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

K11n166 K11n168

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X11,18,12,19 X4,14,5,13 X22,15,1,16 X17,12,18,13 X8,20,9,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 5t-2 + 15t-1 - 21 + 15t - 5t2 + t3

Conway Polynomial:

1 + 4z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{K11n146, ...}

Determinant and Signature:

{63, 2}

Jones Polynomial:

q-1 - 4 + 7q - 9q2 + 12q3 - 10q4 + 9q5 - 7q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-4 - q-2 - 2 + 2q2 - 2q4 + 2q6 + 2q8 + q10 + 4q12 - q14 + 2q16 - 2q18 - 3q20 + q22 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 + a-8z2 - 5a-8z4 + 3a-8z6 - 4a-7z + 6a-7z3 - 10a-7z5 + 5a-7z7 + 3a-6 - 4a-6z2 + 3a-6z4 - 6a-6z6 + 4a-6z8 - 8a-5z + 16a-5z3 - 16a-5z5 + 6a-5z7 + a-5z9 + 5a-4 - 5a-4z2 + 7a-4z4 - 8a-4z6 + 5a-4z8 - 4a-3z + 4a-3z3 - a-3z5 + a-3z7 + a-3z9 + a-2 - a-2z2 + a-2z6 + a-2z8 - a-1z - 4a-1z3 + 4a-1z5 - z2 + z4

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

{4, 7}

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 11167. 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 j = 17                   1 j = 15                 2   j = 13               5 1   j = 11             4 2     j = 9           6 5       j = 7         6 4         j = 5       3 6           j = 3     4 6             j = 1   1 4               j = -1   3                 j = -3 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, 167]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 167]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[11, 18, 12, 19], X[4, 14, 5, 13], X[22, 15, 1, 16], > X[17, 12, 18, 13], X[8, 20, 9, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 167]]
Out[4]=	GaussCode[1, -5, 2, -7, 3, -1, 4, -10, 5, -2, -6, 9, 7, -3, 8, -11, -9, 6, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 167]]
Out[5]=	DTCode[6, 10, 14, 20, 2, -18, 4, 22, -12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 167]][t]
Out[6]=	-3 5 15 2 3 -21 + t - -- + -- + 15 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 167]][z]
Out[7]=	2 4 6 1 + 4 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 146], Knot[11, NonAlternating, 167]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 167]], KnotSignature[Knot[11, NonAlternating, 167]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 167]][q]
Out[10]=	1 2 3 4 5 6 7 8 -4 + - + 7 q - 9 q + 12 q - 10 q + 9 q - 7 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 167]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 167]][q]
Out[12]=	-4 -2 2 4 6 8 10 12 14 16 -2 + q - q + 2 q - 2 q + 2 q + 2 q + q + 4 q - q + 2 q - 18 20 22 24 > 2 q - 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 167]][a, z]
Out[13]=	2 2 2 4 4 4 6 -3 5 -2 2 2 z 8 z 3 z z 4 z 2 z z -- + -- - a + z - ---- + ---- - ---- - -- + ---- - ---- + -- 6 4 6 4 2 6 4 2 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 167]][a, z]
Out[14]=	2 2 2 2 3 3 5 -2 z 4 z 8 z 4 z z 2 z 4 z 5 z z 2 z -- + -- + a + -- - --- - --- - --- - - - z + -- - ---- - ---- - -- - ---- + 6 4 9 7 5 3 a 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 5 5 5 6 z 16 z 4 z 4 z 4 5 z 3 z 7 z z 10 z 16 z > ---- + ----- + ---- - ---- + z - ---- + ---- + ---- + -- - ----- - ----- - 7 5 3 a 8 6 4 9 7 5 a a a a a a a a a 5 5 6 6 6 6 7 7 7 8 8 8 z 4 z 3 z 6 z 8 z z 5 z 6 z z 4 z 5 z z > -- + ---- + ---- - ---- - ---- + -- + ---- + ---- + -- + ---- + ---- + -- + 3 a 8 6 4 2 7 5 3 6 4 2 a a a a a a a a a a a 9 9 z z > -- + -- 5 3 a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 167]], Vassiliev[3][Knot[11, NonAlternating, 167]]}
Out[15]=	{4, 7}
In[16]:=	Kh[Knot[11, NonAlternating, 167]][q, t]
Out[16]=	3 1 3 q 3 5 5 2 7 2 7 3 4 q + 4 q + ----- + --- + - + 6 q t + 3 q t + 6 q t + 6 q t + 4 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 6 q t + 5 q t + 4 q t + 2 q t + 5 q t + q t + 2 q t + 17 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n167

K11n166 K11n168