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

K11n170 K11n172

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X3,11,4,10 X5,17,6,16 X12,8,13,7 X9,19,10,18 X11,3,12,2 X22,14,1,13 X20,16,21,15 X17,5,18,4 X19,9,20,8 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

6t-2 - 16t-1 + 21 - 16t + 6t2

Conway Polynomial:

1 + 8z2 + 6z4

Other knots with the same Alexander/Conway Polynomial:

{K11a238, ...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 9q5 + 11q6 - 11q7 + 10q8 - 7q9 + 4q10 - 2q11

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - q14 + 3q16 - q18 + 2q20 + q22 + 2q26 - 3q28 - 2q34

HOMFLY-PT Polynomial:

- 2a-10 - 2a-10z2 + a-8 + 3a-8z2 + 2a-8z4 + 2a-6 + 6a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

5a-13z - 8a-13z3 + 3a-13z5 + 3a-12z2 - 4a-12z4 + a-12z8 + 3a-11z - 7a-11z3 + a-11z5 + a-11z9 + 2a-10 - 7a-10z2 + 7a-10z4 - 9a-10z6 + 5a-10z8 - a-9z + 5a-9z3 - 12a-9z5 + 6a-9z7 + a-9z9 + a-8 - 2a-8z2 + a-8z4 - 3a-8z6 + 4a-8z8 + a-7z + 2a-7z3 - 7a-7z5 + 6a-7z7 - 2a-6 + 7a-6z2 - 9a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{8, 22}

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=4 is the signature of 11171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   2 j = 21                 2   j = 19               5 2   j = 17             5 2     j = 15           6 5       j = 13         5 5         j = 11       4 6           j = 9     3 5             j = 7     4               j = 5 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, 171]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 171]]
Out[3]=	PD[X[6, 2, 7, 1], X[3, 11, 4, 10], X[5, 17, 6, 16], X[12, 8, 13, 7], > X[9, 19, 10, 18], X[11, 3, 12, 2], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[17, 5, 18, 4], X[19, 9, 20, 8], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 171]]
Out[4]=	GaussCode[1, 6, -2, 9, -3, -1, 4, 10, -5, 2, -6, -4, 7, -11, 8, 3, -9, 5, -10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, NonAlternating, 171]]
Out[5]=	DTCode[6, -10, -16, 12, -18, -2, 22, 20, -4, -8, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 171]][t]
Out[6]=	6 16 2 21 + -- - -- - 16 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 171]][z]
Out[7]=	2 4 1 + 8 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 238], Knot[11, NonAlternating, 171]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 171]], KnotSignature[Knot[11, NonAlternating, 171]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 171]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 3 q + 7 q - 9 q + 11 q - 11 q + 10 q - 7 q + 4 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 171]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 171]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 26 28 q - 2 q + 2 q + q - q + 3 q - q + 2 q + q + 2 q - 3 q - 34 > 2 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 171]][a, z]
Out[13]=	2 2 2 2 4 4 4 -2 -8 2 2 z 3 z 6 z z 2 z 3 z z --- + a + -- - ---- + ---- + ---- + -- + ---- + ---- + -- 10 6 10 8 6 4 8 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 171]][a, z]
Out[14]=	2 2 2 2 2 3 2 -8 2 5 z 3 z z z 3 z 7 z 2 z 7 z z 8 z --- + a - -- + --- + --- - -- + -- + ---- - ---- - ---- + ---- - -- - ---- - 10 6 13 11 9 7 12 10 8 6 4 13 a a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 5 5 7 z 5 z 2 z 2 z 4 z 7 z z 9 z z 3 z z > ---- + ---- + ---- - ---- - ---- + ---- + -- - ---- + -- + ---- + --- - 11 9 7 5 12 10 8 6 4 13 11 a a a a a a a a a a a 5 5 5 6 6 6 7 7 8 8 12 z 7 z 3 z 9 z 3 z 6 z 6 z 6 z z 5 z > ----- - ---- + ---- - ---- - ---- + ---- + ---- + ---- + --- + ---- + 9 7 5 10 8 6 9 7 12 10 a a a a a a a a a a 8 9 9 4 z z z > ---- + --- + -- 8 11 9 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 171]], Vassiliev[3][Knot[11, NonAlternating, 171]]}
Out[15]=	{8, 22}
In[16]:=	Kh[Knot[11, NonAlternating, 171]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 5 q t + 4 q t + 6 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 5 q t + 5 q t + 6 q t + 5 q t + 5 q t + 2 q t + 19 7 19 8 21 8 23 9 > 5 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n171

K11n170 K11n172