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

K11a165 K11a167

Knotscape

This page is passe. Go here instead!    The Knot K11a166 Visit K11a166's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X18,6,19,5 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X8,16,9,15 X6,18,7,17 X14,19,15,20 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 16 2 22 20 8 6 14 12

Alexander Polynomial:

- 4t-2 + 15t-1 - 21 + 15t - 4t2

Conway Polynomial:

1 - z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{1038, ...}

Determinant and Signature:

{59, 2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

q-10 + 2q-4 - q-2 + q2 - 2q4 + q6 - q8 - q14 + 3q16 + q18 + q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + 2a-6 + 2a-6z2 - a-4z4 - a-2 - 3a-2z2 - 2a-2z4 - z2 - z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 3a-9z3 + a-9z5 - a-8 + 4a-8z2 - 6a-8z4 + 2a-8z6 + a-7z - 4a-7z5 + 2a-7z7 - 2a-6 + 6a-6z2 - 4a-6z6 + 2a-6z8 + 2a-5z - 6a-5z3 + 12a-5z5 - 7a-5z7 + 2a-5z9 - 4a-4z2 + 8a-4z4 - 2a-4z8 + a-4z10 + 6a-3z - 26a-3z3 + 35a-3z5 - 18a-3z7 + 4a-3z9 + a-2 - 8a-2z2 + 6a-2z4 - 2a-2z8 + a-2z10 + 4a-1z - 12a-1z3 + 11a-1z5 - 7a-1z7 + 2a-1z9 + 2z2 - 5z6 + 2z8 + 5az3 - 7az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

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

{-1, 2}

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 r = 6 r = 7 j = 17                       1 j = 15                     1   j = 13                   2 1   j = 11                 4 1     j = 9               4 2       j = 7             5 4         j = 5           4 4           j = 3         4 5             j = 1       3 5               j = -1     1 3                 j = -3   1 3                   j = -5   1                     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, Alternating, 166]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 166]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 6, 19, 5], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 13, 21, 14], X[8, 16, 9, 15], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 166]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -4, 9, -3, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 166]]
Out[5]=	DTCode[4, 10, 18, 16, 2, 22, 20, 8, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 166]][t]
Out[6]=	4 15 2 -21 - -- + -- + 15 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 166]][z]
Out[7]=	2 4 1 - z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 38], Knot[11, Alternating, 166]}
In[9]:=	{KnotDet[Knot[11, Alternating, 166]], KnotSignature[Knot[11, Alternating, 166]]}
Out[9]=	{59, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 166]][q]
Out[10]=	-3 2 4 2 3 4 5 6 7 8 -6 + q - -- + - + 8 q - 9 q + 9 q - 8 q + 6 q - 3 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 166]}
In[12]:=	A2Invariant[Knot[11, Alternating, 166]][q]
Out[12]=	-10 2 -2 2 4 6 8 14 16 18 22 24 26 q + -- - q + q - 2 q + q - q - q + 3 q + q + q - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 166]][a, z]
Out[13]=	2 2 4 4 -8 2 -2 2 2 2 z 3 z 2 2 4 z 2 z -a + -- - a + a - z + ---- - ---- + a z - z - -- - ---- 6 6 2 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 166]][a, z]
Out[14]=	2 2 2 -8 2 -2 2 z z 2 z 6 z 4 z 2 4 z 6 z 4 z -a - -- + a - a + -- + -- + --- + --- + --- + 2 z + ---- + ---- - ---- - 6 9 7 5 3 a 8 6 4 a a a a a a a a 2 3 3 3 3 4 4 8 z 2 2 3 z 6 z 26 z 12 z 3 6 z 8 z > ---- + 4 a z - ---- - ---- - ----- - ----- + 5 a z - ---- + ---- + 2 9 5 3 a 8 4 a a a a a a 4 5 5 5 5 5 6 6 z 2 4 z 4 z 12 z 35 z 11 z 5 6 2 z > ---- - 4 a z + -- - ---- + ----- + ----- + ----- - 7 a z - 5 z + ---- - 2 9 7 5 3 a 8 a a a a a a 6 7 7 7 7 8 8 4 z 2 6 2 z 7 z 18 z 7 z 7 8 2 z 2 z > ---- + a z + ---- - ---- - ----- - ---- + 2 a z + 2 z + ---- - ---- - 6 7 5 3 a 6 4 a a a a a a 8 9 9 9 10 10 2 z 2 z 4 z 2 z z z > ---- + ---- + ---- + ---- + --- + --- 2 5 3 a 4 2 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 166]], Vassiliev[3][Knot[11, Alternating, 166]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 166]][q, t]
Out[16]=	3 1 1 1 3 1 3 3 q 3 5 q + 4 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 > 4 q t + 4 q t + 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + 11 5 13 5 13 6 15 6 17 7 > q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a166

K11a165 K11a167