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

10104 10106

KnotPlot

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 20 18 2 8 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 - 8t-2 + 22t-1 - 29 + 22t - 8t2 + t3

Conway Polynomial:

1 - z2 - 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n163, ...}

Determinant and Signature:

{91, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 14q - 15q2 + 15q3 - 12q4 + 8q5 - 4q6 + q7

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

{...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 2q-2 + 2q2 - 3q4 + 3q6 - 2q8 + 2q10 + q12 - 2q14 + 3q16 - 2q18 - q20 + q22

HOMFLY-PT Polynomial:

a-6z2 - 2a-4z2 - 2a-4z4 + a-2 + 2a-2z2 + 2a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-8z4 - 2a-7z3 + 4a-7z5 + 3a-6z2 - 8a-6z4 + 8a-6z6 - a-5z + 6a-5z3 - 13a-5z5 + 10a-5z7 + 5a-4z2 - 9a-4z4 - 3a-4z6 + 7a-4z8 - 3a-3z + 19a-3z3 - 33a-3z5 + 13a-3z7 + 2a-3z9 - a-2 + 4a-2z2 + 2a-2z4 - 19a-2z6 + 11a-2z8 - 4a-1z + 18a-1z3 - 24a-1z5 + 6a-1z7 + 2a-1z9 - 1 + 5z2 - z4 - 7z6 + 4z8 - 2az + 7az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 0}

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 10105. 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 = 15                     1 j = 13                   3   j = 11                 5 1   j = 9               7 3     j = 7             8 5       j = 5           7 7         j = 3         7 8           j = 1       5 8             j = -1     2 6               j = -3   1 5                 j = -5   2                   j = -7 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q-10 - 3q-9 + q-8 + 11q-7 - 19q-6 - 6q-5 + 51q-4 - 44q-3 - 45q-2 + 117q-1 - 49 - 112q + 171q2 - 26q3 - 171q4 + 185q5 + 10q6 - 190q7 + 154q8 + 38q9 - 156q10 + 91q11 + 41q12 - 86q13 + 34q14 + 21q15 - 27q16 + 8q17 + 4q18 - 4q19 + q20
3	q-21 - 3q-20 + q-19 + 5q-18 + 3q-17 - 19q-16 - 9q-15 + 40q-14 + 36q-13 - 71q-12 - 93q-11 + 90q-10 + 200q-9 - 89q-8 - 329q-7 + 16q-6 + 488q-5 + 114q-4 - 622q-3 - 310q-2 + 712q-1 + 545 - 738q - 797q2 + 711q3 + 1023q4 - 623q5 - 1221q6 + 506q7 + 1365q8 - 355q9 - 1463q10 + 204q11 + 1479q12 - 30q13 - 1441q14 - 119q15 + 1307q16 + 262q17 - 1122q18 - 344q19 + 878q20 + 375q21 - 625q22 - 349q23 + 402q24 + 276q25 - 229q26 - 187q27 + 116q28 + 108q29 - 52q30 - 57q31 + 28q32 + 20q33 - 11q34 - 7q35 + 4q36 + 4q37 - 4q38 + q39
4	q-36 - 3q-35 + q-34 + 5q-33 - 3q-32 + 3q-31 - 22q-30 + 3q-29 + 42q-28 + 8q-27 + 10q-26 - 131q-25 - 62q-24 + 150q-23 + 168q-22 + 188q-21 - 401q-20 - 489q-19 + 51q-18 + 543q-17 + 1071q-16 - 357q-15 - 1372q-14 - 1016q-13 + 363q-12 + 2760q-11 + 1024q-10 - 1653q-9 - 3115q-8 - 1629q-7 + 3936q-6 + 3740q-5 + 101q-4 - 4744q-3 - 5330q-2 + 3049q-1 + 6190 + 3766q - 4430q2 - 9063q3 + 220q4 + 6975q5 + 7759q6 - 2333q7 - 11434q8 - 3204q9 + 6174q10 + 10833q11 + 428q12 - 12283q13 - 6213q14 + 4506q15 + 12649q16 + 3161q17 - 11785q18 - 8484q19 + 2218q20 + 12982q21 + 5650q22 - 9721q23 - 9553q24 - 621q25 + 11269q26 + 7270q27 - 6107q28 - 8620q29 - 3134q30 + 7567q31 + 6969q32 - 2233q33 - 5700q34 - 3927q35 + 3454q36 + 4733q37 + 91q38 - 2437q39 - 2840q40 + 863q41 + 2147q42 + 512q43 - 515q44 - 1278q45 + 47q46 + 630q47 + 199q48 + 22q49 - 373q50 - 10q51 + 129q52 + 10q53 + 40q54 - 76q55 + 7q56 + 24q57 - 12q58 + 9q59 - 11q60 + 4q61 + 4q62 - 4q63 + q64

