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10100 10102

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KnotPlot

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 2 16 6 20 8 12

Minimum Braid Representative: Length is 14, 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 2 3 / NotAvailable 2

Alexander Polynomial:

7t-2 - 21t-1 + 29 - 21t + 7t2

Conway Polynomial:

1 + 7z2 + 7z4

Other knots with the same Alexander/Conway Polynomial:

{K11a200, ...}

Determinant and Signature:

{85, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 10q5 + 14q6 - 14q7 + 13q8 - 11q9 + 7q10 - 4q11 + q12

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - 2q14 + 4q16 + 2q20 + 2q22 - q24 + 2q26 - 4q28 - q30 - 3q34 + q36 + q38

HOMFLY-PT Polynomial:

a-12 - 4a-10 - 4a-10z2 + 2a-8 + 5a-8z2 + 3a-8z4 + 2a-6 + 5a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

a-14z2 - 2a-14z4 + a-14z6 - a-13z + 8a-13z3 - 11a-13z5 + 4a-13z7 + a-12 + a-12z2 + 3a-12z4 - 11a-12z6 + 5a-12z8 - 9a-11z + 28a-11z3 - 31a-11z5 + 7a-11z7 + 2a-11z9 + 4a-10 - 9a-10z2 + 15a-10z4 - 24a-10z6 + 11a-10z8 - 8a-9z + 26a-9z3 - 31a-9z5 + 10a-9z7 + 2a-9z9 + 2a-8 - a-8z2 + a-8z4 - 6a-8z6 + 6a-8z8 + 4a-7z3 - 8a-7z5 + 7a-7z7 - 2a-6 + 7a-6z2 - 8a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{7, 17}

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 10101. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 j = 25                     1 j = 23                   3   j = 21                 4 1   j = 19               7 3     j = 17             6 4       j = 15           8 7         j = 13         6 6           j = 11       4 8             j = 9     3 6               j = 7     4                 j = 5 1 3                   j = 3 1

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q4 - 3q5 + 3q6 + 6q7 - 18q8 + 15q9 + 20q10 - 59q11 + 39q12 + 55q13 - 120q14 + 50q15 + 104q16 - 161q17 + 35q18 + 137q19 - 156q20 + 3q21 + 138q22 - 114q23 - 26q24 + 107q25 - 56q26 - 35q27 + 56q28 - 12q29 - 21q30 + 15q31 + q32 - 4q33 + q34
3	q6 - 3q7 + 3q8 + 2q9 - 2q10 - 8q11 + 11q12 + 10q13 - 24q14 - 16q15 + 60q16 + 22q17 - 110q18 - 62q19 + 209q20 + 113q21 - 301q22 - 235q23 + 416q24 + 375q25 - 478q26 - 568q27 + 524q28 + 723q29 - 479q30 - 889q31 + 426q32 + 974q33 - 310q34 - 1032q35 + 192q36 + 1025q37 - 55q38 - 981q39 - 84q40 + 902q41 + 207q42 - 775q43 - 322q44 + 630q45 + 388q46 - 448q47 - 424q48 + 286q49 + 385q50 - 123q51 - 322q52 + 19q53 + 224q54 + 43q55 - 135q56 - 54q57 + 60q58 + 47q59 - 24q60 - 24q61 + 5q62 + 9q63 + q64 - 4q65 + q66

