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

1095 1097

KnotPlot

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KnotPlot

PD Presentation:

X1425 X5,18,6,19 X3948 X9,3,10,2 X11,17,12,16 X7,12,8,13 X15,6,16,7 X17,11,18,10 X13,1,14,20 X19,15,20,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 18 12 2 16 20 6 10 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 Reversible 2 3 3 / NotAvailable 1

Alexander Polynomial:

- t-3 + 7t-2 - 22t-1 + 33 - 22t + 7t2 - t3

Conway Polynomial:

1 - 3z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

q-4 - 4q-3 + 9q-2 - 12q-1 + 15 - 16q + 14q2 - 11q3 + 7q4 - 3q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-12 - 2q-10 + 3q-8 + 2q-6 - 3q-4 + 3q-2 - 3 + q2 - q6 + 3q8 - 3q10 + q12 + q14 - 2q16 + q18 + q20

HOMFLY-PT Polynomial:

a-6 - 2a-4 - 3a-4z2 + 3a-2 + 5a-2z2 + 3a-2z4 - 3 - 6z2 - 3z4 - z6 + 2a2 + a2z2 + a2z4

Kauffman Polynomial:

- a-6 + 3a-6z2 - 3a-6z4 + a-6z6 - 2a-5z + 7a-5z3 - 8a-5z5 + 3a-5z7 - 2a-4 + 6a-4z2 - a-4z4 - 7a-4z6 + 4a-4z8 - 2a-3z + 16a-3z3 - 23a-3z5 + 6a-3z7 + 2a-3z9 - 3a-2 + 10a-2z2 - 4a-2z4 - 17a-2z6 + 11a-2z8 - a-1z + 17a-1z3 - 34a-1z5 + 14a-1z7 + 2a-1z9 - 3 + 12z2 - 17z4 + 7z8 - az + 7az3 - 15az5 + 11az7 - 2a2 + 5a2z2 - 10a2z4 + 9a2z6 - a3z3 + 4a3z5 + a4z4

