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

1021 1023

KnotPlot

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KnotPlot

PD Presentation:

X6271 X16,12,17,11 X12,3,13,4 X2,15,3,16 X14,5,15,6 X18,8,19,7 X20,10,1,9 X8,20,9,19 X4,13,5,14 X10,18,11,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 14 18 20 16 4 2 10 8

Minimum Braid Representative: Length is 11, width is 4 Braid index is 4

A Morse Link Presentation:

3D Invariants:

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

Alexander Polynomial:

- 2t-3 + 6t-2 - 10t-1 + 13 - 10t + 6t2 - 2t3

Conway Polynomial:

1 - 4z2 - 6z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{49, 0}

Jones Polynomial:

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

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

{1035, ...}

A2 (sl(3)) Invariant:

q-12 + q-8 + q-6 - q-4 + 2q-2 - 1 - q4 - 2q6 + q8 - q10 + q12 + q14 + q18

HOMFLY-PT Polynomial:

2a-4 + 3a-4z2 + a-4z4 - 2a-2 - 5a-2z2 - 4a-2z4 - a-2z6 - 1 - 5z2 - 4z4 - z6 + 2a2 + 3a2z2 + a2z4

Kauffman Polynomial:

4a-6z2 - 4a-6z4 + a-6z6 - a-5z + 6a-5z3 - 7a-5z5 + 2a-5z7 + 2a-4 - 6a-4z2 + 6a-4z4 - 6a-4z6 + 2a-4z8 + a-3z - 4a-3z3 - a-3z7 + a-3z9 + 2a-2 - 12a-2z2 + 16a-2z4 - 12a-2z6 + 4a-2z8 + a-1z - a-1z5 + a-1z9 - 1 + 6z2 - z4 - 2z6 + 2z8 - az + 7az3 - 6az5 + 3az7 - 2a2 + 6a2z2 - 6a2z4 + 3a2z6 - 3a3z3 + 2a3z5 - 2a4z2 + a4z4

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

{-4, -2}

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 1022. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

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

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-12 - 2q-11 + q-10 + 4q-9 - 8q-8 + 3q-7 + 11q-6 - 20q-5 + 6q-4 + 24q-3 - 37q-2 + 8q-1 + 39 - 48q + 5q2 + 45q3 - 45q4 - 3q5 + 43q6 - 32q7 - 10q8 + 33q9 - 17q10 - 11q11 + 18q12 - 5q13 - 7q14 + 6q15 - 2q17 + q18
3	q-24 - 2q-23 + q-22 + q-21 + q-20 - 5q-19 + 2q-18 + 4q-17 - 10q-15 + 5q-14 + 11q-13 - 5q-12 - 21q-11 + 14q-10 + 28q-9 - 18q-8 - 46q-7 + 27q-6 + 66q-5 - 32q-4 - 90q-3 + 36q-2 + 108q-1 - 28 - 129q + 25q2 + 136q3 - 11q4 - 139q5 - 3q6 + 134q7 + 17q8 - 122q9 - 36q10 + 111q11 + 47q12 - 91q13 - 60q14 + 73q15 + 66q16 - 52q17 - 67q18 + 32q19 + 62q20 - 15q21 - 49q22 - q23 + 39q24 + 5q25 - 22q26 - 11q27 + 14q28 + 7q29 - 5q30 - 6q31 + 3q32 + 2q33 - 2q35 + q36
4	q-40 - 2q-39 + q-38 + q-37 - 2q-36 + 4q-35 - 7q-34 + 4q-33 + 3q-32 - 8q-31 + 14q-30 - 16q-29 + 7q-28 + 5q-27 - 20q-26 + 33q-25 - 26q-24 + 15q-23 + 2q-22 - 49q-21 + 58q-20 - 27q-19 + 48q-18 - q-17 - 120q-16 + 60q-15 - 17q-14 + 142q-13 + 29q-12 - 236q-11 - 3q-10 - 37q-9 + 294q-8 + 136q-7 - 343q-6 - 124q-5 - 123q-4 + 431q-3 + 289q-2 - 372q-1 - 222 - 253q + 477q2 + 410q3 - 322q4 - 245q5 - 359q6 + 431q7 + 448q8 - 234q9 - 195q10 - 418q11 + 330q12 + 422q13 - 128q14 - 108q15 - 442q16 + 197q17 + 354q18 - 10q19 + q20 - 431q21 + 48q22 + 249q23 + 84q24 + 117q25 - 355q26 - 74q27 + 109q28 + 108q29 + 201q30 - 218q31 - 113q32 - 14q33 + 55q34 + 200q35 - 79q36 - 68q37 - 64q38 - 13q39 + 128q40 - 7q41 - 8q42 - 44q43 - 37q44 + 52q45 + 3q46 + 14q47 - 13q48 - 23q49 + 15q50 - 2q51 + 8q52 - q53 - 8q54 + 4q55 - q56 + 2q57 - 2q59 + q60

