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1080 1082

KnotPlot

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KnotPlot

PD Presentation:

X4251 X8493 X12,6,13,5 X16,9,17,10 X20,17,1,18 X18,13,19,14 X14,19,15,20 X10,15,11,16 X6,12,7,11 X2837

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 16 6 18 10 20 14

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 NegativeAmphicheiral 2 3 3 / NotAvailable 1

Alexander Polynomial:

- t-3 + 8t-2 - 20t-1 + 27 - 20t + 8t2 - t3

Conway Polynomial:

1 + 3z2 + 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{85, 0}

Jones Polynomial:

- q-5 + 3q-4 - 7q-3 + 11q-2 - 13q-1 + 15 - 13q + 11q2 - 7q3 + 3q4 - q5

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

{10109, ...}

A2 (sl(3)) Invariant:

- q-16 + q-12 - 3q-10 + 2q-8 - q-4 + 4q-2 - 1 + 4q2 - q4 + 2q8 - 3q10 + q12 - q16

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + a-2 + 3a-2z2 + 2a-2z4 + 1 - z2 - 2z4 - z6 + a2 + 3a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 3a-4z2 - 5a-4z4 + 3a-4z6 - 2a-3z + 5a-3z3 - 8a-3z5 + 5a-3z7 - a-2 + 6a-2z2 - 9a-2z4 + 4a-2z8 - 8a-1z + 25a-1z3 - 31a-1z5 + 13a-1z7 + a-1z9 + 1 + 6z2 - 8z4 - 6z6 + 8z8 - 8az + 25az3 - 31az5 + 13az7 + az9 - a2 + 6a2z2 - 9a2z4 + 4a2z8 - 2a3z + 5a3z3 - 8a3z5 + 5a3z7 - a4 + 3a4z2 - 5a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

