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1091 1093

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 2 16 8 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 2 3 3 / NotAvailable 1

Alexander Polynomial:

- 2t-3 + 10t-2 - 20t-1 + 25 - 20t + 10t2 - 2t3

Conway Polynomial:

1 + 2z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a153, K11a224, K11n35, K11n43, ...}

Determinant and Signature:

{89, 4}

Jones Polynomial:

1 - 3q + 7q2 - 10q3 + 14q4 - 15q5 + 14q6 - 12q7 + 8q8 - 4q9 + q10

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 2q6 - 2q8 + 4q10 - q12 + q14 + q16 - 3q18 + 2q20 - 3q22 + q24 + q26 - 2q28 + q30

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-12z4 - 2a-11z3 + 4a-11z5 + 2a-10z2 - 8a-10z4 + 8a-10z6 - a-9z + 7a-9z3 - 14a-9z5 + 10a-9z7 + 2a-8z2 - 4a-8z4 - 5a-8z6 + 7a-8z8 - 5a-7z + 21a-7z3 - 32a-7z5 + 12a-7z7 + 2a-7z9 + a-6 - 2a-6z2 + 10a-6z4 - 22a-6z6 + 11a-6z8 - 5a-5z + 18a-5z3 - 22a-5z5 + 5a-5z7 + 2a-5z9 + a-4 + a-4z2 + 2a-4z4 - 8a-4z6 + 4a-4z8 - a-3z + 6a-3z3 - 8a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{2, 3}

