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

1083 1085

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KnotPlot

PD Presentation:

X4251 X10,4,11,3 X8,12,9,11 X20,15,1,16 X16,5,17,6 X12,18,13,17 X14,8,15,7 X18,14,19,13 X6,19,7,20 X2,10,3,9

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 16 14 2 8 18 20 12 6

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 Chiral 1 3 3 / NotAvailable 1

Alexander Polynomial:

2t-3 - 9t-2 + 20t-1 - 25 + 20t - 9t2 + 2t3

Conway Polynomial:

1 + 2z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a46, K11n184, ...}

Determinant and Signature:

{87, 2}

Jones Polynomial:

- q-2 + 3q-1 - 6 + 11q - 13q2 + 15q3 - 14q4 + 11q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

- q-6 + q-4 - q-2 - 1 + 4q2 - q4 + 4q6 + q8 - q10 + q12 - 4q14 + 2q16 - q18 - q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6z2 - a-6z4 - 2a-4 + 2a-4z4 + a-4z6 + 4a-2 + 5a-2z2 + 3a-2z4 + a-2z6 - 1 - 2z2 - z4

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 6a-7z3 - 13a-7z5 + 7a-7z7 - a-6z2 + 2a-6z4 - 8a-6z6 + 6a-6z8 + 11a-5z3 - 20a-5z5 + 8a-5z7 + 2a-5z9 - 2a-4 + a-4z2 + 9a-4z4 - 17a-4z6 + 10a-4z8 + 2a-3z + 4a-3z3 - 11a-3z5 + 5a-3z7 + 2a-3z9 - 4a-2 + 7a-2z2 - 5a-2z4 - 2a-2z6 + 4a-2z8 + 2a-1z - 2a-1z3 - 4a-1z5 + 4a-1z7 - 1 + 4z2 - 6z4 + 3z6 + az - 2az3 + az5

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

{2, 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=2 is the signature of 1084. 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 = 17                     1 j = 15                   3   j = 13                 5 1   j = 11               6 3     j = 9             8 5       j = 7           7 6         j = 5         6 8           j = 3       5 7             j = 1     2 7               j = -1   1 4                 j = -3   2                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 3q-6 + q-5 + 8q-4 - 17q-3 + 5q-2 + 34q-1 - 54 + q + 93q2 - 101q3 - 27q4 + 161q5 - 124q6 - 67q7 + 197q8 - 111q9 - 93q10 + 181q11 - 69q12 - 93q13 + 123q14 - 22q15 - 66q16 + 55q17 + 3q18 - 27q19 + 12q20 + 3q21 - 4q22 + q23
3	- q-15 + 3q-14 - q-13 - 3q-12 - 2q-11 + 11q-10 - 2q-9 - 22q-8 + 6q-7 + 48q-6 - 10q-5 - 99q-4 + 5q-3 + 181q-2 + 29q-1 - 296 - 99q + 413q2 + 245q3 - 547q4 - 408q5 + 612q6 + 637q7 - 663q8 - 834q9 + 632q10 + 1036q11 - 587q12 - 1162q13 + 478q14 + 1262q15 - 371q16 - 1284q17 + 232q18 + 1256q19 - 88q20 - 1172q21 - 51q22 + 1027q23 + 186q24 - 852q25 - 271q26 + 638q27 + 319q28 - 433q29 - 308q30 + 250q31 + 259q32 - 121q33 - 181q34 + 38q35 + 109q36 - 58q38 - 4q39 + 22q40 + 4q41 - 7q42 - 3q43 + 4q44 - q45
4	q-26 - 3q-25 + q-24 + 3q-23 - 3q-22 + 8q-21 - 14q-20 + 6q-19 + 12q-18 - 26q-17 + 21q-16 - 36q-15 + 51q-14 + 67q-13 - 112q-12 - 37q-11 - 140q-10 + 220q-9 + 370q-8 - 178q-7 - 328q-6 - 691q-5 + 383q-4 + 1257q-3 + 304q-2 - 674q-1 - 2200 - 196q + 2521q2 + 1994q3 - 136q4 - 4415q5 - 2287q6 + 3053q7 + 4578q8 + 2073q9 - 6065q10 - 5438q11 + 1978q12 + 6723q13 + 5381q14 - 6212q15 - 8221q16 - 244q17 + 7496q18 + 8380q19 - 5100q20 - 9678q21 - 2557q22 + 6994q23 + 10227q24 - 3391q25 - 9766q26 - 4417q27 + 5586q28 + 10812q29 - 1335q30 - 8620q31 - 5742q32 + 3372q33 + 10090q34 + 891q35 - 6244q36 - 6187q37 + 653q38 + 7892q39 + 2573q40 - 3055q41 - 5232q42 - 1568q43 + 4658q44 + 2830q45 - 298q46 - 3108q47 - 2242q48 + 1718q49 + 1767q50 + 900q51 - 1066q52 - 1526q53 + 201q54 + 551q55 + 735q56 - 76q57 - 583q58 - 96q59 + 12q60 + 268q61 + 74q62 - 128q63 - 26q64 - 40q65 + 51q66 + 24q67 - 21q68 + q69 - 9q70 + 7q71 + 3q72 - 4q73 + q74