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

{3, 0}

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 1081. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                     1 j = 9                   2   j = 7                 5 1   j = 5               6 2     j = 3             7 5       j = 1           8 6         j = -1         6 8           j = -3       5 7             j = -5     2 6               j = -7   1 5                 j = -9   2                   j = -11 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-15 - 3q-14 + 2q-13 + 9q-12 - 21q-11 + 4q-10 + 43q-9 - 60q-8 - 10q-7 + 108q-6 - 98q-5 - 48q-4 + 172q-3 - 109q-2 - 89q-1 + 199 - 89q - 109q2 + 172q3 - 48q4 - 98q5 + 108q6 - 10q7 - 60q8 + 43q9 + 4q10 - 21q11 + 9q12 + 2q13 - 3q14 + q15
3	- q-30 + 3q-29 - 2q-28 - 4q-27 + q-26 + 17q-25 - 5q-24 - 39q-23 + 2q-22 + 83q-21 + 12q-20 - 144q-19 - 64q-18 + 237q-17 + 144q-16 - 320q-15 - 287q-14 + 395q-13 + 472q-12 - 440q-11 - 677q-10 + 430q-9 + 894q-8 - 387q-7 - 1074q-6 + 290q-5 + 1235q-4 - 196q-3 - 1308q-2 + 54q-1 + 1359 + 54q - 1308q2 - 196q3 + 1235q4 + 290q5 - 1074q6 - 387q7 + 894q8 + 430q9 - 677q10 - 440q11 + 472q12 + 395q13 - 287q14 - 320q15 + 144q16 + 237q17 - 64q18 - 144q19 + 12q20 + 83q21 + 2q22 - 39q23 - 5q24 + 17q25 + q26 - 4q27 - 2q28 + 3q29 - q30
4	q-50 - 3q-49 + 2q-48 + 4q-47 - 6q-46 + 3q-45 - 16q-44 + 16q-43 + 34q-42 - 29q-41 - 14q-40 - 87q-39 + 62q-38 + 185q-37 - 25q-36 - 96q-35 - 399q-34 + 58q-33 + 615q-32 + 278q-31 - 116q-30 - 1255q-29 - 429q-28 + 1230q-27 + 1314q-26 + 525q-25 - 2569q-24 - 1990q-23 + 1284q-22 + 3006q-21 + 2539q-20 - 3483q-19 - 4520q-18 - 33q-17 + 4439q-16 + 5724q-15 - 3114q-14 - 6965q-13 - 2568q-12 + 4730q-11 + 8927q-10 - 1545q-9 - 8332q-8 - 5289q-7 + 3864q-6 + 11073q-5 + 460q-4 - 8389q-3 - 7331q-2 + 2332q-1 + 11797 + 2332q - 7331q2 - 8389q3 + 460q4 + 11073q5 + 3864q6 - 5289q7 - 8332q8 - 1545q9 + 8927q10 + 4730q11 - 2568q12 - 6965q13 - 3114q14 + 5724q15 + 4439q16 - 33q17 - 4520q18 - 3483q19 + 2539q20 + 3006q21 + 1284q22 - 1990q23 - 2569q24 + 525q25 + 1314q26 + 1230q27 - 429q28 - 1255q29 - 116q30 + 278q31 + 615q32 + 58q33 - 399q34 - 96q35 - 25q36 + 185q37 + 62q38 - 87q39 - 14q40 - 29q41 + 34q42 + 16q43 - 16q44 + 3q45 - 6q46 + 4q47 + 2q48 - 3q49 + q50
5	- q-75 + 3q-74 - 2q-73 - 4q-72 + 6q-71 + 2q-70 - 4q-69 + 5q-68 - 11q-67 - 22q-66 + 23q-65 + 43q-64 + 11q-63 - 14q-62 - 90q-61 - 112q-60 + 40q-59 + 238q-58 + 245q-57 + 2q-56 - 416q-55 - 645q-54 - 200q-53 + 710q-52 + 1312q-51 + 783q-50 - 897q-49 - 2429q-48 - 2020q-47 + 699q-46 + 3831q-45 + 4318q-44 + 432q-43 - 5363q-42 - 7663q-41 - 3090q-40 + 6098q-39 + 12133q-38 + 7872q-37 - 5472q-36 - 16917q-35 - 14774q-34 + 2252q-33 + 21133q-32 + 23676q-31 + 3836q-30 - 23638q-29 - 33459q-28 - 12909q-27 + 23384q-26 + 43029q-25 + 24403q-24 - 20125q-23 - 51135q-22 - 36996q-21 + 13867q-20 + 56767q-19 + 49718q-18 - 5493q-17 - 59785q-16 - 60976q-15 - 4204q-14 + 60057q-13 + 70514q-12 + 13932q-11 - 58298q-10 - 77413q-9 - 23291q-8 + 54796q-7 + 82432q-6 + 31414q-5 - 50368q-4 - 84899q-3 - 38762q-2 + 44856q-1 + 86051 + 44856q - 38762q2 - 84899q3 - 50368q4 + 31414q5 + 82432q6 + 54796q7 - 23291q8 - 77413q9 - 58298q10 + 13932q11 + 70514q12 + 60057q13 - 4204q14 - 60976q15 - 59785q16 - 5493q17 + 49718q18 + 56767q19 + 13867q20 - 36996q21 - 51135q22 - 20125q23 + 24403q24 + 43029q25 + 23384q26 - 12909q27 - 33459q28 - 23638q29 + 3836q30 + 23676q31 + 21133q32 + 2252q33 - 14774q34 - 16917q35 - 5472q36 + 7872q37 + 12133q38 + 6098q39 - 3090q40 - 7663q41 - 5363q42 + 432q43 + 4318q44 + 3831q45 + 699q46 - 2020q47 - 2429q48 - 897q49 + 783q50 + 1312q51 + 710q52 - 200q53 - 645q54 - 416q55 + 2q56 + 245q57 + 238q58 + 40q59 - 112q60 - 90q61 - 14q62 + 11q63 + 43q64 + 23q65 - 22q66 - 11q67 + 5q68 - 4q69 + 2q70 + 6q71 - 4q72 - 2q73 + 3q74 - q75

