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

K11a164 K11a166

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X18,6,19,5 X16,8,17,7 X2,10,3,9 X20,11,21,12 X22,13,1,14 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, -6, 11, -7}

DT (Dowker-Thistlethwaite) Code:

4 10 18 16 2 20 22 8 6 14 12

Alexander Polynomial:

- 2t-3 + 9t-2 - 18t-1 + 23 - 18t + 9t2 - 2t3

Conway Polynomial:

1 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{1087, 1098, K11a58, K11n72, ...}

Determinant and Signature:

{81, 0}

Jones Polynomial:

q-4 - 3q-3 + 6q-2 - 9q-1 + 12 - 13q + 12q2 - 10q3 + 8q4 - 4q5 + 2q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - q-10 + q-8 + q-6 - 2q-4 + 3q-2 - 1 - q4 - 3q6 + 2q8 + 3q12 + 3q14 - q16 - q20 - q22

HOMFLY-PT Polynomial:

- 2a-6 - a-6z2 + 5a-4 + 6a-4z2 + 2a-4z4 - 3a-2 - 4a-2z2 - 3a-2z4 - a-2z6 - 3z2 - 3z4 - z6 + a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- 2a-7z + 7a-7z3 - 5a-7z5 + a-7z7 + 2a-6 - 8a-6z2 + 13a-6z4 - 9a-6z6 + 2a-6z8 - 2a-5z + 4a-5z3 + 2a-5z5 - 6a-5z7 + 2a-5z9 + 5a-4 - 18a-4z2 + 21a-4z4 - 13a-4z6 + a-4z8 + a-4z10 - 4a-3z3 + 11a-3z5 - 14a-3z7 + 5a-3z9 + 3a-2 - 10a-2z2 + 15a-2z4 - 15a-2z6 + 4a-2z8 + a-2z10 - 2a-1z + 9a-1z3 - 8a-1z5 - a-1z7 + 3a-1z9 + 4z2 - 6z6 + 5z8 - 2az + 7az3 - 9az5 + 6az7 - a2 + 3a2z2 - 6a2z4 + 5a2z6 - 3a3z3 + 3a3z5 - a4z2 + a4z4

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

{0, 3}

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 r = 6 r = 7 j = 15                       1 j = 13                     1   j = 11                   3 1   j = 9                 5 1     j = 7               5 3       j = 5             7 5         j = 3           6 5           j = 1         6 7             j = -1       4 7               j = -3     2 5                 j = -5   1 4                   j = -7   2                     j = -9 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, 165]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 165]]
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[20, 11, 21, 12], X[22, 13, 1, 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, 165]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -8, 5, -2, 6, -11, 7, -10, 8, -4, 9, -3, 10, > -6, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 165]]
Out[5]=	DTCode[4, 10, 18, 16, 2, 20, 22, 8, 6, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 165]][t]
Out[6]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 165]][z]
Out[7]=	4 6 1 - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[9]:=	{KnotDet[Knot[11, Alternating, 165]], KnotSignature[Knot[11, Alternating, 165]]}
Out[9]=	{81, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 165]][q]
Out[10]=	-4 3 6 9 2 3 4 5 6 7 12 + q - -- + -- - - - 13 q + 12 q - 10 q + 8 q - 4 q + 2 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 165]}
In[12]:=	A2Invariant[Knot[11, Alternating, 165]][q]
Out[12]=	-12 -10 -8 -6 2 3 4 6 8 12 14 -1 + q - q + q + q - -- + -- - q - 3 q + 2 q + 3 q + 3 q - 4 2 q q 16 20 22 > q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 165]][a, z]
Out[13]=	2 2 2 4 4 -2 5 3 2 2 z 6 z 4 z 2 2 4 2 z 3 z -- + -- - -- + a - 3 z - -- + ---- - ---- + 2 a z - 3 z + ---- - ---- + 6 4 2 6 4 2 4 2 a a a a a a a a 6 2 4 6 z > a z - z - -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 165]][a, z]
Out[14]=	2 2 2 2 5 3 2 2 z 2 z 2 z 2 8 z 18 z 10 z -- + -- + -- - a - --- - --- - --- - 2 a z + 4 z - ---- - ----- - ----- + 6 4 2 7 5 a 6 4 2 a a a a a a a a 3 3 3 3 4 2 2 4 2 7 z 4 z 4 z 9 z 3 3 3 13 z > 3 a z - a z + ---- + ---- - ---- + ---- + 7 a z - 3 a z + ----- + 7 5 3 a 6 a a a a 4 4 5 5 5 5 21 z 15 z 2 4 4 4 5 z 2 z 11 z 8 z 5 > ----- + ----- - 6 a z + a z - ---- + ---- + ----- - ---- - 9 a z + 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 7 3 5 6 9 z 13 z 15 z 2 6 z 6 z 14 z z > 3 a z - 6 z - ---- - ----- - ----- + 5 a z + -- - ---- - ----- - -- + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 7 8 2 z z 4 z 2 z 5 z 3 z z z > 6 a z + 5 z + ---- + -- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 165]], Vassiliev[3][Knot[11, Alternating, 165]]}
Out[15]=	{0, 3}
In[16]:=	Kh[Knot[11, Alternating, 165]][q, t]
Out[16]=	7 1 2 1 4 2 5 4 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 7 q t + 6 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 9 5 > 5 q t + 7 q t + 5 q t + 5 q t + 3 q t + 5 q t + q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a165

K11a164 K11a166