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1076 1078

KnotPlot

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KnotPlot

PD Presentation:

X1425 X3849 X13,17,14,16 X5,15,6,14 X15,7,16,6 X9,19,10,18 X11,1,12,20 X19,11,20,10 X17,13,18,12 X7283

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 18 20 16 6 12 10

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 + 14t-1 - 17 + 14t - 7t2 + 2t3

Conway Polynomial:

1 + 4z2 + 5z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1065, K11n71, K11n75, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

- q-2 + 2q-1 - 4 + 8q - 9q2 + 11q3 - 10q4 + 8q5 - 6q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

- q-6 - q-2 - 1 + 3q2 + 4q6 + 2q8 + q12 - 3q14 + q16 - q18 - q20 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 - a-4 + 2a-4z2 + 3a-4z4 + a-4z6 + 5a-2 + 7a-2z2 + 4a-2z4 + a-2z6 - 2 - 3z2 - z4

Kauffman Polynomial:

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

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

{4, 5}

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 1077. 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                   2   j = 13                 4 1   j = 11               4 2     j = 9             6 4       j = 7           5 4         j = 5         4 6           j = 3       4 5             j = 1     1 5               j = -1   1 3                 j = -3   1                   j = -5 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-7 - 2q-6 + 5q-4 - 9q-3 + q-2 + 18q-1 - 26 - q + 47q2 - 50q3 - 13q4 + 82q5 - 65q6 - 31q7 + 102q8 - 63q9 - 43q10 + 95q11 - 43q12 - 44q13 + 67q14 - 18q15 - 33q16 + 33q17 - 2q18 - 15q19 + 9q20 + q21 - 3q22 + q23
3	- q-15 + 2q-14 - q-12 - 3q-11 + 6q-10 - 7q-8 - 3q-7 + 18q-6 - 31q-4 - 5q-3 + 57q-2 + 13q-1 - 91 - 32q + 127q2 + 79q3 - 180q4 - 119q5 + 202q6 + 200q7 - 240q8 - 255q9 + 235q10 + 335q11 - 245q12 - 371q13 + 212q14 + 420q15 - 194q16 - 427q17 + 147q18 + 427q19 - 104q20 - 404q21 + 54q22 + 362q23 - q24 - 313q25 - 35q26 + 245q27 + 67q28 - 182q29 - 76q30 + 118q31 + 76q32 - 72q33 - 57q34 + 33q35 + 41q36 - 14q37 - 24q38 + 5q39 + 12q40 - 2q41 - 4q42 - q43 + 3q44 - q45
4	q-26 - 2q-25 + q-23 - q-22 + 6q-21 - 8q-20 + 2q-19 + 4q-18 - 10q-17 + 16q-16 - 19q-15 + 13q-14 + 19q-13 - 36q-12 + 18q-11 - 51q-10 + 48q-9 + 82q-8 - 67q-7 - 14q-6 - 167q-5 + 93q-4 + 256q-3 - 21q-2 - 63q-1 - 470 + 33q + 540q2 + 242q3 + 20q4 - 978q5 - 308q6 + 771q7 + 743q8 + 410q9 - 1491q10 - 928q11 + 733q12 + 1277q13 + 1078q14 - 1769q15 - 1585q16 + 424q17 + 1605q18 + 1761q19 - 1751q20 - 2024q21 + 9q22 + 1652q23 + 2241q24 - 1513q25 - 2163q26 - 386q27 + 1455q28 + 2443q29 - 1101q30 - 2010q31 - 733q32 + 1043q33 + 2368q34 - 560q35 - 1583q36 - 968q37 + 472q38 + 1993q39 - 30q40 - 948q41 - 972q42 - 79q43 + 1368q44 + 275q45 - 318q46 - 701q47 - 373q48 + 698q49 + 273q50 + 61q51 - 330q52 - 347q53 + 242q54 + 117q55 + 134q56 - 81q57 - 179q58 + 58q59 + 12q60 + 71q61 - 4q62 - 60q63 + 17q64 - 8q65 + 20q66 + 3q67 - 15q68 + 5q69 - 3q70 + 4q71 + q72 - 3q73 + q74
5	- q-40 + 2q-39 - q-37 + q-36 - 2q-35 - 4q-34 + 6q-33 + 2q-32 - 4q-31 + 7q-30 - 2q-29 - 14q-28 + 4q-27 + 3q-26 - 5q-25 + 24q-24 + 15q-23 - 25q-22 - 25q-21 - 24q-20 - 13q-19 + 70q-18 + 88q-17 + 10q-16 - 91q-15 - 159q-14 - 110q-13 + 125q-12 + 308q-11 + 239q-10 - 84q-9 - 477q-8 - 539q-7 - 44q-6 + 664q-5 + 955q-4 + 389q-3 - 757q-2 - 1535q-1 - 1026 + 672q + 2194q2 + 1955q3 - 234q4 - 2737q5 - 3276q6 - 650q7 + 3194q8 + 4656q9 + 1975q10 - 3079q11 - 6212q12 - 3770q13 + 2753q14 + 7434q15 + 5683q16 - 1680q17 - 8495q18 - 7756q19 + 557q20 + 8976q21 + 9537q22 + 1042q23 - 9210q24 - 11129q25 - 2385q26 + 8941q27 + 12221q28 + 3907q29 - 8592q30 - 13037q31 - 4975q32 + 7900q33 + 13400q34 + 6107q35 - 7221q36 - 13563q37 - 6836q38 + 6299q39 + 13343q40 + 7612q41 - 5309q42 - 12923q43 - 8139q44 + 4116q45 + 12138q46 + 8581q47 - 2752q48 - 11056q49 - 8821q50 + 1304q51 + 9623q52 + 8749q53 + 194q54 - 7873q55 - 8382q56 - 1529q57 + 5966q58 + 7549q59 + 2610q60 - 3981q61 - 6437q62 - 3224q63 + 2174q64 + 5010q65 + 3410q66 - 688q67 - 3574q68 - 3102q69 - 327q70 + 2171q71 + 2549q72 + 879q73 - 1117q74 - 1809q75 - 992q76 + 339q77 + 1126q78 + 889q79 + 36q80 - 587q81 - 618q82 - 198q83 + 239q84 + 375q85 + 195q86 - 63q87 - 191q88 - 138q89 + 6q90 + 76q91 + 69q92 + 25q93 - 33q94 - 41q95 - 4q96 + 15q97 + 3q98 + 10q99 + q100 - 15q101 + q102 + 7q103 - 2q104 + 3q106 - 4q107 - q108 + 3q109 - q110

