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

K11a36 K11a38

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X16,9,17,10 X20,12,21,11 X18,14,19,13 X6,15,7,16 X22,17,1,18 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 16 20 18 6 22 12 10

Alexander Polynomial:

- 2t-3 + 9t-2 - 21t-1 + 29 - 21t + 9t2 - 2t3

Conway Polynomial:

1 - 3z2 - 3z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{93, 0}

Jones Polynomial:

q-4 - 3q-3 + 7q-2 - 10q-1 + 13 - 15q + 14q2 - 12q3 + 9q4 - 5q5 + 3q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-12 - q-10 + 2q-8 + 2q-6 - 2q-4 + 3q-2 - 2 - q2 - q4 - 3q6 + 2q8 - q10 + 2q12 + 3q14 - q16 + q18 - q22

HOMFLY-PT Polynomial:

- a-6 - a-6z2 + 4a-4 + 5a-4z2 + 2a-4z4 - 3a-2 - 5a-2z2 - 3a-2z4 - a-2z6 - 1 - 4z2 - 3z4 - z6 + 2a2 + 2a2z2 + a2z4

Kauffman Polynomial:

- a-7z + 4a-7z3 - 4a-7z5 + a-7z7 + a-6 - 8a-6z2 + 17a-6z4 - 13a-6z6 + 3a-6z8 - a-5z + 5a-5z3 + 3a-5z5 - 9a-5z7 + 3a-5z9 + 4a-4 - 27a-4z2 + 50a-4z4 - 35a-4z6 + 6a-4z8 + a-4z10 + 3a-3z - 7a-3z3 + 16a-3z5 - 20a-3z7 + 7a-3z9 + 3a-2 - 22a-2z2 + 40a-2z4 - 34a-2z6 + 9a-2z8 + a-2z10 + 2a-1z - a-1z3 - 3a-1z5 - 3a-1z7 + 4a-1z9 - 1 + 4z2 - 2z4 - 6z6 + 6z8 - az + 5az3 - 9az5 + 7az7 - 2a2 + 6a2z2 - 8a2z4 + 6a2z6 - 2a3z3 + 3a3z5 - a4z2 + a4z4

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

{-3, 0}

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=0 is the signature of 1137. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   3 1   j = 9                 6 2     j = 7               6 3       j = 5             8 6         j = 3           7 6           j = 1         6 8             j = -1       5 8               j = -3     2 5                 j = -5   1 5                   j = -7   2                     j = -9 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, 37]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 37]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[20, 12, 21, 11], X[18, 14, 19, 13], X[6, 15, 7, 16], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 37]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -5, 9, -7, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 37]]
Out[5]=	DTCode[4, 8, 14, 2, 16, 20, 18, 6, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 37]][t]
Out[6]=	2 9 21 2 3 29 - -- + -- - -- - 21 t + 9 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 37]][z]
Out[7]=	2 4 6 1 - 3 z - 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 37]}
In[9]:=	{KnotDet[Knot[11, Alternating, 37]], KnotSignature[Knot[11, Alternating, 37]]}
Out[9]=	{93, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 37]][q]
Out[10]=	-4 3 7 10 2 3 4 5 6 7 13 + q - -- + -- - -- - 15 q + 14 q - 12 q + 9 q - 5 q + 3 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 37]}
In[12]:=	A2Invariant[Knot[11, Alternating, 37]][q]
Out[12]=	-12 -10 2 2 2 3 2 4 6 8 10 12 -2 + q - q + -- + -- - -- + -- - q - q - 3 q + 2 q - q + 2 q + 8 6 4 2 q q q q 14 16 18 22 > 3 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 37]][a, z]
Out[13]=	2 2 2 4 -6 4 3 2 2 z 5 z 5 z 2 2 4 2 z -1 - a + -- - -- + 2 a - 4 z - -- + ---- - ---- + 2 a z - 3 z + ---- - 4 2 6 4 2 4 a a a a a a 4 6 3 z 2 4 6 z > ---- + a z - z - -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 37]][a, z]
Out[14]=	2 2 -6 4 3 2 z z 3 z 2 z 2 8 z 27 z -1 + a + -- + -- - 2 a - -- - -- + --- + --- - a z + 4 z - ---- - ----- - 4 2 7 5 3 a 6 4 a a a a a a a 2 3 3 3 3 22 z 2 2 4 2 4 z 5 z 7 z z 3 3 3 > ----- + 6 a z - a z + ---- + ---- - ---- - -- + 5 a z - 2 a z - 2 7 5 3 a a a a a 4 4 4 5 5 5 4 17 z 50 z 40 z 2 4 4 4 4 z 3 z 16 z > 2 z + ----- + ----- + ----- - 8 a z + a z - ---- + ---- + ----- - 6 4 2 7 5 3 a a a a a a 5 6 6 6 7 3 z 5 3 5 6 13 z 35 z 34 z 2 6 z > ---- - 9 a z + 3 a z - 6 z - ----- - ----- - ----- + 6 a z + -- - a 6 4 2 7 a a a a 7 7 7 8 8 8 9 9 9 z 20 z 3 z 7 8 3 z 6 z 9 z 3 z 7 z > ---- - ----- - ---- + 7 a z + 6 z + ---- + ---- + ---- + ---- + ---- + 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 4 z z z > ---- + --- + --- a 4 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 37]], Vassiliev[3][Knot[11, Alternating, 37]]}
Out[15]=	{-3, 0}
In[16]:=	Kh[Knot[11, Alternating, 37]][q, t]
Out[16]=	8 1 2 1 5 2 5 5 3 - + 6 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 8 q t + 7 q t + q 9 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 6 q t + 8 q t + 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 11 5 11 6 13 6 15 7 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a37

K11a36 K11a38