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

K11a237 K11a239

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X14,4,15,3 X16,6,17,5 X20,8,21,7 X18,10,19,9 X22,12,1,11 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, -3, 9, -5, 10, -4, 11, -6}

DT (Dowker-Thistlethwaite) Code:

4 14 16 20 18 22 2 12 10 8 6

Alexander Polynomial:

6t-2 - 16t-1 + 21 - 16t + 6t2

Conway Polynomial:

1 + 8z2 + 6z4

Other knots with the same Alexander/Conway Polynomial:

{K11n171, ...}

Determinant and Signature:

{65, 4}

Jones Polynomial:

q2 - 2q3 + 5q4 - 7q5 + 9q6 - 10q7 + 10q8 - 8q9 + 6q10 - 4q11 + 2q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - q8 + 2q10 + q12 - q14 + 2q16 - q18 + q22 + 2q26 - q28 + q32 - 2q34 - q40

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

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

{8, 24}

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 11238. 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                 3 1     j = 19               5 3       j = 17             5 3         j = 15           5 5           j = 13         4 5             j = 11       3 5               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, 238]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 238]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[16, 6, 17, 5], X[20, 8, 21, 7], > X[18, 10, 19, 9], X[22, 12, 1, 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, 238]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -3, 9, -5, 10, > -4, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 238]]
Out[5]=	DTCode[4, 14, 16, 20, 18, 22, 2, 12, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 238]][t]
Out[6]=	6 16 2 21 + -- - -- - 16 t + 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 238]][z]
Out[7]=	2 4 1 + 8 z + 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 238], Knot[11, NonAlternating, 171]}
In[9]:=	{KnotDet[Knot[11, Alternating, 238]], KnotSignature[Knot[11, Alternating, 238]]}
Out[9]=	{65, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 238]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 12 q - 2 q + 5 q - 7 q + 9 q - 10 q + 10 q - 8 q + 6 q - 4 q + 2 q - 13 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 238]}
In[12]:=	A2Invariant[Knot[11, Alternating, 238]][q]
Out[12]=	6 8 10 12 14 16 18 22 26 28 32 34 q - q + 2 q + q - q + 2 q - q + q + 2 q - q + q - 2 q - 40 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 238]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -12 -8 -4 z z 3 z 3 z 2 z z 2 z 2 z z -a + a + a - --- + --- + ---- + ---- + ---- + --- + ---- + ---- + -- 12 10 8 6 4 10 8 6 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 238]][a, z]
Out[14]=	2 2 2 2 2 -12 -8 -4 3 z z z 3 z 4 z 3 z z 5 z z -a + a + a - --- - --- - --- - --- - ---- + ---- - --- - ---- + -- - 15 13 11 9 14 12 10 8 6 a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 2 z 7 z 2 z 2 z 5 z 2 z 11 z 3 z 3 z 7 z > ---- + ---- + ---- - ---- + ---- - ---- + ----- + ---- + ---- + ---- - 4 15 13 11 9 5 14 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 6 3 z z 5 z 4 z 11 z 2 z 2 z 2 z 9 z 5 z 4 z > ---- + -- - ---- + ---- + ----- - ---- - ---- + ---- - ---- - ---- - ---- - 6 4 15 13 11 9 7 5 14 12 10 a a a a a a a a a a a 6 6 7 7 7 7 7 8 8 8 9 5 z 3 z z 7 z 13 z 2 z 3 z 2 z z 3 z 2 z > ---- + ---- + --- - ---- - ----- - ---- + ---- + ---- - --- + ---- + ---- + 8 6 15 13 11 9 7 14 12 8 13 a a a a a a a a a a a 9 9 10 10 4 z 2 z z z > ---- + ---- + --- + --- 11 9 12 10 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 238]], Vassiliev[3][Knot[11, Alternating, 238]]}
Out[15]=	{8, 24}
In[16]:=	Kh[Knot[11, Alternating, 238]][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 + 5 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 4 q t + 5 q t + 5 q t + 5 q t + 5 q t + 3 q t + 19 7 19 8 21 8 21 9 23 9 23 10 25 10 > 5 q t + 3 q t + 3 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 K11a238

K11a237 K11a239