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

K11n138 K11n140

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,3,13,4 X16,6,17,5 X14,8,15,7 X9,21,10,20 X11,19,12,18 X2,13,3,14 X6,16,7,15 X17,22,18,1 X19,11,20,10 X21,9,22,8

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-1 + 5 - 2t

Conway Polynomial:

1 - 2z2

Other knots with the same Alexander/Conway Polynomial:

{61, 946, K11n67, K11n97, ...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

1 + q2 - q3 + q4 - 2q5 + q6 - q7 + q8

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

{...}

A2 (sl(3)) Invariant:

q-2 + 1 + q2 + q4 + q6 - q12 - q14 - q16 - q18 + q24 + q26

HOMFLY-PT Polynomial:

a-8 - a-6 - a-6z2 - a-4 - a-4z2 + a-2 + 1

Kauffman Polynomial:

a-8 - 10a-8z2 + 15a-8z4 - 7a-8z6 + a-8z8 + 6a-7z - 15a-7z3 + 16a-7z5 - 7a-7z7 + a-7z9 + a-6 - 15a-6z2 + 25a-6z4 - 13a-6z6 + 2a-6z8 + 10a-5z - 20a-5z3 + 17a-5z5 - 7a-5z7 + a-5z9 - a-4 - 5a-4z2 + 10a-4z4 - 6a-4z6 + a-4z8 + 4a-3z - 5a-3z3 + a-3z5 - a-2 + 1

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

{-2, -5}

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 11139. 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 j = 17                 1 j = 15                   j = 13             1 1   j = 11           1       j = 9           1       j = 7       1 1         j = 5                   j = 3     1             j = 1 1                 j = -1 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, 139]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 139]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 6, 17, 5], X[14, 8, 15, 7], > X[9, 21, 10, 20], X[11, 19, 12, 18], X[2, 13, 3, 14], X[6, 16, 7, 15], > X[17, 22, 18, 1], X[19, 11, 20, 10], X[21, 9, 22, 8]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 139]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -8, 4, 11, -5, 10, -6, -2, 7, -4, 8, -3, -9, 6, -10, > 5, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 139]]
Out[5]=	DTCode[4, 12, 16, 14, -20, -18, 2, 6, -22, -10, -8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 139]][t]
Out[6]=	2 5 - - - 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 139]][z]
Out[7]=	2 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], > Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 139]], KnotSignature[Knot[11, NonAlternating, 139]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 139]][q]
Out[10]=	2 3 4 5 6 7 8 1 + q - q + q - 2 q + q - q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 139]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 139]][q]
Out[12]=	-2 2 4 6 12 14 16 18 24 26 1 + q + q + q + q - q - q - q - q + q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 139]][a, z]
Out[13]=	2 2 -8 -6 -4 -2 z z 1 + a - a - a + a - -- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 139]][a, z]
Out[14]=	2 2 2 3 -8 -6 -4 -2 6 z 10 z 4 z 10 z 15 z 5 z 15 z 1 + a + a - a - a + --- + ---- + --- - ----- - ----- - ---- - ----- - 7 5 3 8 6 4 7 a a a a a a a 3 3 4 4 4 5 5 5 6 6 20 z 5 z 15 z 25 z 10 z 16 z 17 z z 7 z 13 z > ----- - ---- + ----- + ----- + ----- + ----- + ----- + -- - ---- - ----- - 5 3 8 6 4 7 5 3 8 6 a a a a a a a a a a 6 7 7 8 8 8 9 9 6 z 7 z 7 z z 2 z z z z > ---- - ---- - ---- + -- + ---- + -- + -- + -- 4 7 5 8 6 4 7 5 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 139]], Vassiliev[3][Knot[11, NonAlternating, 139]]}
Out[15]=	{-2, -5}
In[16]:=	Kh[Knot[11, NonAlternating, 139]][q, t]
Out[16]=	1 3 2 7 3 7 4 9 5 11 5 13 6 13 7 17 8 - + q + q t + q t + q t + q t + q t + q t + q t + q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n139

K11n138 K11n140