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

1075 1077

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 20 14 16 2 10 8 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 Reversible 2--3 3 3 / NotAvailable 1

Alexander Polynomial:

- 2t-3 + 7t-2 - 12t-1 + 15 - 12t + 7t2 - 2t3

Conway Polynomial:

1 - 2z2 - 5z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{57, 4}

Jones Polynomial:

1 - q + 4q2 - 6q3 + 8q4 - 10q5 + 9q6 - 8q7 + 6q8 - 3q9 + q10

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

{...}

A2 (sl(3)) Invariant:

1 + q2 + 2q4 + 3q6 - q8 + q10 - 3q12 - 2q14 - 2q18 + 2q20 - q22 + q24 + q26 - q28 + q30

HOMFLY-PT Polynomial:

a-8 + 2a-8z2 + a-8z4 - 2a-6z2 - 3a-6z4 - a-6z6 - 4a-4 - 6a-4z2 - 4a-4z4 - a-4z6 + 4a-2 + 4a-2z2 + a-2z4

Kauffman Polynomial:

- a-12z2 + a-12z4 - 3a-11z3 + 3a-11z5 + 3a-10z2 - 7a-10z4 + 5a-10z6 - 2a-9z + 7a-9z3 - 8a-9z5 + 5a-9z7 + a-8 - 4a-8z2 + 4a-8z4 - 3a-8z6 + 3a-8z8 + 2a-7z - 3a-7z3 - 2a-7z5 + 2a-7z7 + a-7z9 - 7a-6z2 + 10a-6z4 - 9a-6z6 + 4a-6z8 + 8a-5z - 15a-5z3 + 7a-5z5 - 2a-5z7 + a-5z9 - 4a-4 + 9a-4z2 - 7a-4z4 + a-4z8 + 4a-3z - 2a-3z3 - 2a-3z5 + a-3z7 - 4a-2 + 8a-2z2 - 5a-2z4 + a-2z6

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

{-2, -6}

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=4 is the signature of 1076. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 21                     1 j = 19                   2   j = 17                 4 1   j = 15               4 2     j = 13             5 4       j = 11           5 4         j = 9         3 5           j = 7       3 5             j = 5     1 3               j = 3   1 4                 j = 1                       j = -1 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-2 - q-1 + 4q - 5q2 - 2q3 + 15q4 - 15q5 - 11q6 + 38q7 - 24q8 - 32q9 + 65q10 - 24q11 - 55q12 + 81q13 - 16q14 - 68q15 + 77q16 - 5q17 - 62q18 + 55q19 + 3q20 - 40q21 + 26q22 + 5q23 - 16q24 + 7q25 + 2q26 - 3q27 + q28
3	q-6 - q-5 + 3q-2 - 4q-1 + 2q + 8q2 - 13q3 - 7q4 + 12q5 + 29q6 - 31q7 - 39q8 + 23q9 + 83q10 - 30q11 - 112q12 + 5q13 + 162q14 + 18q15 - 194q16 - 64q17 + 228q18 + 109q19 - 248q20 - 156q21 + 252q22 + 210q23 - 259q24 - 240q25 + 239q26 + 277q27 - 226q28 - 286q29 + 189q30 + 292q31 - 157q32 - 272q33 + 114q34 + 241q35 - 73q36 - 199q37 + 40q38 + 150q39 - 13q40 - 107q41 + 3q42 + 64q43 + 6q44 - 39q45 - 5q46 + 21q47 + 3q48 - 10q49 - q50 + 3q51 + 2q52 - 3q53 + q54
4	q-12 - q-11 - q-8 + 4q-7 - 4q-6 + q-5 + 3q-4 - 6q-3 + 9q-2 - 13q-1 + 2 + 16q - 8q2 + 18q3 - 45q4 - 13q5 + 43q6 + 19q7 + 67q8 - 113q9 - 92q10 + 40q11 + 83q12 + 238q13 - 151q14 - 255q15 - 102q16 + 88q17 + 561q18 - 25q19 - 387q20 - 412q21 - 119q22 + 895q23 + 301q24 - 311q25 - 756q26 - 562q27 + 1062q28 + 692q29 - 3q30 - 976q31 - 1094q32 + 1021q33 + 1004q34 + 410q35 - 1040q36 - 1557q37 + 850q38 + 1189q39 + 795q40 - 982q41 - 1861q42 + 604q43 + 1230q44 + 1093q45 - 809q46 - 1951q47 + 306q48 + 1084q49 + 1238q50 - 506q51 - 1762q52 + 10q53 + 738q54 + 1158q55 - 150q56 - 1309q57 - 159q58 + 315q59 + 847q60 + 94q61 - 756q62 - 150q63 + 18q64 + 461q65 + 140q66 - 333q67 - 52q68 - 73q69 + 183q70 + 81q71 - 121q72 + 6q73 - 51q74 + 59q75 + 29q76 - 42q77 + 13q78 - 20q79 + 15q80 + 9q81 - 12q82 + 5q83 - 5q84 + 3q85 + 2q86 - 3q87 + q88

