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

K11a242 K11a244

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X18,6,19,5 X20,8,21,7 X22,10,1,9 X16,12,17,11 X2,14,3,13 X12,16,13,15 X10,18,11,17 X8,20,9,19 X6,22,7,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 18 20 22 16 2 12 10 8 6

Alexander Polynomial:

6t-2 - 17t-1 + 23 - 17t + 6t2

Conway Polynomial:

1 + 7z2 + 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{69, 4}

Jones Polynomial:

q2 - 2q3 + 5q4 - 7q5 + 10q6 - 11q7 + 10q8 - 9q9 + 7q10 - 4q11 + 2q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - q8 + 2q10 + q12 - q14 + 3q16 - 2q24 + q26 - q28 + q30 + 2q32 - 2q34 - q40

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-15z + 7a-15z3 - 5a-15z5 + a-15z7 - 5a-14z2 + 12a-14z4 - 9a-14z6 + 2a-14z8 + a-13z - 2a-13z3 + 6a-13z5 - 7a-13z7 + 2a-13z9 - a-12 + 2a-12z2 - 3a-12z6 - a-12z8 + a-12z10 + 3a-11z - 9a-11z3 + 12a-11z5 - 12a-11z7 + 4a-11z9 - a-10 + 5a-10z2 - 7a-10z4 + a-10z10 + 2a-9z - 2a-9z5 - a-9z7 + 2a-9z9 - a-8 + 2a-8z2 + a-8z4 - 3a-8z6 + 3a-8z8 + 2a-7z - 2a-7z3 - a-7z5 + 3a-7z7 - a-6 + 2a-6z2 - 3a-6z4 + 3a-6z6 - 2a-5z3 + 2a-5z5 + a-4 - 2a-4z2 + a-4z4

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

{7, 20}

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 11243. 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 r = 10 r = 11 j = 27                       1 j = 25                     1   j = 23                   3 1   j = 21                 4 1     j = 19               5 3       j = 17             5 4         j = 15           6 5           j = 13         4 5             j = 11       3 6               j = 9     2 4                 j = 7     3                   j = 5 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, Alternating, 243]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 243]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[18, 6, 19, 5], X[20, 8, 21, 7], > X[22, 10, 1, 9], X[16, 12, 17, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 243]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -6, 9, -3, 10, > -4, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 243]]
Out[5]=	DTCode[4, 14, 18, 20, 22, 16, 2, 12, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 243]][t]
Out[6]=	6 17 2 23 + -- - -- - 17 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 243]][z]
Out[7]=	2 4 1 + 7 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 243]}
In[9]:=	{KnotDet[Knot[11, Alternating, 243]], KnotSignature[Knot[11, Alternating, 243]]}
Out[9]=	{69, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 243]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 2 q + 5 q - 7 q + 10 q - 11 q + 10 q - 9 q + 7 q - 4 q + 12 13 > 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 243]}
In[12]:=	A2Invariant[Knot[11, Alternating, 243]][q]
Out[12]=	6 8 10 12 14 16 24 26 28 30 32 34 q - q + 2 q + q - q + 3 q - 2 q + q - q + q + 2 q - 2 q - 40 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 243]][a, z]
Out[13]=	2 2 2 2 2 4 4 -12 -10 -8 -6 -4 z z 2 z 3 z 2 z z 2 z -a + a - a + a + a - --- + --- + ---- + ---- + ---- + --- + ---- + 12 10 8 6 4 10 8 a a a a a a a 4 4 2 z z > ---- + -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 243]][a, z]
Out[14]=	2 2 -12 -10 -8 -6 -4 2 z z 3 z 2 z 2 z 5 z 2 z -a - a - a - a + a - --- + --- + --- + --- + --- - ---- + ---- + 15 13 11 9 7 14 12 a a a a a a a 2 2 2 2 3 3 3 3 3 4 5 z 2 z 2 z 2 z 7 z 2 z 9 z 2 z 2 z 12 z > ---- + ---- + ---- - ---- + ---- - ---- - ---- - ---- - ---- + ----- - 10 8 6 4 15 13 11 7 5 14 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 6 7 z z 3 z z 5 z 6 z 12 z 2 z z 2 z 9 z > ---- + -- - ---- + -- - ---- + ---- + ----- - ---- - -- + ---- - ---- - 10 8 6 4 15 13 11 9 7 5 14 a a a a a a a a a a a 6 6 6 7 7 7 7 7 8 8 8 3 z 3 z 3 z z 7 z 12 z z 3 z 2 z z 3 z > ---- - ---- + ---- + --- - ---- - ----- - -- + ---- + ---- - --- + ---- + 12 8 6 15 13 11 9 7 14 12 8 a a a a a a a a a a a 9 9 9 10 10 2 z 4 z 2 z z z > ---- + ---- + ---- + --- + --- 13 11 9 12 10 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 243]], Vassiliev[3][Knot[11, Alternating, 243]]}
Out[15]=	{7, 20}
In[16]:=	Kh[Knot[11, Alternating, 243]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 2 q t + 3 q t + 2 q t + 4 q t + 3 q t + 6 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 4 q t + 5 q t + 6 q t + 5 q t + 5 q t + 4 q t + 19 7 19 8 21 8 21 9 23 9 23 10 25 10 > 5 q t + 3 q t + 4 q t + q t + 3 q t + q t + q t + 27 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a243

K11a242 K11a244