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 1092. 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                   3   j = 17                 5 1   j = 15               7 3     j = 13             7 5       j = 11           8 7         j = 9         6 7           j = 7       4 8             j = 5     3 6               j = 3   1 5                 j = 1   2                   j = -1 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-2 - 3q-1 + 1 + 11q - 18q2 - 6q3 + 48q4 - 41q5 - 41q6 + 109q7 - 47q8 - 103q9 + 161q10 - 28q11 - 159q12 + 176q13 + 5q14 - 178q15 + 148q16 + 32q17 - 147q18 + 90q19 + 36q20 - 83q21 + 35q22 + 20q23 - 27q24 + 8q25 + 4q26 - 4q27 + q28
3	q-6 - 3q-5 + q-4 + 5q-3 + 3q-2 - 18q-1 - 9 + 37q + 34q2 - 65q3 - 83q4 + 81q5 + 178q6 - 84q7 - 289q8 + 23q9 + 435q10 + 78q11 - 550q12 - 254q13 + 651q14 + 446q15 - 675q16 - 680q17 + 676q18 + 871q19 - 598q20 - 1065q21 + 510q22 + 1197q23 - 383q24 - 1290q25 + 247q26 + 1320q27 - 101q28 - 1285q29 - 44q30 + 1187q31 + 162q32 - 1017q33 - 255q34 + 815q35 + 289q36 - 587q37 - 284q38 + 390q39 + 230q40 - 228q41 - 163q42 + 119q43 + 98q44 - 55q45 - 53q46 + 29q47 + 19q48 - 11q49 - 7q50 + 4q51 + 4q52 - 4q53 + q54
4	q-12 - 3q-11 + q-10 + 5q-9 - 3q-8 + 3q-7 - 21q-6 + 3q-5 + 39q-4 + 6q-3 + 11q-2 - 120q-1 - 53 + 137q + 141q2 + 164q3 - 361q4 - 411q5 + 73q6 + 460q7 + 903q8 - 369q9 - 1170q10 - 773q11 + 370q12 + 2348q13 + 692q14 - 1515q15 - 2522q16 - 1173q17 + 3490q18 + 2925q19 - 240q20 - 4038q21 - 4213q22 + 3016q23 + 5105q24 + 2702q25 - 4042q26 - 7464q27 + 862q28 + 6014q29 + 6095q30 - 2514q31 - 9709q32 - 1940q33 + 5589q34 + 8837q35 - 320q36 - 10679q37 - 4508q38 + 4362q39 + 10549q40 + 1939q41 - 10458q42 - 6525q43 + 2543q44 + 11021q45 + 4062q46 - 8868q47 - 7594q48 + 180q49 + 9769q50 + 5562q51 - 5888q52 - 7049q53 - 2052q54 + 6790q55 + 5569q56 - 2531q57 - 4828q58 - 2978q59 + 3315q60 + 3950q61 - 322q62 - 2175q63 - 2318q64 + 969q65 + 1884q66 + 286q67 - 506q68 - 1113q69 + 119q70 + 583q71 + 139q72 + 8q73 - 342q74 + 6q75 + 122q76 + 2q77 + 38q78 - 72q79 + 8q80 + 23q81 - 12q82 + 9q83 - 11q84 + 4q85 + 4q86 - 4q87 + 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, 92]]
Out[2]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 16, 1, 15], > X[16, 12, 17, 11], X[18, 7, 19, 8], X[12, 18, 13, 17], X[6, 19, 7, 20], > X[8, 14, 9, 13], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 92]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -8, 6, -9, 10, -2, 5, -7, 9, -3, 4, -5, 7, -6, 8, > -4]
In[4]:=	DTCode[Knot[10, 92]]
Out[4]=	DTCode[4, 10, 14, 18, 2, 16, 8, 20, 12, 6]
In[5]:=	br = BR[Knot[10, 92]]
Out[5]=	BR[4, {1, 1, 1, 2, 2, -3, 2, -1, 2, -3, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 92]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 92]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 92]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 92]][t]
Out[10]=	2 10 20 2 3 25 - -- + -- - -- - 20 t + 10 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 92]][z]
Out[11]=	2 4 6 1 + 2 z - 2 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 92], Knot[11, Alternating, 153], Knot[11, Alternating, 224], > Knot[11, NonAlternating, 35], Knot[11, NonAlternating, 43]}
In[13]:=	{KnotDet[Knot[10, 92]], KnotSignature[Knot[10, 92]]}
Out[13]=	{89, 4}
In[14]:=	Jones[Knot[10, 92]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 10 q + 14 q - 15 q + 14 q - 12 q + 8 q - 4 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 92]}
In[16]:=	A2Invariant[Knot[10, 92]][q]
Out[16]=	2 4 6 8 10 12 14 16 18 20 22 1 - q + q + 2 q - 2 q + 4 q - q + q + q - 3 q + 2 q - 3 q + 24 26 28 30 > q + q - 2 q + q
In[17]:=	HOMFLYPT[Knot[10, 92]][a, z]
Out[17]=	2 2 2 4 4 4 4 6 6 -6 -4 -2 z z 2 z z 2 z 2 z z z z -a + a + a + -- - -- + ---- + -- - ---- - ---- + -- - -- - -- 8 6 2 8 6 4 2 6 4 a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 92]][a, z]
Out[18]=	2 2 2 2 2 3 -6 -4 -2 z 5 z 5 z z 2 z 2 z 2 z z 3 z 2 z a + a - a - -- - --- - --- - -- + ---- + ---- - ---- + -- + ---- - ---- + 9 7 5 3 10 8 6 4 2 11 a a a a a a a a a a 3 3 3 3 4 4 4 4 4 4 7 z 21 z 18 z 6 z z 8 z 4 z 10 z 2 z 3 z > ---- + ----- + ----- + ---- + --- - ---- - ---- + ----- + ---- - ---- + 9 7 5 3 12 10 8 6 4 2 a a a a a a a a a a 5 5 5 5 5 6 6 6 6 6 4 z 14 z 32 z 22 z 8 z 8 z 5 z 22 z 8 z z > ---- - ----- - ----- - ----- - ---- + ---- - ---- - ----- - ---- + -- + 11 9 7 5 3 10 8 6 4 2 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 10 z 12 z 5 z 3 z 7 z 11 z 4 z 2 z 2 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, 92]], Vassiliev[3][Knot[10, 92]]}
Out[19]=	{2, 3}
In[20]:=	Kh[Knot[10, 92]][q, t]
Out[20]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 5 q + 3 q + ---- + --- + -- + 6 q t + 4 q t + 8 q t + 6 q t + 7 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 8 q t + 7 q t + 7 q t + 5 q t + 7 q t + 3 q t + 17 6 17 7 19 7 21 8 > 5 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 92], 2][q]
Out[21]=	-2 3 2 3 4 5 6 7 8 1 + q - - + 11 q - 18 q - 6 q + 48 q - 41 q - 41 q + 109 q - 47 q - q 9 10 11 12 13 14 15 16 > 103 q + 161 q - 28 q - 159 q + 176 q + 5 q - 178 q + 148 q + 17 18 19 20 21 22 23 24 > 32 q - 147 q + 90 q + 36 q - 83 q + 35 q + 20 q - 27 q + 25 26 27 28 > 8 q + 4 q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 1092

1091 1093