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, 84]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[8, 12, 9, 11], X[20, 15, 1, 16], > X[16, 5, 17, 6], X[12, 18, 13, 17], X[14, 8, 15, 7], X[18, 14, 19, 13], > X[6, 19, 7, 20], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 84]]
Out[3]=	GaussCode[1, -10, 2, -1, 5, -9, 7, -3, 10, -2, 3, -6, 8, -7, 4, -5, 6, -8, 9, > -4]
In[4]:=	DTCode[Knot[10, 84]]
Out[4]=	DTCode[4, 10, 16, 14, 2, 8, 18, 20, 12, 6]
In[5]:=	br = BR[Knot[10, 84]]
Out[5]=	BR[4, {1, 1, 1, 2, -1, -3, 2, 2, -3, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 84]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 84]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 84]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 1, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 84]][t]
Out[10]=	2 9 20 2 3 -25 + -- - -- + -- + 20 t - 9 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 84]][z]
Out[11]=	2 4 6 1 + 2 z + 3 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 84], Knot[11, Alternating, 46], Knot[11, NonAlternating, 184]}
In[13]:=	{KnotDet[Knot[10, 84]], KnotSignature[Knot[10, 84]]}
Out[13]=	{87, 2}
In[14]:=	Jones[Knot[10, 84]][q]
Out[14]=	-2 3 2 3 4 5 6 7 8 -6 - q + - + 11 q - 13 q + 15 q - 14 q + 11 q - 8 q + 4 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 84]}
In[16]:=	A2Invariant[Knot[10, 84]][q]
Out[16]=	-6 -4 -2 2 4 6 8 10 12 14 16 -1 - q + q - q + 4 q - q + 4 q + q - q + q - 4 q + 2 q - 18 20 22 24 > q - q + 2 q - q
In[17]:=	HOMFLYPT[Knot[10, 84]][a, z]
Out[17]=	2 2 4 4 4 6 6 2 4 2 z 5 z 4 z 2 z 3 z z z -1 - -- + -- - 2 z - -- + ---- - z - -- + ---- + ---- + -- + -- 4 2 6 2 6 4 2 4 2 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 84]][a, z]
Out[18]=	2 2 2 2 3 3 2 4 z 2 z 2 z 2 z z z 7 z z 6 z -1 - -- - -- - -- + --- + --- + a z + 4 z + -- - -- + -- + ---- - -- + ---- + 4 2 7 3 a 8 6 4 2 9 7 a a a a a a a a a a 3 3 3 4 4 4 4 5 11 z 4 z 2 z 3 4 6 z 2 z 9 z 5 z z > ----- + ---- - ---- - 2 a z - 6 z - ---- + ---- + ---- - ---- + -- - 5 3 a 8 6 4 2 9 a a a a a a a 5 5 5 5 6 6 6 6 13 z 20 z 11 z 4 z 5 6 4 z 8 z 17 z 2 z > ----- - ----- - ----- - ---- + a z + 3 z + ---- - ---- - ----- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 9 9 7 z 8 z 5 z 4 z 6 z 10 z 4 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- + ----- + ---- + ---- + ---- 7 5 3 a 6 4 2 5 3 a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 84]], Vassiliev[3][Knot[10, 84]]}
Out[19]=	{2, 2}
In[20]:=	Kh[Knot[10, 84]][q, t]
Out[20]=	3 1 2 1 4 2 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 7 q t + 6 q t + 8 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 7 q t + 6 q t + 8 q t + 5 q t + 6 q t + 3 q t + 5 q t + 13 6 15 6 17 7 > q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 84], 2][q]
Out[21]=	-7 3 -5 8 17 5 34 2 3 4 -54 + q - -- + q + -- - -- + -- + -- + q + 93 q - 101 q - 27 q + 6 4 3 2 q q q q q 5 6 7 8 9 10 11 12 > 161 q - 124 q - 67 q + 197 q - 111 q - 93 q + 181 q - 69 q - 13 14 15 16 17 18 19 20 > 93 q + 123 q - 22 q - 66 q + 55 q + 3 q - 27 q + 12 q + 21 22 23 > 3 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1084

1083 1085