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

K11n168 K11n170

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 X20,14,21,13 X22,16,1,15 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, -7, 11, -8}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-3 - 6t-2 + 6t-1 - 5 + 6t - 6t2 + 3t3

Conway Polynomial:

1 + 9z2 + 12z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{35, 6}

Jones Polynomial:

q3 - 2q4 + 4q5 - 4q6 + 6q7 - 6q8 + 5q9 - 4q10 + 2q11 - q12

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + q14 + 2q18 + 2q20 + q22 + 3q24 - q26 - 2q30 - 2q32 - q34 - q36 + q38

HOMFLY-PT Polynomial:

a-12 - 6a-10 - 8a-10z2 - 2a-10z4 + 6a-8 + 14a-8z2 + 10a-8z4 + 2a-8z6 + 3a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

- a-15z + a-15z3 - 2a-14z2 + 2a-14z4 + 2a-13z - 4a-13z3 + 3a-13z5 + a-12 - 3a-12z2 + 6a-12z4 - 3a-12z6 + a-12z8 - 2a-11z + 2a-11z3 + 3a-11z5 - 3a-11z7 + a-11z9 + 6a-10 - 22a-10z2 + 30a-10z4 - 18a-10z6 + 4a-10z8 - 5a-9z + 11a-9z3 - 7a-9z5 - a-9z7 + a-9z9 + 6a-8 - 18a-8z2 + 22a-8z4 - 14a-8z6 + 3a-8z8 + 4a-7z3 - 7a-7z5 + 2a-7z7 + 3a-6z2 - 4a-6z4 + a-6z6

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

{9, 25}

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=6 is the signature of 11169. 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 = 25                   1 j = 23                 1   j = 21               3 1   j = 19             2 1     j = 17           4 3       j = 15         2 2         j = 13       2 4           j = 11     2 2             j = 9     2               j = 7 1 2                 j = 5 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, 169]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 169]]
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[20, 14, 21, 13], X[22, 16, 1, 15], > X[17, 5, 18, 4], X[19, 9, 20, 8], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 169]]
Out[4]=	GaussCode[1, 6, -2, 9, -3, -1, 4, 10, -5, 2, -6, -4, 7, -11, 8, 3, -9, 5, -10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 169]]
Out[5]=	DTCode[6, -10, -16, 12, -18, -2, 20, 22, -4, -8, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 169]][t]
Out[6]=	3 6 6 2 3 -5 + -- - -- + - + 6 t - 6 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 169]][z]
Out[7]=	2 4 6 1 + 9 z + 12 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 169]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 169]], KnotSignature[Knot[11, NonAlternating, 169]]}
Out[9]=	{35, 6}
In[10]:=	J=Jones[Knot[11, NonAlternating, 169]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 4 q - 4 q + 6 q - 6 q + 5 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 169]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 169]][q]
Out[12]=	10 12 14 18 20 22 24 26 30 32 34 q - q + q + 2 q + 2 q + q + 3 q - q - 2 q - 2 q - q - 36 38 > q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 169]][a, z]
Out[13]=	2 2 2 4 4 4 6 6 -12 6 6 8 z 14 z 3 z 2 z 10 z 4 z 2 z z a - --- + -- - ---- + ----- + ---- - ---- + ----- + ---- + ---- + -- 10 8 10 8 6 10 8 6 8 6 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 169]][a, z]
Out[14]=	2 2 2 2 2 -12 6 6 z 2 z 2 z 5 z 2 z 3 z 22 z 18 z 3 z a + --- + -- - --- + --- - --- - --- - ---- - ---- - ----- - ----- + ---- + 10 8 15 13 11 9 14 12 10 8 6 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 4 z 4 z 2 z 11 z 4 z 2 z 6 z 30 z 22 z 4 z > --- - ---- + ---- + ----- + ---- + ---- + ---- + ----- + ----- - ---- + 15 13 11 9 7 14 12 10 8 6 a a a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 3 z 3 z 7 z 7 z 3 z 18 z 14 z z 3 z z 2 z > ---- + ---- - ---- - ---- - ---- - ----- - ----- + -- - ---- - -- + ---- + 13 11 9 7 12 10 8 6 11 9 7 a a a a a a a a a a a 8 8 8 9 9 z 4 z 3 z z z > --- + ---- + ---- + --- + -- 12 10 8 11 9 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 169]], Vassiliev[3][Knot[11, NonAlternating, 169]]}
Out[15]=	{9, 25}
In[16]:=	Kh[Knot[11, NonAlternating, 169]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + 4 q t + 15 4 15 5 17 5 17 6 19 6 19 7 21 7 > 2 q t + 2 q t + 4 q t + 3 q t + 2 q t + q t + 3 q t + 21 8 23 8 25 9 > q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n169

K11n168 K11n170