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

{-3, -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 1096. 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                   2   j = 9                 5 1   j = 7               6 2     j = 5             8 5       j = 3           8 6         j = 1         7 8           j = -1       6 9             j = -3     3 6               j = -5   1 6                 j = -7   3                   j = -9 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-12 - 4q-11 + 5q-10 + 8q-9 - 31q-8 + 23q-7 + 40q-6 - 94q-5 + 41q-4 + 102q-3 - 165q-2 + 34q-1 + 167 - 198q + 3q2 + 198q3 - 175q4 - 33q5 + 179q6 - 112q7 - 54q8 + 119q9 - 44q10 - 45q11 + 51q12 - 6q13 - 19q14 + 11q15 + q16 - 3q17 + q18
3	q-24 - 4q-23 + 5q-22 + 4q-21 - 11q-20 - 13q-19 + 29q-18 + 37q-17 - 70q-16 - 79q-15 + 129q-14 + 158q-13 - 194q-12 - 309q-11 + 279q-10 + 497q-9 - 314q-8 - 754q-7 + 324q-6 + 1004q-5 - 249q-4 - 1259q-3 + 143q-2 + 1437q-1 + 20 - 1554q - 191q2 + 1583q3 + 365q4 - 1532q5 - 532q6 + 1424q7 + 662q8 - 1241q9 - 773q10 + 1023q11 + 831q12 - 771q13 - 833q14 + 511q15 + 772q16 - 273q17 - 654q18 + 83q19 + 501q20 + 36q21 - 332q22 - 98q23 + 199q24 + 93q25 - 92q26 - 72q27 + 36q28 + 40q29 - 9q30 - 19q31 + 3q32 + 5q33 + q34 - 3q35 + q36
4	q-40 - 4q-39 + 5q-38 + 4q-37 - 15q-36 + 7q-35 - 7q-34 + 36q-33 + 11q-32 - 105q-31 + 16q-30 + 23q-29 + 198q-28 + 45q-27 - 467q-26 - 116q-25 + 157q-24 + 852q-23 + 326q-22 - 1399q-21 - 928q-20 + 161q-19 + 2500q-18 + 1643q-17 - 2690q-16 - 3140q-15 - 928q-14 + 4894q-13 + 4797q-12 - 3110q-11 - 6376q-10 - 3997q-9 + 6528q-8 + 9152q-7 - 1544q-6 - 8871q-5 - 8311q-4 + 6116q-3 + 12686q-2 + 1487q-1 - 9232 - 11969q + 3978q2 + 14001q3 + 4493q4 - 7654q5 - 13785q6 + 1213q7 + 13171q8 + 6672q9 - 4977q10 - 13811q11 - 1582q12 + 10792q13 + 7965q14 - 1705q15 - 12277q16 - 4118q17 + 7178q18 + 8110q19 + 1697q20 - 9164q21 - 5652q22 + 2914q23 + 6571q24 + 4120q25 - 4968q26 - 5259q27 - 588q28 + 3623q29 + 4417q30 - 1252q31 - 3151q32 - 1960q33 + 825q34 + 2848q35 + 527q36 - 943q37 - 1427q38 - 438q39 + 1058q40 + 568q41 + 78q42 - 486q43 - 413q44 + 183q45 + 168q46 + 153q47 - 61q48 - 132q49 + 10q50 + 8q51 + 42q52 + 3q53 - 22q54 + 3q55 - 3q56 + 5q57 + q58 - 3q59 + 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, 96]]
Out[2]=	PD[X[1, 4, 2, 5], X[5, 18, 6, 19], X[3, 9, 4, 8], X[9, 3, 10, 2], > X[11, 17, 12, 16], X[7, 12, 8, 13], X[15, 6, 16, 7], X[17, 11, 18, 10], > X[13, 1, 14, 20], X[19, 15, 20, 14]]
In[3]:=	GaussCode[Knot[10, 96]]
Out[3]=	GaussCode[-1, 4, -3, 1, -2, 7, -6, 3, -4, 8, -5, 6, -9, 10, -7, 5, -8, 2, -10, > 9]
In[4]:=	DTCode[Knot[10, 96]]
Out[4]=	DTCode[4, 8, 18, 12, 2, 16, 20, 6, 10, 14]
In[5]:=	br = BR[Knot[10, 96]]
Out[5]=	BR[5, {-1, 2, 1, -3, 2, 1, -3, 4, -3, 2, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 96]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 96]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 96]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 96]][t]
Out[10]=	-3 7 22 2 3 33 - t + -- - -- - 22 t + 7 t - t 2 t t
In[11]:=	Conway[Knot[10, 96]][z]
Out[11]=	2 4 6 1 - 3 z + z - z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 96]}
In[13]:=	{KnotDet[Knot[10, 96]], KnotSignature[Knot[10, 96]]}
Out[13]=	{93, 0}
In[14]:=	Jones[Knot[10, 96]][q]
Out[14]=	-4 4 9 12 2 3 4 5 6 15 + q - -- + -- - -- - 16 q + 14 q - 11 q + 7 q - 3 q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 96]}
In[16]:=	A2Invariant[Knot[10, 96]][q]
Out[16]=	-12 2 3 2 3 3 2 6 8 10 12 14 -3 + q - --- + -- + -- - -- + -- + q - q + 3 q - 3 q + q + q - 10 8 6 4 2 q q q q q 16 18 20 > 2 q + q + q
In[17]:=	HOMFLYPT[Knot[10, 96]][a, z]
Out[17]=	2 2 4 -6 2 3 2 2 3 z 5 z 2 2 4 3 z 2 4 -3 + a - -- + -- + 2 a - 6 z - ---- + ---- + a z - 3 z + ---- + a z - 4 2 4 2 2 a a a a a 6 > z
In[18]:=	Kauffman[Knot[10, 96]][a, z]
Out[18]=	2 2 2 -6 2 3 2 2 z 2 z z 2 3 z 6 z 10 z -3 - a - -- - -- - 2 a - --- - --- - - - a z + 12 z + ---- + ---- + ----- + 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 4 4 2 2 7 z 16 z 17 z 3 3 3 4 3 z z > 5 a z + ---- + ----- + ----- + 7 a z - a z - 17 z - ---- - -- - 5 3 a 6 4 a a a a 4 5 5 5 6 4 z 2 4 4 4 8 z 23 z 34 z 5 3 5 z > ---- - 10 a z + a z - ---- - ----- - ----- - 15 a z + 4 a z + -- - 2 5 3 a 6 a a a a 6 6 7 7 7 8 7 z 17 z 2 6 3 z 6 z 14 z 7 8 4 z > ---- - ----- + 9 a z + ---- + ---- + ----- + 11 a z + 7 z + ---- + 4 2 5 3 a 4 a a a a a 8 9 9 11 z 2 z 2 z > ----- + ---- + ---- 2 3 a a a
In[19]:=	{Vassiliev[2][Knot[10, 96]], Vassiliev[3][Knot[10, 96]]}
Out[19]=	{-3, -2}
In[20]:=	Kh[Knot[10, 96]][q, t]
Out[20]=	9 1 3 1 6 3 6 6 3 - + 7 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 8 q t + 8 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 > 6 q t + 8 q t + 5 q t + 6 q t + 2 q t + 5 q t + q t + 11 5 13 6 > 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 96], 2][q]
Out[21]=	-12 4 5 8 31 23 40 94 41 102 165 34 167 + q - --- + --- + -- - -- + -- + -- - -- + -- + --- - --- + -- - 198 q + 11 10 9 8 7 6 5 4 3 2 q q q q q q q q q q q 2 3 4 5 6 7 8 9 > 3 q + 198 q - 175 q - 33 q + 179 q - 112 q - 54 q + 119 q - 10 11 12 13 14 15 16 17 18 > 44 q - 45 q + 51 q - 6 q - 19 q + 11 q + q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1096

1095 1097