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, 81]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 6, 13, 5], X[16, 9, 17, 10], > X[20, 17, 1, 18], X[18, 13, 19, 14], X[14, 19, 15, 20], X[10, 15, 11, 16], > X[6, 12, 7, 11], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[10, 81]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -9, 10, -2, 4, -8, 9, -3, 6, -7, 8, -4, 5, -6, 7, > -5]
In[4]:=	DTCode[Knot[10, 81]]
Out[4]=	DTCode[4, 8, 12, 2, 16, 6, 18, 10, 20, 14]
In[5]:=	br = BR[Knot[10, 81]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, -3, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 81]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 81]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 81]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{NegativeAmphicheiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 81]][t]
Out[10]=	-3 8 20 2 3 27 - t + -- - -- - 20 t + 8 t - t 2 t t
In[11]:=	Conway[Knot[10, 81]][z]
Out[11]=	2 4 6 1 + 3 z + 2 z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 81]}
In[13]:=	{KnotDet[Knot[10, 81]], KnotSignature[Knot[10, 81]]}
Out[13]=	{85, 0}
In[14]:=	Jones[Knot[10, 81]][q]
Out[14]=	-5 3 7 11 13 2 3 4 5 15 - q + -- - -- + -- - -- - 13 q + 11 q - 7 q + 3 q - q 4 3 2 q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 81], Knot[10, 109]}
In[16]:=	A2Invariant[Knot[10, 81]][q]
Out[16]=	-16 -12 3 2 -4 4 2 4 8 10 12 16 -1 - q + q - --- + -- - q + -- + 4 q - q + 2 q - 3 q + q - q 10 8 2 q q q
In[17]:=	HOMFLYPT[Knot[10, 81]][a, z]
Out[17]=	2 2 4 -4 -2 2 4 2 z 3 z 2 2 4 2 4 2 z 1 - a + a + a - a - z - -- + ---- + 3 a z - a z - 2 z + ---- + 4 2 2 a a a 2 4 6 > 2 a z - z
In[18]:=	Kauffman[Knot[10, 81]][a, z]
Out[18]=	-4 -2 2 4 z 2 z 8 z 3 5 2 1 - a - a - a - a + -- - --- - --- - 8 a z - 2 a z + a z + 6 z + 5 3 a a a 2 2 3 3 3 3 z 6 z 2 2 4 2 2 z 5 z 25 z 3 3 3 > ---- + ---- + 6 a z + 3 a z - ---- + ---- + ----- + 25 a z + 5 a z - 4 2 5 3 a a a a a 4 4 5 5 5 5 3 4 5 z 9 z 2 4 4 4 z 8 z 31 z > 2 a z - 8 z - ---- - ---- - 9 a z - 5 a z + -- - ---- - ----- - 4 2 5 3 a a a a a 6 7 7 5 3 5 5 5 6 3 z 4 6 5 z 13 z > 31 a z - 8 a z + a z - 6 z + ---- + 3 a z + ---- + ----- + 4 3 a a a 8 9 7 3 7 8 4 z 2 8 z 9 > 13 a z + 5 a z + 8 z + ---- + 4 a z + -- + a z 2 a a
In[19]:=	{Vassiliev[2][Knot[10, 81]], Vassiliev[3][Knot[10, 81]]}
Out[19]=	{3, 0}
In[20]:=	Kh[Knot[10, 81]][q, t]
Out[20]=	8 1 2 1 5 2 6 5 7 6 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 7 4 9 4 > 6 q t + 7 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 2 q t + 11 5 > q t
In[21]:=	ColouredJones[Knot[10, 81], 2][q]
Out[21]=	-15 3 2 9 21 4 43 60 10 108 98 48 172 199 + q - --- + --- + --- - --- + --- + -- - -- - -- + --- - -- - -- + --- - 14 13 12 11 10 9 8 7 6 5 4 3 q q q q q q q q q q q q 109 89 2 3 4 5 6 7 > --- - -- - 89 q - 109 q + 172 q - 48 q - 98 q + 108 q - 10 q - 2 q q 8 9 10 11 12 13 14 15 > 60 q + 43 q + 4 q - 21 q + 9 q + 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1081

1080 1082