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

K11n148 K11n150

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X11,21,12,20 X13,1,14,22 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, 6, -11, 7}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 4t-3 + 6t-2 - 4t-1 + 3 - 4t + 6t2 - 4t3 + t4

Conway Polynomial:

1 + 2z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{33, 4}

Jones Polynomial:

- q-1 + 3 - 3q + 5q2 - 5q3 + 5q4 - 5q5 + 3q6 - 2q7 + q8

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 + q2 + 2q4 + 2q6 + q8 + q10 - 3q12 - 2q16 + q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 2a-6 - 6a-6z2 - 5a-6z4 - a-6z6 + 7a-4z2 + 11a-4z4 + 6a-4z6 + a-4z8 + 2a-2 - 2a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-10z2 - a-9z + 2a-9z3 + a-8 - 4a-8z2 + 3a-8z4 + a-7z - 2a-7z3 - a-7z5 + a-7z7 + 2a-6 - 15a-6z2 + 22a-6z4 - 14a-6z6 + 3a-6z8 + 5a-5z - 8a-5z3 + 9a-5z5 - 8a-5z7 + 2a-5z9 - 16a-4z2 + 39a-4z4 - 29a-4z6 + 6a-4z8 + 3a-3z - a-3z3 + 6a-3z5 - 8a-3z7 + 2a-3z9 - 2a-2 - 6a-2z2 + 20a-2z4 - 15a-2z6 + 3a-2z8 + 3a-1z3 - 4a-1z5 + a-1z7

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

{0, -2}

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 11149. 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 = 17                   1 j = 15                 1   j = 13               2 1   j = 11             3 1     j = 9           2 2       j = 7         3 3         j = 5       2 2           j = 3     2 4             j = 1   1 1               j = -1   2                 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, 149]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 149]]
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, 21, 12, 20], X[13, 1, 14, 22], 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, 149]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -2, 5, -3, -6, 11, -7, 10, 8, -5, 9, -4, -10, > 6, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 149]]
Out[5]=	DTCode[6, 8, 10, 18, 16, -20, -22, 4, 2, -14, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 149]][t]
Out[6]=	-4 4 6 4 2 3 4 3 + t - -- + -- - - - 4 t + 6 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 149]][z]
Out[7]=	4 6 8 1 + 2 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 149]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 149]], KnotSignature[Knot[11, NonAlternating, 149]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 149]][q]
Out[10]=	1 2 3 4 5 6 7 8 3 - - - 3 q + 5 q - 5 q + 5 q - 5 q + 3 q - 2 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 149]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 149]][q]
Out[12]=	-2 2 4 6 8 10 12 16 24 26 28 1 - q + q + 2 q + 2 q + q + q - 3 q - 2 q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 149]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 6 -8 2 2 z 6 z 7 z 2 z 5 z 11 z 4 z z 6 z a - -- + -- + -- - ---- + ---- - ---- - ---- + ----- - ---- - -- + ---- - 6 2 8 6 4 2 6 4 2 6 4 a a a a a a a a a a a 6 8 z z > -- + -- 2 4 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 149]][a, z]
Out[14]=	2 2 2 2 2 -8 2 2 z z 5 z 3 z z 4 z 15 z 16 z 6 z a + -- - -- - -- + -- + --- + --- + --- - ---- - ----- - ----- - ---- + 6 2 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 3 4 4 4 4 5 5 2 z 2 z 8 z z 3 z 3 z 22 z 39 z 20 z z 9 z > ---- - ---- - ---- - -- + ---- + ---- + ----- + ----- + ----- - -- + ---- + 9 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a a 5 5 6 6 6 7 7 7 7 8 8 6 z 4 z 14 z 29 z 15 z z 8 z 8 z z 3 z 6 z > ---- - ---- - ----- - ----- - ----- + -- - ---- - ---- + -- + ---- + ---- + 3 a 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 3 z 2 z 2 z > ---- + ---- + ---- 2 5 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 149]], Vassiliev[3][Knot[11, NonAlternating, 149]]}
Out[15]=	{0, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 149]][q, t]
Out[16]=	3 3 5 1 2 q q 2 q 5 7 7 2 4 q + 2 q + ----- + ---- + -- + - + ---- + 2 q t + 3 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 17 6 > 2 q t + 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n149

K11n148 K11n150