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, 77]]
Out[2]=	PD[X[1, 4, 2, 5], X[3, 8, 4, 9], X[13, 17, 14, 16], X[5, 15, 6, 14], > X[15, 7, 16, 6], X[9, 19, 10, 18], X[11, 1, 12, 20], X[19, 11, 20, 10], > X[17, 13, 18, 12], X[7, 2, 8, 3]]
In[3]:=	GaussCode[Knot[10, 77]]
Out[3]=	GaussCode[-1, 10, -2, 1, -4, 5, -10, 2, -6, 8, -7, 9, -3, 4, -5, 3, -9, 6, -8, > 7]
In[4]:=	DTCode[Knot[10, 77]]
Out[4]=	DTCode[4, 8, 14, 2, 18, 20, 16, 6, 12, 10]
In[5]:=	br = BR[Knot[10, 77]]
Out[5]=	BR[4, {1, 1, 1, 1, 2, -1, -3, 2, 2, -3, -3}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 77]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 77]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 77]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 77]][t]
Out[10]=	2 7 14 2 3 -17 + -- - -- + -- + 14 t - 7 t + 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 77]][z]
Out[11]=	2 4 6 1 + 4 z + 5 z + 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], > Knot[11, NonAlternating, 75]}
In[13]:=	{KnotDet[Knot[10, 77]], KnotSignature[Knot[10, 77]]}
Out[13]=	{63, 2}
In[14]:=	Jones[Knot[10, 77]][q]
Out[14]=	-2 2 2 3 4 5 6 7 8 -4 - q + - + 8 q - 9 q + 11 q - 10 q + 8 q - 6 q + 3 q - q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 77]}
In[16]:=	A2Invariant[Knot[10, 77]][q]
Out[16]=	-6 -2 2 6 8 12 14 16 18 20 22 24 -1 - q - q + 3 q + 4 q + 2 q + q - 3 q + q - q - q + q - q
In[17]:=	HOMFLYPT[Knot[10, 77]][a, z]
Out[17]=	2 2 2 4 4 4 6 -6 -4 5 2 2 z 2 z 7 z 4 z 3 z 4 z z -2 - a - a + -- - 3 z - ---- + ---- + ---- - z - -- + ---- + ---- + -- + 2 6 4 2 6 4 2 4 a a a a a a a a 6 z > -- 2 a
In[18]:=	Kauffman[Knot[10, 77]][a, z]
Out[18]=	2 2 -6 -4 5 z z z 3 z 4 z 2 2 z 2 z -2 + a - a - -- + -- - -- - -- + --- + --- + 2 a z + 4 z + ---- - ---- - 2 9 7 5 3 a 8 6 a a a a a a a 2 2 3 3 3 3 4 4 z 7 z 2 z 2 z 6 z 5 z 3 4 6 z 8 z > -- + ---- - ---- + ---- + ---- - ---- - 3 a z - 5 z - ---- + ---- - 4 2 9 7 5 a 8 4 a a a a a a a 4 5 5 5 5 5 6 6 6 3 z z 7 z 9 z 3 z z 5 6 3 z 3 z 9 z > ---- + -- - ---- - ---- - ---- - -- + a z + 2 z + ---- - ---- - ---- - 2 9 7 5 3 a 8 6 4 a a a a a a a a 6 7 7 7 7 8 8 8 9 9 z 4 z 4 z 2 z 2 z 3 z 5 z 2 z z z > -- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + -- + -- 2 7 5 3 a 6 4 2 5 3 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 77]], Vassiliev[3][Knot[10, 77]]}
Out[19]=	{4, 5}
In[20]:=	Kh[Knot[10, 77]][q, t]
Out[20]=	3 1 1 1 3 q 3 5 5 2 5 q + 4 q + ----- + ----- + ---- + --- + - + 5 q t + 4 q t + 6 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 > 5 q t + 4 q t + 6 q t + 4 q t + 4 q t + 2 q t + 4 q t + 13 6 15 6 17 7 > q t + 2 q t + q t
In[21]:=	ColouredJones[Knot[10, 77], 2][q]
Out[21]=	-7 2 5 9 -2 18 2 3 4 5 -26 + q - -- + -- - -- + q + -- - q + 47 q - 50 q - 13 q + 82 q - 6 4 3 q q q q 6 7 8 9 10 11 12 13 > 65 q - 31 q + 102 q - 63 q - 43 q + 95 q - 43 q - 44 q + 14 15 16 17 18 19 20 21 22 > 67 q - 18 q - 33 q + 33 q - 2 q - 15 q + 9 q + q - 3 q + 23 > q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1077

1076 1078