 6 6 7 7 7 8 8 8 9 9 16 z z z 4 z 5 z 2 z 5 z 3 z z z > ----- - -- + -- + ---- + ---- + ---- + ---- + ---- + -- + -- 4 2 7 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 28]], Vassiliev[3][Knot[10, 28]]}
Out[19]=	{3, 4}
In[20]:=	Kh[Knot[10, 28]][q, t]
Out[20]=	4 1 2 1 3 2 3 3 2 - + 4 q + ----- + ----- + ----- + ---- + --- + 5 q t + 3 q t + 4 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 > 5 q t + 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + 11 6 13 6 15 7 > q t + q t + q t
In[21]:=	ColouredJones[Knot[10, 28], 2][q]
Out[21]=	-9 3 2 5 13 9 10 28 21 2 3 15 + q - -- + -- + -- - -- + -- + -- - -- + -- - 44 q + 29 q + 23 q - 8 7 6 5 4 3 2 q q q q q q q q 4 5 6 7 8 9 10 11 12 > 54 q + 26 q + 32 q - 53 q + 16 q + 34 q - 41 q + 4 q + 29 q - 13 14 15 16 17 18 19 20 21 > 24 q - 4 q + 18 q - 9 q - 5 q + 7 q - q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1028

1027 1029