© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n165 K11n167

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X5,15,6,14 X18,7,19,8 X2,10,3,9 X20,11,21,12 X22,14,1,13 X15,5,16,4 X12,17,13,18 X8,19,9,20 X16,22,17,21

Gauss Code:

{1, -5, 2, 8, -3, -1, 4, -10, 5, -2, 6, -9, 7, 3, -8, -11, 9, -4, 10, -6, 11, -7}

DT (Dowker-Thistlethwaite) Code:

6 10 -14 18 2 20 22 -4 12 8 16

Alexander Polynomial:

- t-4 + 4t-3 - 8t-2 + 11t-1 - 11 + 11t - 8t2 + 4t3 - t4

Conway Polynomial:

1 - z2 - 4z4 - 4z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{59, 2}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 8 + 10q - 10q2 + 9q3 - 7q4 + 4q5 - q6

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

{K11n124, ...}

A2 (sl(3)) Invariant:

q-8 - q-6 + 2q-4 + 1 + q2 - 3q4 + 2q6 - 3q8 + 2q10 + 2q16 - q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 4a-4 + 8a-4z2 + 5a-4z4 + a-4z6 - 5a-2 - 13a-2z2 - 13a-2z4 - 6a-2z6 - a-2z8 + 3 + 5z2 + 4z4 + z6

Kauffman Polynomial:

- a-7z + a-7z3 + a-6 - 3a-6z2 + 4a-6z4 - 3a-5z + 7a-5z3 - 3a-5z5 + 2a-5z7 + 4a-4 - 17a-4z2 + 26a-4z4 - 14a-4z6 + 4a-4z8 - 3a-3z + 7a-3z3 - 3a-3z5 - 2a-3z7 + 2a-3z9 + 5a-2 - 23a-2z2 + 35a-2z4 - 27a-2z6 + 8a-2z8 - 2a-1z + 7a-1z3 - 9a-1z5 - a-1z7 + 2a-1z9 + 3 - 7z2 + 10z4 - 12z6 + 4z8 - az + 6az3 - 9az5 + 3az7 + 2a2z2 - 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 11166. 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 j = 13                   1 j = 11                 3   j = 9               4 1   j = 7             5 3     j = 5           5 4       j = 3         5 5         j = 1       4 6           j = -1     2 4             j = -3   1 4               j = -5   2                 j = -7 1

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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 166]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 166]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[5, 15, 6, 14], X[18, 7, 19, 8], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[22, 14, 1, 13], X[15, 5, 16, 4], > X[12, 17, 13, 18], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 166]]
Out[4]=	GaussCode[1, -5, 2, 8, -3, -1, 4, -10, 5, -2, 6, -9, 7, 3, -8, -11, 9, -4, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 166]]
Out[5]=	DTCode[6, 10, -14, 18, 2, 20, 22, -4, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 166]][t]
Out[6]=	-4 4 8 11 2 3 4 -11 - t + -- - -- + -- + 11 t - 8 t + 4 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 166]][z]
Out[7]=	2 4 6 8 1 - z - 4 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 166]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 166]], KnotSignature[Knot[11, NonAlternating, 166]]}
Out[9]=	{59, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 166]][q]
Out[10]=	-3 3 6 2 3 4 5 6 -8 + q - -- + - + 10 q - 10 q + 9 q - 7 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 124], Knot[11, NonAlternating, 166]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 166]][q]
Out[12]=	-8 -6 2 2 4 6 8 10 16 18 20 22 1 + q - q + -- + q - 3 q + 2 q - 3 q + 2 q + 2 q - q + q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 166]][a, z]
Out[13]=	2 2 2 4 4 6 -6 4 5 2 z 8 z 13 z 4 5 z 13 z 6 z 3 - a + -- - -- + 5 z - -- + ---- - ----- + 4 z + ---- - ----- + z + -- - 4 2 6 4 2 4 2 4 a a a a a a a a 6 8 6 z z > ---- - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 166]][a, z]
Out[14]=	2 2 2 -6 4 5 z 3 z 3 z 2 z 2 3 z 17 z 23 z 3 + a + -- + -- - -- - --- - --- - --- - a z - 7 z - ---- - ----- - ----- + 4 2 7 5 3 a 6 4 2 a a a a a a a a 3 3 3 3 4 4 4 2 2 z 7 z 7 z 7 z 3 4 4 z 26 z 35 z > 2 a z + -- + ---- + ---- + ---- + 6 a z + 10 z + ---- + ----- + ----- - 7 5 3 a 6 4 2 a a a a a a 5 5 5 6 6 2 4 3 z 3 z 9 z 5 6 14 z 27 z 2 6 > 3 a z - ---- - ---- - ---- - 9 a z - 12 z - ----- - ----- + a z + 5 3 a 4 2 a a a a 7 7 7 8 8 9 9 2 z 2 z z 7 8 4 z 8 z 2 z 2 z > ---- - ---- - -- + 3 a z + 4 z + ---- + ---- + ---- + ---- 5 3 a 4 2 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 166]], Vassiliev[3][Knot[11, NonAlternating, 166]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 166]][q, t]
Out[16]=	3 1 2 1 4 2 4 4 q 3 6 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 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 13 5 > 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n166

K11n165 K11n167