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, 101]]
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[12, 20, 13, 19], X[18, 8, 19, 7], X[6, 14, 7, 13], > X[8, 18, 9, 17], X[2, 10, 3, 9]]
In[3]:=	GaussCode[Knot[10, 101]]
Out[3]=	GaussCode[1, -10, 2, -1, 3, -8, 7, -9, 10, -2, 5, -6, 8, -3, 4, -5, 9, -7, 6, > -4]
In[4]:=	DTCode[Knot[10, 101]]
Out[4]=	DTCode[4, 10, 14, 18, 2, 16, 6, 20, 8, 12]
In[5]:=	br = BR[Knot[10, 101]]
Out[5]=	BR[5, {1, 1, 1, 2, -1, 3, -2, 1, 3, 2, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 14}
In[7]:=	BraidIndex[Knot[10, 101]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[10, 101]]]
Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 101]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, {2, 3}, 2, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 101]][t]
Out[10]=	7 21 2 29 + -- - -- - 21 t + 7 t 2 t t
In[11]:=	Conway[Knot[10, 101]][z]
Out[11]=	2 4 1 + 7 z + 7 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 101], Knot[11, Alternating, 200]}
In[13]:=	{KnotDet[Knot[10, 101]], KnotSignature[Knot[10, 101]]}
Out[13]=	{85, 4}
In[14]:=	Jones[Knot[10, 101]][q]
Out[14]=	2 3 4 5 6 7 8 9 10 11 12 q - 3 q + 7 q - 10 q + 14 q - 14 q + 13 q - 11 q + 7 q - 4 q + q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 101]}
In[16]:=	A2Invariant[Knot[10, 101]][q]
Out[16]=	6 8 10 12 14 16 20 22 24 26 28 q - 2 q + 2 q + q - 2 q + 4 q + 2 q + 2 q - q + 2 q - 4 q - 30 34 36 38 > q - 3 q + q + q
In[17]:=	HOMFLYPT[Knot[10, 101]][a, z]
Out[17]=	2 2 2 2 4 4 4 -12 4 2 2 4 z 5 z 5 z z 3 z 3 z z a - --- + -- + -- - ---- + ---- + ---- + -- + ---- + ---- + -- 10 8 6 10 8 6 4 8 6 4 a a a a a a a a a a
In[18]:=	Kauffman[Knot[10, 101]][a, z]
Out[18]=	2 2 2 2 2 2 -12 4 2 2 z 9 z 8 z z z 9 z z 7 z z a + --- + -- - -- - --- - --- - --- + --- + --- - ---- - -- + ---- - -- + 10 8 6 13 11 9 14 12 10 8 6 4 a a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 4 8 z 28 z 26 z 4 z 2 z 2 z 3 z 15 z z 8 z z > ---- + ----- + ----- + ---- - ---- - ---- + ---- + ----- + -- - ---- + -- - 13 11 9 7 5 14 12 10 8 6 4 a a a a a a a a a a a 5 5 5 5 5 6 6 6 6 6 11 z 31 z 31 z 8 z 3 z z 11 z 24 z 6 z 6 z > ----- - ----- - ----- - ---- + ---- + --- - ----- - ----- - ---- + ---- + 13 11 9 7 5 14 12 10 8 6 a a a a a a a a a a 7 7 7 7 8 8 8 9 9 4 z 7 z 10 z 7 z 5 z 11 z 6 z 2 z 2 z > ---- + ---- + ----- + ---- + ---- + ----- + ---- + ---- + ---- 13 11 9 7 12 10 8 11 9 a a a a a a a a a
In[19]:=	{Vassiliev[2][Knot[10, 101]], Vassiliev[3][Knot[10, 101]]}
Out[19]=	{7, 17}
In[20]:=	Kh[Knot[10, 101]][q, t]
Out[20]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 6 q t + 4 q t + 8 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 6 q t + 6 q t + 8 q t + 7 q t + 6 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 25 10 > 7 q t + 3 q t + 4 q t + q t + 3 q t + q t
In[21]:=	ColouredJones[Knot[10, 101], 2][q]
Out[21]=	4 5 6 7 8 9 10 11 12 13 q - 3 q + 3 q + 6 q - 18 q + 15 q + 20 q - 59 q + 39 q + 55 q - 14 15 16 17 18 19 20 21 > 120 q + 50 q + 104 q - 161 q + 35 q + 137 q - 156 q + 3 q + 22 23 24 25 26 27 28 29 > 138 q - 114 q - 26 q + 107 q - 56 q - 35 q + 56 q - 12 q - 30 31 32 33 34 > 21 q + 15 q + q - 4 q + q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10101

10100 10102