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

K11a236 K11a238

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 16 18 22 20 2 12 8 6 10

Alexander Polynomial:

8t-2 - 23t-1 + 31 - 23t + 8t2

Conway Polynomial:

1 + 9z2 + 8z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{93, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 10q5 + 14q6 - 15q7 + 14q8 - 12q9 + 9q10 - 5q11 + 2q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - 2q14 + 4q16 + q20 + q22 - 2q24 + 2q26 - 2q28 + q30 + 2q32 - 3q34 - q40

HOMFLY-PT Polynomial:

- a-12 - a-12z2 + a-10z4 + 4a-8z2 + 3a-8z4 + 2a-6 + 5a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

- 4a-15z + 8a-15z3 - 5a-15z5 + a-15z7 - 4a-14z2 + 10a-14z4 - 8a-14z6 + 2a-14z8 - 3a-13z + 10a-13z3 - 4a-13z5 - 4a-13z7 + 2a-13z9 - a-12 + 4a-12z2 + 3a-12z4 - 10a-12z6 + 2a-12z8 + a-12z10 - 3a-11z + 12a-11z3 - 9a-11z5 - 7a-11z7 + 5a-11z9 + 2a-10z2 - 13a-10z6 + 6a-10z8 + a-10z10 - 4a-9z + 16a-9z3 - 21a-9z5 + 5a-9z7 + 3a-9z9 + 2a-8z2 - 2a-8z4 - 5a-8z6 + 6a-8z8 + 4a-7z3 - 8a-7z5 + 7a-7z7 - 2a-6 + 7a-6z2 - 8a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{9, 27}

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 11237. 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                   4 1   j = 21                 5 1     j = 19               7 4       j = 17             7 5         j = 15           8 7           j = 13         6 7             j = 11       4 8               j = 9     3 6                 j = 7     4                   j = 5 1 3                     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, 237]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 237]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[18, 8, 19, 7], > X[22, 10, 1, 9], X[20, 12, 21, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], > X[8, 18, 9, 17], X[6, 20, 7, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 237]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -9, 5, -11, 6, -8, 7, -2, 8, -3, 9, -4, 10, > -6, 11, -5]
In[5]:=	DTCode[Knot[11, Alternating, 237]]
Out[5]=	DTCode[4, 14, 16, 18, 22, 20, 2, 12, 8, 6, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 237]][t]
Out[6]=	8 23 2 31 + -- - -- - 23 t + 8 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 237]][z]
Out[7]=	2 4 1 + 9 z + 8 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 237]}
In[9]:=	{KnotDet[Knot[11, Alternating, 237]], KnotSignature[Knot[11, Alternating, 237]]}
Out[9]=	{93, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 237]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 3 q + 7 q - 10 q + 14 q - 15 q + 14 q - 12 q + 9 q - 5 q + 12 13 > 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 237]}
In[12]:=	A2Invariant[Knot[11, Alternating, 237]][q]
Out[12]=	6 8 10 12 14 16 20 22 24 26 28 q - 2 q + 2 q + q - 2 q + 4 q + q + q - 2 q + 2 q - 2 q + 30 32 34 40 > q + 2 q - 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 237]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 2 z 4 z 5 z z z 3 z 3 z z -a + -- - --- + ---- + ---- + -- + --- + ---- + ---- + -- 6 12 8 6 4 10 8 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 237]][a, z]
Out[14]=	2 2 2 2 2 2 -12 2 4 z 3 z 3 z 4 z 4 z 4 z 2 z 2 z 7 z z -a - -- - --- - --- - --- - --- - ---- + ---- + ---- + ---- + ---- - -- + 6 15 13 11 9 14 12 10 8 6 4 a a a a a a a a a a a 3 3 3 3 3 3 4 4 4 4 8 z 10 z 12 z 16 z 4 z 2 z 10 z 3 z 2 z 8 z > ---- + ----- + ----- + ----- + ---- - ---- + ----- + ---- - ---- - ---- + 15 13 11 9 7 5 14 12 8 6 a a a a a a a a a a 4 5 5 5 5 5 5 6 6 6 z 5 z 4 z 9 z 21 z 8 z 3 z 8 z 10 z 13 z > -- - ---- - ---- - ---- - ----- - ---- + ---- - ---- - ----- - ----- - 4 15 13 11 9 7 5 14 12 10 a a a a a a a a a a 6 6 7 7 7 7 7 8 8 8 8 5 z 6 z z 4 z 7 z 5 z 7 z 2 z 2 z 6 z 6 z > ---- + ---- + --- - ---- - ---- + ---- + ---- + ---- + ---- + ---- + ---- + 8 6 15 13 11 9 7 14 12 10 8 a a a a a a a a a a a 9 9 9 10 10 2 z 5 z 3 z z z > ---- + ---- + ---- + --- + --- 13 11 9 12 10 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 237]], Vassiliev[3][Knot[11, Alternating, 237]]}
Out[15]=	{9, 27}
In[16]:=	Kh[Knot[11, Alternating, 237]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 6 q t + 4 q t + 8 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 6 q t + 7 q t + 8 q t + 7 q t + 7 q t + 5 q t + 19 7 19 8 21 8 21 9 23 9 23 10 25 10 > 7 q t + 4 q t + 5 q t + q t + 4 q t + q t + q t + 27 11 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a237

K11a236 K11a238