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, 76]]
Out[2]=	PD[X[4, 2, 5, 1], X[14, 10, 15, 9], X[12, 3, 13, 4], X[2, 13, 3, 14], > X[18, 6, 19, 5], X[20, 8, 1, 7], X[6, 20, 7, 19], X[8, 18, 9, 17], > X[16, 12, 17, 11], X[10, 16, 11, 15]]
In[3]:=	GaussCode[Knot[10, 76]]
Out[3]=	GaussCode[1, -4, 3, -1, 5, -7, 6, -8, 2, -10, 9, -3, 4, -2, 10, -9, 8, -5, 7, > -6]
In[4]:=	DTCode[Knot[10, 76]]
Out[4]=	DTCode[4, 12, 18, 20, 14, 16, 2, 10, 8, 6]
In[5]:=	br = BR[Knot[10, 76]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, -3, 2, 2, 2, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 76]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 76]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 76]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 76]][t]
Out[10]=	2 7 12 2 3 15 - -- + -- - -- - 12 t + 7 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 76]][z]
Out[11]=	2 4 6 1 - 2 z - 5 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 76]}
In[13]:=	{KnotDet[Knot[10, 76]], KnotSignature[Knot[10, 76]]}
Out[13]=	{57, 4}
In[14]:=	Jones[Knot[10, 76]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - q + 4 q - 6 q + 8 q - 10 q + 9 q - 8 q + 6 q - 3 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 76]}
In[16]:=	A2Invariant[Knot[10, 76]][q]
Out[16]=	2 4 6 8 10 12 14 18 20 22 24 1 + q + 2 q + 3 q - q + q - 3 q - 2 q - 2 q + 2 q - q + q + 26 28 30 > q - q + q
In[17]:=	HOMFLYPT[Knot[10, 76]][a, z]
Out[17]=	2 2 2 2 4 4 4 4 6 6 -8 4 4 2 z 2 z 6 z 4 z z 3 z 4 z z z z a - -- + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + -- - -- - -- 4 2 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 76]][a, z]
Out[18]=	2 2 2 2 2 -8 4 4 2 z 2 z 8 z 4 z z 3 z 4 z 7 z 9 z a - -- - -- - --- + --- + --- + --- - --- + ---- - ---- - ---- + ---- + 4 2 9 7 5 3 12 10 8 6 4 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 8 z 3 z 7 z 3 z 15 z 2 z z 7 z 4 z 10 z > ---- - ---- + ---- - ---- - ----- - ---- + --- - ---- + ---- + ----- - 2 11 9 7 5 3 12 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 5 6 6 6 6 7 z 5 z 3 z 8 z 2 z 7 z 2 z 5 z 3 z 9 z z > ---- - ---- + ---- - ---- - ---- + ---- - ---- + ---- - ---- - ---- + -- + 4 2 11 9 7 5 3 10 8 6 2 a a a a a a a a a a a 7 7 7 7 8 8 8 9 9 5 z 2 z 2 z z 3 z 4 z z z z > ---- + ---- - ---- + -- + ---- + ---- + -- + -- + -- 9 7 5 3 8 6 4 7 5 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 76]], Vassiliev[3][Knot[10, 76]]}
Out[19]=	{-2, -6}
In[20]:=	Kh[Knot[10, 76]][q, t]
Out[20]=	3 3 5 1 q 5 7 7 2 9 2 9 3 4 q + q + ---- + -- + 3 q t + 3 q t + 5 q t + 3 q t + 5 q t + 2 t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 5 q t + 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 17 6 17 7 19 7 21 8 > 4 q t + q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 76], 2][q]
Out[21]=	-2 1 2 3 4 5 6 7 8 9 q - - + 4 q - 5 q - 2 q + 15 q - 15 q - 11 q + 38 q - 24 q - 32 q + q 10 11 12 13 14 15 16 17 > 65 q - 24 q - 55 q + 81 q - 16 q - 68 q + 77 q - 5 q - 18 19 20 21 22 23 24 25 > 62 q + 55 q + 3 q - 40 q + 26 q + 5 q - 16 q + 7 q + 26 27 28 > 2 q - 3 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1076

1075 1077