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

K11n164 K11n166

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X14,5,15,6 X16,8,17,7 X20,9,21,10 X11,5,12,4 X18,13,19,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 20t-1 + 29 - 20t + 7t2 - t3

Conway Polynomial:

1 - z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1060, ...}

Determinant and Signature:

{85, 0}

Jones Polynomial:

2q-4 - 6q-3 + 10q-2 - 13q-1 + 15 - 14q + 12q2 - 8q3 + 4q4 - q5

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

{...}

A2 (sl(3)) Invariant:

q-14 + 2q-12 - 3q-10 + q-8 - q-6 - 3q-4 + 4q-2 - 1 + 4q2 - q4 - q6 + 2q8 - 3q10 + 2q12 + q14 - q16

HOMFLY-PT Polynomial:

- a-4z2 + 2a-2z2 + 2a-2z4 + 2 - z2 - 2z4 - z6 - 2a2 - a2z2 + a2z4 + a4

Kauffman Polynomial:

- a-5z3 + a-5z5 + 2a-4z2 - 6a-4z4 + 4a-4z6 + 5a-3z3 - 12a-3z5 + 7a-3z7 + 2a-2z2 - 3a-2z4 - 6a-2z6 + 6a-2z8 + a-1z + 5a-1z3 - 17a-1z5 + 8a-1z7 + 2a-1z9 + 2 - 4z2 + 5z4 - 12z6 + 9z8 + 3az - 8az3 + az5 + 2az7 + 2az9 + 2a2 - 7a2z2 + 5a2z4 - 2a2z6 + 3a2z8 + 2a3z - 7a3z3 + 5a3z5 + a3z7 + a4 - 3a4z2 + 3a4z4

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

{-1, 1}

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=0 is the signature of 11165. 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 = 11                   1 j = 9                 3   j = 7               5 1   j = 5             7 3     j = 3           7 5       j = 1         8 7         j = -1       6 8           j = -3     4 7             j = -5   2 6               j = -7   4                 j = -9 2

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, 165]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 165]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[11, 5, 12, 4], X[18, 13, 19, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 165]]
Out[4]=	GaussCode[1, -8, -2, 6, 3, -1, 4, -11, 5, 2, -6, -10, 7, -3, 8, -4, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 165]]
Out[5]=	DTCode[6, -10, 14, 16, 20, -4, 18, 2, 22, 12, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 165]][t]
Out[6]=	-3 7 20 2 3 29 - t + -- - -- - 20 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 165]][z]
Out[7]=	2 4 6 1 - z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 60], Knot[11, NonAlternating, 165]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 165]], KnotSignature[Knot[11, NonAlternating, 165]]}
Out[9]=	{85, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 165]][q]
Out[10]=	2 6 10 13 2 3 4 5 15 + -- - -- + -- - -- - 14 q + 12 q - 8 q + 4 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 165]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 165]][q]
Out[12]=	-14 2 3 -8 -6 3 4 2 4 6 8 10 -1 + q + --- - --- + q - q - -- + -- + 4 q - q - q + 2 q - 3 q + 12 10 4 2 q q q q 12 14 16 > 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 165]][a, z]
Out[13]=	2 2 4 2 4 2 z 2 z 2 2 4 2 z 2 4 6 2 - 2 a + a - z - -- + ---- - a z - 2 z + ---- + a z - z 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 165]][a, z]
Out[14]=	2 2 2 4 z 3 2 2 z 2 z 2 2 4 2 2 + 2 a + a + - + 3 a z + 2 a z - 4 z + ---- + ---- - 7 a z - 3 a z - a 4 2 a a 3 3 3 4 4 z 5 z 5 z 3 3 3 4 6 z 3 z 2 4 > -- + ---- + ---- - 8 a z - 7 a z + 5 z - ---- - ---- + 5 a z + 5 3 a 4 2 a a a a 5 5 5 6 6 4 4 z 12 z 17 z 5 3 5 6 4 z 6 z > 3 a z + -- - ----- - ----- + a z + 5 a z - 12 z + ---- - ---- - 5 3 a 4 2 a a a a 7 7 8 9 2 6 7 z 8 z 7 3 7 8 6 z 2 8 2 z > 2 a z + ---- + ---- + 2 a z + a z + 9 z + ---- + 3 a z + ---- + 3 a 2 a a a 9 > 2 a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 165]], Vassiliev[3][Knot[11, NonAlternating, 165]]}
Out[15]=	{-1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 165]][q, t]
Out[16]=	8 2 4 2 6 4 7 6 3 - + 8 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 11 5 > 5 q t + 7 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n165

K11n164 K11n166