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, 105]]
Out[2]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[20, 8, 1, 7], X[16, 5, 17, 6], > X[6, 15, 7, 16], X[10, 17, 11, 18], X[18, 9, 19, 10], X[8, 14, 9, 13], > X[14, 20, 15, 19], X[2, 12, 3, 11]]
In[3]:=	GaussCode[Knot[10, 105]]
Out[3]=	GaussCode[1, -10, 2, -1, 4, -5, 3, -8, 7, -6, 10, -2, 8, -9, 5, -4, 6, -7, 9, > -3]
In[4]:=	DTCode[Knot[10, 105]]
Out[4]=	DTCode[4, 12, 16, 20, 18, 2, 8, 6, 10, 14]
In[5]:=	br = BR[Knot[10, 105]]
Out[5]=	BR[5, {1, 1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 12}
In[7]:=	BraidIndex[Knot[10, 105]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 105]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 105]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 2, 3, 3, NotAvailable, 1}
In[10]:=	alex = Alexander[Knot[10, 105]][t]
Out[10]=	-3 8 22 2 3 -29 + t - -- + -- + 22 t - 8 t + t 2 t t
In[11]:=	Conway[Knot[10, 105]][z]
Out[11]=	2 4 6 1 - z - 2 z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 105], Knot[11, NonAlternating, 163]}
In[13]:=	{KnotDet[Knot[10, 105]], KnotSignature[Knot[10, 105]]}
Out[13]=	{91, 2}
In[14]:=	Jones[Knot[10, 105]][q]
Out[14]=	-3 3 7 2 3 4 5 6 7 -11 + q - -- + - + 14 q - 15 q + 15 q - 12 q + 8 q - 4 q + q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 105]}
In[16]:=	A2Invariant[Knot[10, 105]][q]
Out[16]=	-10 -6 3 2 2 4 6 8 10 12 14 q - q + -- - -- + 2 q - 3 q + 3 q - 2 q + 2 q + q - 2 q + 4 2 q q 16 18 20 22 > 3 q - 2 q - q + q
In[17]:=	HOMFLYPT[Knot[10, 105]][a, z]
Out[17]=	2 2 2 4 4 6 -2 2 2 z 2 z 2 z 2 2 4 2 z 2 z z -1 + a + a - 3 z + -- - ---- + ---- + a z - 2 z - ---- + ---- + -- 6 4 2 4 2 2 a a a a a a
In[18]:=	Kauffman[Knot[10, 105]][a, z]
Out[18]=	2 2 2 -2 2 z 3 z 4 z 2 3 z 5 z 4 z 2 2 -1 - a - a - -- - --- - --- - 2 a z + 5 z + ---- + ---- + ---- + 3 a z - 5 3 a 6 4 2 a a a a a 3 3 3 3 4 4 4 4 2 z 6 z 19 z 18 z 3 4 z 8 z 9 z 2 z > ---- + ---- + ----- + ----- + 7 a z - z + -- - ---- - ---- + ---- - 7 5 3 a 8 6 4 2 a a a a a a a 5 5 5 5 6 6 2 4 4 z 13 z 33 z 24 z 5 6 8 z 3 z > 3 a z + ---- - ----- - ----- - ----- - 8 a z - 7 z + ---- - ---- - 7 5 3 a 6 4 a a a a a 6 7 7 7 8 8 19 z 2 6 10 z 13 z 6 z 7 8 7 z 11 z > ----- + a z + ----- + ----- + ---- + 3 a z + 4 z + ---- + ----- + 2 5 3 a 4 2 a a a a a 9 9 2 z 2 z > ---- + ---- 3 a a
In[19]:=	{Vassiliev[2][Knot[10, 105]], Vassiliev[3][Knot[10, 105]]}
Out[19]=	{-1, 0}
In[20]:=	Kh[Knot[10, 105]][q, t]
Out[20]=	3 1 2 1 5 2 6 5 q 3 8 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 7 q t + 7 q t + 8 q t + 5 q t + 7 q t + 3 q t + 5 q t + 11 5 13 5 15 6 > q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 105], 2][q]
Out[21]=	-10 3 -8 11 19 6 51 44 45 117 2 -49 + q - -- + q + -- - -- - -- + -- - -- - -- + --- - 112 q + 171 q - 9 7 6 5 4 3 2 q q q q q q q q 3 4 5 6 7 8 9 10 > 26 q - 171 q + 185 q + 10 q - 190 q + 154 q + 38 q - 156 q + 11 12 13 14 15 16 17 18 > 91 q + 41 q - 86 q + 34 q + 21 q - 27 q + 8 q + 4 q - 19 20 > 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10105

10104 10106