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

K11n149 K11n151

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X11,1,12,22 X13,21,14,20 X4,16,5,15 X2,17,3,18 X19,15,20,14 X21,13,22,12

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-3 - 9t-2 + 17t-1 - 19 + 17t - 9t2 + 2t3

Conway Polynomial:

1 - z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a258, ...}

Determinant and Signature:

{75, 2}

Jones Polynomial:

- q-2 + 4q-1 - 7 + 11q - 12q2 + 13q3 - 12q4 + 8q5 - 5q6 + 2q7

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

{K11n66, ...}

A2 (sl(3)) Invariant:

- q-6 + 2q-4 + 3q2 - 2q4 + 3q6 - q8 + q12 - 3q14 + 2q16 - 2q18 - q20 + q22 + q26

HOMFLY-PT Polynomial:

a-8 - 2a-6 - 3a-6z2 - a-6z4 + a-4 + 3a-4z2 + 3a-4z4 + a-4z6 + 2a-2z4 + a-2z6 + 1 - z2 - z4

Kauffman Polynomial:

a-8 - 5a-8z2 + 3a-8z4 + 4a-7z - 7a-7z3 + 3a-7z5 + a-7z7 + 2a-6 - 12a-6z2 + 15a-6z4 - 7a-6z6 + 3a-6z8 + 7a-5z - 12a-5z3 + 8a-5z5 - 2a-5z7 + 2a-5z9 + a-4 - 10a-4z2 + 19a-4z4 - 17a-4z6 + 8a-4z8 + 3a-3z - 2a-3z3 - 6a-3z5 + 3a-3z7 + 2a-3z9 - a-2z2 - 6a-2z6 + 5a-2z8 + 2a-1z3 - 10a-1z5 + 6a-1z7 + 1 + 2z2 - 7z4 + 4z6 - az3 + az5

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

{-1, -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=2 is the signature of 11150. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                   2 j = 13                 3   j = 11               5 2   j = 9             7 3     j = 7           6 5       j = 5         6 7         j = 3       5 6           j = 1     3 7             j = -1   1 4               j = -3   3                 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, 150]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 150]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[18, 8, 19, 7], > X[16, 9, 17, 10], X[11, 1, 12, 22], X[13, 21, 14, 20], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[19, 15, 20, 14], X[21, 13, 22, 12]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 150]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -2, 5, -3, -6, 11, -7, 10, 8, -5, 9, -4, -10, > 7, -11, 6]
In[5]:=	DTCode[Knot[11, NonAlternating, 150]]
Out[5]=	DTCode[6, 8, 10, 18, 16, -22, -20, 4, 2, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 150]][t]
Out[6]=	2 9 17 2 3 -19 + -- - -- + -- + 17 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 150]][z]
Out[7]=	2 4 6 1 - z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 258], Knot[11, NonAlternating, 150]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 150]], KnotSignature[Knot[11, NonAlternating, 150]]}
Out[9]=	{75, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 150]][q]
Out[10]=	-2 4 2 3 4 5 6 7 -7 - q + - + 11 q - 12 q + 13 q - 12 q + 8 q - 5 q + 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 66], Knot[11, NonAlternating, 150]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 150]][q]
Out[12]=	-6 2 2 4 6 8 12 14 16 18 20 22 -q + -- + 3 q - 2 q + 3 q - q + q - 3 q + 2 q - 2 q - q + q + 4 q 26 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 150]][a, z]
Out[13]=	2 2 4 4 4 6 6 -8 2 -4 2 3 z 3 z 4 z 3 z 2 z z z 1 + a - -- + a - z - ---- + ---- - z - -- + ---- + ---- + -- + -- 6 6 4 6 4 2 4 2 a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 150]][a, z]
Out[14]=	2 2 2 2 -8 2 -4 4 z 7 z 3 z 2 5 z 12 z 10 z z 1 + a + -- + a + --- + --- + --- + 2 z - ---- - ----- - ----- - -- - 6 7 5 3 8 6 4 2 a a a a a a a a 3 3 3 3 4 4 4 5 7 z 12 z 2 z 2 z 3 4 3 z 15 z 19 z 3 z > ---- - ----- - ---- + ---- - a z - 7 z + ---- + ----- + ----- + ---- + 7 5 3 a 8 6 4 7 a a a a a a a 5 5 5 6 6 6 7 7 8 z 6 z 10 z 5 6 7 z 17 z 6 z z 2 z > ---- - ---- - ----- + a z + 4 z - ---- - ----- - ---- + -- - ---- + 5 3 a 6 4 2 7 5 a a a a a a a 7 7 8 8 8 9 9 3 z 6 z 3 z 8 z 5 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- + ---- + ---- 3 a 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 150]], Vassiliev[3][Knot[11, NonAlternating, 150]]}
Out[15]=	{-1, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 150]][q, t]
Out[16]=	3 1 3 1 4 3 q 3 5 5 2 7 q + 5 q + ----- + ----- + ---- + --- + --- + 6 q t + 6 q t + 7 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 13 5 > 6 q t + 5 q t + 7 q t + 3 q t + 5 q t + 2 q t + 3 q t + 15 6 > 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n150

K11n149 K11n151