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, 22]]
Out[2]=	PD[X[6, 2, 7, 1], X[16, 12, 17, 11], X[12, 3, 13, 4], X[2, 15, 3, 16], > X[14, 5, 15, 6], X[18, 8, 19, 7], X[20, 10, 1, 9], X[8, 20, 9, 19], > X[4, 13, 5, 14], X[10, 18, 11, 17]]
In[3]:=	GaussCode[Knot[10, 22]]
Out[3]=	GaussCode[1, -4, 3, -9, 5, -1, 6, -8, 7, -10, 2, -3, 9, -5, 4, -2, 10, -6, 8, > -7]
In[4]:=	DTCode[Knot[10, 22]]
Out[4]=	DTCode[6, 12, 14, 18, 20, 16, 4, 2, 10, 8]
In[5]:=	br = BR[Knot[10, 22]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, -3, 2, -3, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 22]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 22]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 22]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 2, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 22]][t]
Out[10]=	2 6 10 2 3 13 - -- + -- - -- - 10 t + 6 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 22]][z]
Out[11]=	2 4 6 1 - 4 z - 6 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 22]}
In[13]:=	{KnotDet[Knot[10, 22]], KnotSignature[Knot[10, 22]]}
Out[13]=	{49, 0}
In[14]:=	Jones[Knot[10, 22]][q]
Out[14]=	-4 2 4 6 2 3 4 5 6 8 + q - -- + -- - - - 8 q + 7 q - 6 q + 4 q - 2 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 22], Knot[10, 35]}
In[16]:=	A2Invariant[Knot[10, 22]][q]
Out[16]=	-12 -8 -6 -4 2 4 6 8 10 12 14 18 -1 + q + q + q - q + -- - q - 2 q + q - q + q + q + q 2 q
In[17]:=	HOMFLYPT[Knot[10, 22]][a, z]
Out[17]=	2 2 4 4 2 2 2 2 3 z 5 z 2 2 4 z 4 z 2 4 -1 + -- - -- + 2 a - 5 z + ---- - ---- + 3 a z - 4 z + -- - ---- + a z - 4 2 4 2 4 2 a a a a a a 6 6 z > z - -- 2 a
In[18]:=	Kauffman[Knot[10, 22]][a, z]
Out[18]=	2 2 2 2 2 2 z z z 2 4 z 6 z 12 z -1 + -- + -- - 2 a - -- + -- + - - a z + 6 z + ---- - ---- - ----- + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 4 4 2 2 4 2 6 z 4 z 3 3 3 4 4 z 6 z > 6 a z - 2 a z + ---- - ---- + 7 a z - 3 a z - z - ---- + ---- + 5 3 6 4 a a a a 4 5 5 6 6 16 z 2 4 4 4 7 z z 5 3 5 6 z 6 z > ----- - 6 a z + a z - ---- - -- - 6 a z + 2 a z - 2 z + -- - ---- - 2 5 a 6 4 a a a a 6 7 7 8 8 9 9 12 z 2 6 2 z z 7 8 2 z 4 z z z > ----- + 3 a z + ---- - -- + 3 a z + 2 z + ---- + ---- + -- + -- 2 5 3 4 2 3 a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 22]], Vassiliev[3][Knot[10, 22]]}
Out[19]=	{-4, -2}
In[20]:=	Kh[Knot[10, 22]][q, t]
Out[20]=	5 1 1 1 3 1 3 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 4 q t + 4 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 11 5 > 3 q t + 4 q t + 3 q t + 3 q t + q t + 3 q t + q t + q t + 13 6 > q t
In[21]:=	ColouredJones[Knot[10, 22], 2][q]
Out[21]=	-12 2 -10 4 8 3 11 20 6 24 37 8 39 + q - --- + q + -- - -- + -- + -- - -- + -- + -- - -- + - - 48 q + 11 9 8 7 6 5 4 3 2 q q q q q q q q q q 2 3 4 5 6 7 8 9 10 > 5 q + 45 q - 45 q - 3 q + 43 q - 32 q - 10 q + 33 q - 17 q - 11 12 13 14 15 17 18 > 11 q + 18 q - 5 q - 7 q + 6 q - 2 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1022

1021 1023